By Editor
- What is a logarithm?
Logarithms are a group of functions that take a number as input and produce another number. There i…
- Nate Soares
- Metric
A metric is a function that defines a distance between elements in a set and follows some basic rules.
- Bryce Woodworth
- Bayes' rule: Probability form
The original formulation of Bayes' rule.
- Nate Soares - Bayes' rule: Vector form
For when you want to apply Bayes' rule to lots of evidence and lots of variables, all in one go. (This is more or less how spam filters work.)
- Eliezer Yudkowsky - Bayesian view of scientific virtues
Why is it that science relies on bold, precise, and falsifiable predictions? Because of Bayes' rule, of course.
- Eliezer Yudkowsky - Complexity theory
Study of the computational resources needed to compute something
- Jaime Sevilla Molina - Conditional probability: Refresher
Is P(yellow | banana) the probability that a banana is yellow, or the probability that a yellow thing is a banana?
- Nate Soares - Properties of the logarithm
- $\log_b(x \cdot y) = \log_b(x) + \log_b(y)$ for any $b$, this is the defining characteristic of …
- Nate Soares - The characteristic of the logarithm
Any time you find an output that adds whenever the input multiplies, you're probably looking at a (…
- Nate Soares - Waterfall diagrams and relative odds
A way to visualize Bayes' rule that yields an easier way to solve some problems
- Eliezer Yudkowsky
- Frequency diagrams: A first look at Bayes
The most straightforward visualization of Bayes' rule.
- Nate Soares
- Relative likelihood
How relatively likely an observation is, given two or more hypotheses, determines the strength and direction of evidence.
- Eliezer Yudkowsky
- Probability
The degree to which someone believes something, measured on a scale from 0 to 1, allowing us to do math to it.
- Eliezer Yudkowsky
- Decision problem
Formalization of general problems
- Jaime Sevilla Molina - Turing machine
A Turing Machine is a simple mathematical model of computation that is powerful enough to describe any computation a computer can do.
- Eric Leese
- Mindcrime: Introduction
The more predictive accuracy we want from a model, the more detailed the model becomes. A very roug…
- Eliezer Yudkowsky
wiki
- AI alignment
The great civilizational problem of creating artificially intelligent computer systems such that running them is a good idea.
- Eliezer Yudkowsky - AI alignment open problem
Tag for open problems under AI alignment.
- Eliezer Yudkowsky - AIXI
How to build an (evil) superintelligent AI using unlimited computing power and one page of Python code.
- Eliezer Yudkowsky - Abelian group
A group where the operation commutes. Named after Niels Henrik Abel.
- Nate Soares - Ability to read algebra
Do you have sufficient mathematical ability that you can read a sentence that uses some algebra or invokes a mathematical idea, without slowing down too much?
- Eliezer Yudkowsky - Ability to read calculus
Can you take integral signs and differentiations in stride?
- Eliezer Yudkowsky - Ability to read logic
Can you read sentences symbolically stating "For all x: exists y: phi(x, y) or not theta(y)" without slowing down too much?
- Eliezer Yudkowsky - Abstract algebra
The study of groups, fields, vector spaces, arithmetics, algebras, and more.
- Nate Soares - Advanced agent properties
How smart does a machine intelligence need to be, for its niceness to become an issue? "Advanced" is a broad term to cover cognitive abilities such that we'd need to start considering AI alignment.
- Eliezer Yudkowsky - Advanced safety
An agent is *really* safe when it has the capacity to do anything, but chooses to do what the programmer wants.
- Eliezer Yudkowsky - Algebraic structure
Roughly speaking, an algebraic structure is a set $X$, known as the underlying set, paired with a co…
- Nate Soares - Algorithmic complexity
When you compress the information, what you are left with determines the complexity.
- Eliezer Yudkowsky - Almost all real-world domains are rich
Anything you're trying to accomplish in the real world can potentially be accomplished in a *lot* of different ways.
- Eliezer Yudkowsky - Arbital playpen
Want to test a feature? Feel free to edit this page! asdfasfdasfda
- Eliezer Yudkowsky - Arbital practices
Guidelines and rules for interacting on Arbital.
- Eliezer Yudkowsky - Arbital scope
What kind of content is Arbital looking for?
- Eric Bruylant - Arbital's next pillar: Discussion
Arbital has a prototype discussion platform. If you're excited about this and want to participate, y…
- Eric Bruylant - Arbital: Solving online explanations
An explanation of Arbital's mid-term goals
- Eliezer Yudkowsky - Arithmetical hierarchy: If you don't read logic
The arithmetical hierarchy is a way of stratifying statements by how many "for every number" and "th…
- Eliezer Yudkowsky - Axiom of Choice: Guide
Learn about the most controversial axiom of the 20th century.
- Mark Chimes - Bayes' rule
Bayes' rule is the core theorem of probability theory saying how to revise our beliefs when we make a new observation.
- Eliezer Yudkowsky - Bayes' rule: Definition
Bayes' rule is the mathematics of probability theory governing how to update your beliefs in the lig…
- Nate Soares - Bayes' rule: Functional form
Bayes' rule for to continuous variables.
- Eliezer Yudkowsky - Bayes' rule: Guide
The Arbital guide to Bayes' rule
- Eliezer Yudkowsky - Bayes' rule: Log-odds form
A simple transformation of Bayes' rule reveals tools for measuring degree of belief, and strength of evidence.
- Eliezer Yudkowsky - Bayes' rule: Odds form
The simplest and most easily understandable form of Bayes' rule uses relative odds.
- Eliezer Yudkowsky - Bayes' rule: Proportional form
The fastest way to say something both convincing and true about belief-updating.
- Eliezer Yudkowsky - Bayes' rule: Vector form
For when you want to apply Bayes' rule to lots of evidence and lots of variables, all in one go. (This is more or less how spam filters work.)
- Eliezer Yudkowsky - Bayesian reasoning
A probability-theory-based view of the world; a coherent way of changing probabilistic beliefs based on evidence.
- Eliezer Yudkowsky - Behaviorist genie
An advanced agent that's forbidden to model minds in too much detail.
- Eliezer Yudkowsky - Belief revision as probability elimination
Update your beliefs by throwing away large chunks of probability mass.
- Eliezer Yudkowsky - Binary function
A binary function $f$ is a function of two inputs (i.e., a function with arity 2). For example, $+,$…
- Nate Soares - Blue oysters
A probability problem about blue oysters.
- Nate Soares - Boxed AI
Idea: what if we limit how AI can interact with the world. That'll make it safe, right??
- Eliezer Yudkowsky - Calories-In-Calories-Out
CICO is a proposed conceptual decomposition of the causes of changes in human body mass, particularl…
- Eliezer Yudkowsky - Central examples
The "central examples" for a subject are examples that are referred to over and over again in the co…
- Eliezer Yudkowsky - Central examples
List of central examples in Value Alignment Theory domain.
- Eliezer Yudkowsky - Claim explainer: donor lotteries and returns to scale
Sometimes, new technical developments in the discourse around effective altruism can be difficult to…
- Benjamin Hoffman - Cognitive uncontainability
'Cognitive uncontainability' is when we can't hold all of an agent's possibilities inside our own minds.
- Eliezer Yudkowsky - Complexity of value
There's no simple way to describe the goals we want Artificial Intelligences to want.
- Eliezer Yudkowsky - Conceivability
A hypothetical scenario is 'conceivable' or 'imaginable' when it is not *immediately* incoherent, al…
- Eliezer Yudkowsky - Consequentialist cognition
The cognitive ability to foresee the consequences of actions, prefer some outcomes to others, and output actions leading to the preferred outcomes.
- Eliezer Yudkowsky - Context disaster
Some possible designs cause your AI to behave nicely while developing, and behave a lot less nicely when it's smarter.
- Eliezer Yudkowsky - Contributing to Arbital
Want to help Arbital become awesome?
- Eric Bruylant - Coordinative AI development hypothetical
What would safe AI development look like if we didn't have to worry about anything else?
- Eliezer Yudkowsky - Corrigibility
"I can't let you do that, Dave."
- Nate Soares - Definition
Meta tag used to mark pages that strictly define a particular term or phrase.
- Eliezer Yudkowsky - Diamond maximizer
How would you build an agent that made as much diamond material as possible, given vast computing power but an otherwise rich and complicated environment?
- Eliezer Yudkowsky - Distant superintelligences can coerce the most probable environment of your AI
Distant superintelligences may be able to hack your local AI, if your AI's preference framework depends on its most probable environment.
- Eliezer Yudkowsky - Edge instantiation
When you ask the AI to make people happy, and it tiles the universe with the smallest objects that can be happy.
- Eliezer Yudkowsky - Eliezer office hours 2016-04-19
### Steph's notes from the actual meeting:
- "good question" -> people upvoting your question is a g…
- Eric Rogstad - Eliezer's vision for Arbital
Why are we building this? What's the goal?
- Eric Bruylant - Epistemic and instrumental efficiency
An efficient agent never makes a mistake you can predict. You can never successfully predict a directional bias in its estimates.
- Eliezer Yudkowsky - Epistemology
What is truth?
- Eliezer Yudkowsky - Executable philosophy
Philosophical discourse aimed at producing a trustworthy answer or meta-answer, in limited time, which can used in constructing an Artificial Intelligence.
- Eliezer Yudkowsky - Expected utility
Scoring actions based on the average score of their probable consequences.
- Eliezer Yudkowsky - Expected utility agent
If you're not some kind of expected utility agent, you're going in circles.
- Eliezer Yudkowsky - Extraordinary claims require extraordinary evidence
The people who adamantly claim they were abducted by aliens do provide some evidence for aliens. They just don't provide quantitatively enough evidence.
- Eliezer Yudkowsky - Finishing your Bayesian path on Arbital
The page that comes at the end of reading the Arbital Guide to Bayes' rule
- Eliezer Yudkowsky - Glossary (Value Alignment Theory)
Words that have a special meaning in the context of creating nice AIs.
- Eliezer Yudkowsky - Group theory
What kinds of symmetry can an object have?
- Nate Soares - Guarded definition
A guarded definition is one where at least one position suspects there will be pressure to stretch a…
- Eliezer Yudkowsky - Happiness maximizer
It is sometimes proposed that we build an AI intended to maximize human happiness. (One early propo…
- Eliezer Yudkowsky - How to author on Arbital!
Want to contribute pages to Arbital? Here's our current version of the ad-hoc guide to being an author!
- Eliezer Yudkowsky - Identifying ambiguous inductions
What do a "red strawberry", a "red apple", and a "red cherry" have in common that a "yellow carrot" doesn't? Are they "red fruits" or "red objects"?
- Eliezer Yudkowsky - Immediate goods
One of the potential views on 'value' in the value alignment problem is that what we should want fro…
- Eliezer Yudkowsky - In order to have a shared conversation, it would help to have a Schelling set of posts. - Eric Rogstad
- In order to think accumulatively we must have a shared conversation. - Eric Rogstad
- Instrumental
What is "instrumental" in the context of Value Alignment Theory?
- Eliezer Yudkowsky - Instrumental convergence
Some strategies can help achieve most possible simple goals. E.g., acquiring more computing power or more material resources. By default, unless averted, we can expect advanced AIs to do that.
- Eliezer Yudkowsky - Instrumental pressure
A consequentialist agent will want to bring about certain instrumental events that will help to fulfill its goals.
- Eliezer Yudkowsky - Intended goal
Definition. An "intended goal" refers to the intuitive intention in the mind of a human programmer …
- Eliezer Yudkowsky - Intension vs. extension
"Red is a light with a wavelength of 700 nm" vs. "Look at this red apple, red car, and red cup."
- Eliezer Yudkowsky - Interest in mathematical foundations in Bayesianism
"Want" this requisite if you prefer to see extra information about the mathematical foundations in Bayesianism.
- Eliezer Yudkowsky - Introduction to Bayes' rule: Odds form
Bayes' rule is simple, if you think in terms of relative odds.
- Eliezer Yudkowsky - Introductory guide to logarithms
Welcome to the Arbital introduction to logarithms! In modern education, logarithms are often mention…
- Nate Soares - Lattice (Order Theory)
A poset that is closed under binary joins and meets.
- Kevin Clancy - Likelihood function
Let's say you have a piece of evidence $e$ and a set of hypotheses $\mathcal H.$ Each $H_i \in \math…
- Nate Soares - Linguistic conventions in value alignment
How and why to use precise language and words with special meaning when talking about value alignment.
- Eliezer Yudkowsky - Link glossary pages for overloaded words
If your subject is using what sound like ordinary-language words in a special sense, create a glossa…
- Eliezer Yudkowsky - List of Eliezer's current most desired fixes and features
A place for Eliezer to note down his current list of personally-wanted features for editing and writing.
- Eliezer Yudkowsky - List: value-alignment subjects
Bullet point list of core VAT subjects.
- Eliezer Yudkowsky - Log as generalized length
To estimate the log (base 10) of a number, count how many digits it has.
- Nate Soares - Logical game
Game's mathematical structure at its purest form.
- Eliezer Yudkowsky - Look where I'm pointing, not at my finger
When trying to communicate the concept "glove", getting the AGI to focus on "gloves" rather than "my user's decision to label something a glove" or "anything that depresses the glove-labeling button".
- Eliezer Yudkowsky - Meta tags
What are meta tags and when to use them?
- Eliezer Yudkowsky - Methodology of foreseeable difficulties
Building a nice AI is likely to be hard enough, and contain enough gotchas that won't show up in the AI's early days, that we need to foresee problems coming in advance.
- Eliezer Yudkowsky - Methodology of unbounded analysis
What we do and don't understand how to do, using unlimited computing power, is a critical distinction and important frontier.
- Eliezer Yudkowsky - Mindcrime
Might a machine intelligence contain vast numbers of unhappy conscious subprocesses?
- Eliezer Yudkowsky - Mindcrime: Introduction
The more predictive accuracy we want from a model, the more detailed the model becomes. A very roug…
- Eliezer Yudkowsky - Natural language understanding of "right" will yield normativity
What will happen if you tell an advanced agent to do the "right" thing?
- Eliezer Yudkowsky - Nearest unblocked strategy
If you patch an agent's preference framework to avoid an undesirable solution, what can you expect to happen?
- Eliezer Yudkowsky - Object-level vs. indirect goals
Difference between "give Alice the apple" and "give Alice what she wants".
- Eliezer Yudkowsky - Odds
Odds express a relative probability.
- Eliezer Yudkowsky - Omnipotence test for AI safety
Would your AI produce disastrous outcomes if it suddenly gained omnipotence and omniscience? If so, why did you program something that *wants* to hurt you and is held back only by lacking the power?
- Eliezer Yudkowsky - Ontology identification problem
How do we link an agent's utility function to its model of the world, when we don't know what that model will look like?
- Eliezer Yudkowsky - Ontology identification problem: Technical tutorial
Technical tutorial for ontology identification problem.
- Eliezer Yudkowsky - Oracle
System designed to safely answer questions.
- Eliezer Yudkowsky - Orbit-stabiliser theorem
The Orbit-Stabiliser theorem tells us a lot about how a group acts on a given element.
- Patrick Stevens - Order of a group
The order $|G|$ of a group $G$ is the size of its underlying set. For example, if $G=(X,\bullet)$ an…
- Nate Soares - Order theory
The study of binary relations that are reflexive, transitive, and antisymmetic.
- Kevin Clancy - Orthogonality Thesis
Will smart AIs automatically become benevolent, or automatically become hostile? Or do different AI designs imply different goals?
- Eliezer Yudkowsky - Our community has a shot at seriously helping with x-risk. - Eric Rogstad
- Paperclip maximizer
This agent will not stop until the entire universe is filled with paperclips.
- Eliezer Yudkowsky - Patch resistance
One does not simply solve the value alignment problem.
- Eliezer Yudkowsky - Path: Multiple angles on Bayes's Rule
A learning-path placeholder page for learning multiple angles on Bayes's Rule.
- Eliezer Yudkowsky - People
A category for human beings!
- Eliezer Yudkowsky - Perfect rolling sphere
If you don't understand something, start by assuming it's a perfect rolling sphere.
- Eliezer Yudkowsky - Pivotal event
Which types of AIs, if they work, can do things that drastically change the nature of the further game?
- Eliezer Yudkowsky - Preference framework
What's the thing an agent uses to compare its preferences?
- Eliezer Yudkowsky - Principia Qualia: blueprint for a new cause area, consciousness research with an eye toward ethics and x-risk
Claims discussed in this post:
- [claim([6xd])]
- [claim([6x4])]
- [claim([6xf])]
- Eric Rogstad - Probability interpretations: Examples
Consider evaluating, in June of 2016, the question: "What is the probability of Hillary Clinton wi…
- Nate Soares - Probability notation for Bayes' rule: Intro (Math 1)
How to read, and identify, the probabilities in Bayesian problems.
- Eliezer Yudkowsky - Programmer
Who is building these advanced agents?
- Eliezer Yudkowsky - Programmer deception
Programmer deception is when the AI's decision process leads it to optimize for an instrumental goal…
- Eliezer Yudkowsky - Project outline: Intro to the Universal Property
Outline detailing all the work required for a proposed Arbital Project
- Eric Rogstad - Proposed B-Class
Pages which have been proposed for B-Class status.
- Eric Bruylant - Rationality
The subject domain for [ epistemic] and [ instrumental] rationality.
- Eliezer Yudkowsky - Real number
A **real number** is any number that can be used to represent a physical quantity.
Intuitively, rea…
- Michael Cohen - Relevant limited AI
Can we have a limited AI, that's nonetheless relevant?
- Eliezer Yudkowsky - Relevant powerful agent
An agent is relevant if it completely changes the course of history.
- Eliezer Yudkowsky - Relevant powerful agents will be highly optimized
The probability that an agent that is cognitively powerful enough to be relevant to existential outc…
- Eliezer Yudkowsky - Researchers in value alignment theory
Who's working full-time in value alignment theory?
- Eliezer Yudkowsky - Rich domain
A domain is 'rich', relative to our own intelligence, to the extent that (1) its [ search space] is …
- Eliezer Yudkowsky - Safe impact measure
What can we measure to make sure an agent is acting in a safe manner?
- Eliezer Yudkowsky - Set
An unordered collection of distinct objects.
- Nate Soares - Set builder notation
$\{ 2n \mid n \in \mathbb N \}$ denotes the set of all even numbers, using set builder notation. Set…
- Nate Soares - Shift towards the hypothesis of least surprise
When you see new evidence, ask: which hypothesis is *least surprised?*
- Nate Soares - Solomonoff induction
A simple way to superintelligently predict sequences of data, given unlimited computing power.
- Eliezer Yudkowsky - Solomonoff induction: Intro Dialogue (Math 2)
An introduction to Solomonoff induction for the unfamiliar reader who isn't bad at math
- Eliezer Yudkowsky - Some computations are people
It's possible to have a conscious person being simulated inside a computer or other substrate.
- Eliezer Yudkowsky - Standard agent properties
What's a Standard Agent, and what can it do?
- Eliezer Yudkowsky - Strained argument
A phenomenological feeling associated with a step of reasoning going from X to Y where it feels like…
- Eliezer Yudkowsky - Strong cognitive uncontainability
An advanced agent can win in ways humans can't understand in advance.
- Eliezer Yudkowsky - Stub
This page only gives a very brief overview of the topic. If you're able to, please help expand or improve it!
- Eliezer Yudkowsky - Sufficiently optimized agents appear coherent
If you could think as well as a superintelligence, you'd be at least that smart yourself.
- Eliezer Yudkowsky - Task-directed AGI
An advanced AI that's meant to pursue a series of limited-scope goals given it by the user. In Bostrom's terminology, a Genie.
- Eliezer Yudkowsky - The End (of the basic log tutorial)
That concludes our introductory tutorial on logarithms! You have made it to the end.
Throughout thi…
- Nate Soares - The characteristic of the logarithm
Any time you find an output that adds whenever the input multiplies, you're probably looking at a (…
- Nate Soares - The log lattice
Log as the change in the cost of communicating and other pages give physical interpretations of what…
- Nate Soares - The rocket alignment problem
If people talked about the problem of space travel the way they talked about AI...
- Eliezer Yudkowsky - The world needs something like a miracle of good thought if it is to have decent odds of overcoming x-risk. - Eric Rogstad
- Thinking accumulatively is one of the keys to solving existential risks. - Eric Rogstad
- Turing machine
A Turing Machine is a simple mathematical model of computation that is powerful enough to describe any computation a computer can do.
- Eric Leese - Type theory
Modern foundations for formal mathematics.
- Jack Gallagher - Uncountability
Some infinities are bigger than others. Uncountable infinities are larger than countable infinities.
- Jason Gross - Underlying set
What do a Group, a Partially ordered set, and a [ topological space] have in common? Each is a Set …
- Nate Soares - Unforeseen maximum
When you tell AI to produce world peace and it kills everyone. (Okay, some SF writers saw that one coming.)
- Eliezer Yudkowsky - Union
The union of two sets is the set of elements which are in one or the other, or both
- M Yass - Utility
What is "utility" in the context of Value Alignment Theory?
- Eliezer Yudkowsky - Value
The word 'value' in the phrase 'value alignment' is a metasyntactic variable that indicates the speaker's future goals for intelligent life.
- Eliezer Yudkowsky - Value achievement dilemma
How can Earth-originating intelligent life achieve most of its potential value, whether by AI or otherwise?
- Eliezer Yudkowsky - Value alignment problem
You want to build an advanced AI with the right values... but how?
- Eliezer Yudkowsky - Value-laden
Cure cancer, but avoid any bad side effects? Categorizing "bad side effects" requires knowing what's "bad". If an agent needs to load complex human goals to evaluate something, it's "value-laden".
- Eliezer Yudkowsky - Vingean uncertainty
You can't predict the exact actions of an agent smarter than you - so is there anything you _can_ say about them?
- Eliezer Yudkowsky - Waterfall diagrams and relative odds
A way to visualize Bayes' rule that yields an easier way to solve some problems
- Eliezer Yudkowsky - We do not currently have a single shared conversation. - Eric Rogstad
- Why argument structure is important
How might we make collaborative truth-seeking both fun and easy?
- Andrea Gallagher - Work in progress
This page is being actively worked on by an editor. Check with them before making major changes.
- Eliezer Yudkowsky - Zermelo-Fraenkel provability oracle
We might be able to build a system that can safely inform us that a theorem has a proof in set theory, but we can't see how to use that capability to save the world.
- Eliezer Yudkowsky
no-type
- playpen subpage
playpen subpage clickbait
- Robert Lecnik
- Probability notation for Bayes' rule
The probability notation used in Bayesian reasoning
- Eliezer Yudkowsky
- Non-adversarial principle
At no point in constructing an Artificial General Intelligence should we construct a computation that tries to hurt us, and then try to stop it from hurting us.
- Eliezer Yudkowsky
- Predictions For 2017
Scott Alexander made 105 predictions for 2017. Most of them are not personal and are listed below. …
- Alexei Andreev
- Arbital
Arbital is the place for crowdsourced, intuitive math explanations.
- Alexei Andreev - Arbital playpen
Want to test a feature? Feel free to edit this page! asdfasfdasfda
- Eliezer Yudkowsky
- Frequency diagrams: A first look at Bayes
The most straightforward visualization of Bayes' rule.
- Nate Soares
- Arbital
Arbital is the place for crowdsourced, intuitive math explanations.
- Alexei Andreev - Fixed point theorem of provability logic
Deal with those pesky self-referential sentences!
- Jaime Sevilla Molina - Group presentation
Presentations are a fairly compact way of expressing groups.
- Patrick Stevens - Set product
A fundamental way of combining sets is to take their product, making a set that contains all tuples of elements from the originals.
- Patrick Stevens
- Existential risk
> Because of accelerating technological progress, humankind may be rapidly approaching a critical ph…
- Eric Rogstad
- AIXI
How to build an (evil) superintelligent AI using unlimited computing power and one page of Python code.
- Eliezer Yudkowsky
- High-speed intro to Bayes's rule
A high-speed introduction to Bayes's Rule on one page, for the impatient and mathematically adept.
- Eliezer Yudkowsky
- Conditional probability: Refresher
Is P(yellow | banana) the probability that a banana is yellow, or the probability that a yellow thing is a banana?
- Nate Soares
- Ackermann function
The slowest-growing fast-growing function.
- Alex Appel - Project proposal: Complex numbers
Project proposal for complex numbers
- Alexei Andreev - Transcendental number
A transcendental number is one which is not the root of any integer-coefficient polynomial.
- Patrick Stevens
- Bayes' rule: Guide
The Arbital guide to Bayes' rule
- Eliezer Yudkowsky
- Poset: Exercises
Try these exercises to test your poset knowledge.
# Corporate Ladder
Imagine a company with five …
- Kevin Clancy
- Group: Exercises
Test your understanding of the definition of a group with these exercises.
- Qiaochu Yuan
- Bayes' rule: Vector form
For when you want to apply Bayes' rule to lots of evidence and lots of variables, all in one go. (This is more or less how spam filters work.)
- Eliezer Yudkowsky - Church-Turing thesis
A thesis about computational models
- Jaime Sevilla Molina - High-speed intro to Bayes's rule
A high-speed introduction to Bayes's Rule on one page, for the impatient and mathematically adept.
- Eliezer Yudkowsky
- Vitamin D helps prevent cancer
There have been a lot of studies performed that show that vitamin D helps prevent cancer, but overal…
- Alexei Andreev
- Do-What-I-Mean hierarchy
Successive levels of "Do What I Mean" or AGIs that understand their users increasingly well
- Eliezer Yudkowsky
- Bayes' rule
Bayes' rule is the core theorem of probability theory saying how to revise our beliefs when we make a new observation.
- Eliezer Yudkowsky
- Predictions For 2017
Scott Alexander made 105 predictions for 2017. Most of them are not personal and are listed below. …
- Alexei Andreev
- Axiom of Choice
The most controversial axiom of the 20th century.
- Mark Chimes - Category theory
How mathematical objects are related to others in the same category.
- Mark Chimes - Group
The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.
- Nate Soares - Group: Examples
Why would anyone have invented groups, anyway? What were the historically motivating examples, and what examples are important today?
- Qiaochu Yuan - Introductory guide to logarithms
Welcome to the Arbital introduction to logarithms! In modern education, logarithms are often mention…
- Nate Soares - Isomorphism
A morphism between two objects which describes how they are "essentially equivalent" for the purposes of the theory under consideration.
- Mark Chimes - Uncountability (Math 3)
Formal definition of uncountability, and foundational considerations.
- Patrick Stevens
- Frequency diagrams: A first look at Bayes
The most straightforward visualization of Bayes' rule.
- Nate Soares
- Bayesian view of scientific virtues
Why is it that science relies on bold, precise, and falsifiable predictions? Because of Bayes' rule, of course.
- Eliezer Yudkowsky
- Report likelihoods not p-values: FAQ
This page answers frequently asked questions about the Report likelihoods, not p-values proposal for…
- Nate Soares
- Shift towards the hypothesis of least surprise
When you see new evidence, ask: which hypothesis is *least surprised?*
- Nate Soares
- Factorial
The number of ways you can order things. (Alternately subtitled: Is that exclamation point a factorial, or are you just excited to see me?)
- Michael Cohen
- Complexity theory: Complexity zoo
Pass and see the exotic beasts coming from the lands of afar!
- Jaime Sevilla Molina - Death in Damascus
Death tells you that It is coming for you tomorrow. You can stay in Damascus or flee to Aleppo. Whichever decision you actually make is the wrong one. This gives some decision theories trouble.
- Eliezer Yudkowsky
- Associative operation
An **associative operation** $\bullet : X \times X \to X$ is a binary operation such that for all $x…
- Nate Soares - Cardinality
The "size" of a set, or the "number of elements" that it has.
- Joe Zeng - Colon-to notation
Find out what the notation "f : X -> Y" means that everyone keeps using.
- Qiaochu Yuan - Commutative operation
A commutative function $f$ is a function that takes multiple inputs from a set $X$ and produces an o…
- Nate Soares - Cyclic Group Intro (Math 0)
A finite cyclic group is a little bit like a clock.
- Mark Chimes - Empty set
The empty set does what it says on the tin: it is the set which is empty.
- Patrick Stevens - Free group
The free group is "the purest way to make a group containing a given set".
- Patrick Stevens - Free groups are torsion-free
An easy way to determine that many groups are not free: free groups contain no non-identity elements of finite order.
- Patrick Stevens - Generalized element
A category-theoretic generalization of the notion of element of a set.
- Luke Sciarappa - Group
The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.
- Nate Soares - Integer
An **integer** is a Number that can be represented as either a Natural number or its [-additive\_inv…
- Michael Cohen - Inverse function
The inverse of a function returns an input of the original function when fed the original's corresponding output.
- Michael Cohen - Isomorphism
A morphism between two objects which describes how they are "essentially equivalent" for the purposes of the theory under consideration.
- Mark Chimes - Kripke model
The semantics of modal logic
- Jaime Sevilla Molina - Logical system
Logical systems (a.k.a. formal systems) are mathematical abstractions that aim to capture the notion…
- Jaime Sevilla Molina - Math 3 example statements
If you can read these formulas, you're in Math 3!
- Joe Zeng - Mathematical induction
Proving a statement about all positive integers by knocking them down like dominoes.
- Douglas Weathers - Number
An abstract object that expresses quantity or value of some sort.
- Joe Zeng - Proof of Löb's theorem
Proving that I am Santa Claus
- Jaime Sevilla Molina - Provability logic
Learn how to reason about provability!
- Jaime Sevilla Molina - Real number (as Dedekind cut)
A way to construct the real numbers that follows the intuition of filling in the gaps.
- Joe Zeng - Relation
A **relation** is a set of [tuple\_mathematics tuples], all of which have the same [tuple\_arity ar…
- Kevin Clancy - Stabiliser is a subgroup
Given a group acting on a set, each element of the set induces a subgroup of the group.
- Patrick Stevens - The reals (constructed as Dedekind cuts) form a field
The reals are an archetypal example of a field, but if we are to construct them from simpler objects, we need to show that our construction does indeed have the right properties.
- Patrick Stevens - Uncountability (Math 3)
Formal definition of uncountability, and foundational considerations.
- Patrick Stevens
- Solutions Paper - Solving Employment in the Light of Automation
GeM Labs solution space to tackle labour market challenges that arise from the rise of automation and artificial intelligence.
- Geneva Macro Labs
wiki
- A-Class
This page is well-written, high-quality, and essentially complete.
- Eric Bruylant - Arbital
Arbital is the place for crowdsourced, intuitive math explanations.
- Alexei Andreev - Arbital lens
A lens is a page that presents another page's content from a different angle.
- Alexei Andreev - B-Class
This page is mostly complete and without major problems, but has not had detailed feedback from the target audience and reviewers.
- Eric Bruylant - Bayes' rule: Definition
Bayes' rule is the mathematics of probability theory governing how to update your beliefs in the lig…
- Nate Soares - Bayes' rule: Probability form
The original formulation of Bayes' rule.
- Nate Soares - C-Class
This page has substantial content, but may not thoroughly cover the topic, may not meet style and prose standards, or may not explain the concept in a way the target audience will reliably understand.
- Eric Bruylant - Corrigibility
"I can't let you do that, Dave."
- Nate Soares - Featured
This page is has been selected by Arbital to be featured and promoted.
- Eric Bruylant - Frequency diagrams: A first look at Bayes
The most straightforward visualization of Bayes' rule.
- Nate Soares - Likelihood
"Likelihood", when speaking of Bayesian reasoning, denotes *the probability of an observation, sup…
- Nate Soares - Machine Intelligence Research Institute
Where to work if you're doing more formal or technical work on AI safety, of a kind not easily milked for publications.
- Alexei Andreev - More about Arbital
Lots more information about Arbital vision.
- Alexei Andreev - Odds: Introduction
What's the difference between probabilities and odds? Why is a 20% probability of success equivalent to 1 : 4 odds favoring success?
- Nate Soares - Odds: Refresher
A quick review of the notations and mathematical behaviors for odds (e.g. odds of 1 : 2 for drawing a red ball vs. green ball from a barrel).
- Nate Soares - Odds: Technical explanation
Formal definitions, alternate representations, and uses of odds and odds ratios (like a 1 : 2 chance of drawing a red ball vs. green ball from a barrel).
- Alexei Andreev - Page's title should always be capitalized
Vote "agree" if you think Arbital should enforce the first letter of a page title to always be capit…
- Alexei Andreev - Proof of Bayes' rule: Probability form
Let $\mathbf H$ be a [random\_variable variable] in $\mathbb P$ for the true hypothesis, and let $H_…
- Nate Soares
no-type
wiki
- Path to Acceleration
(Back to Accelerator home)
So, you know why we're doing this and what we're aiming for, a natural …
- Eric Bruylant
no-type
- Arbital page
The Arbital is a series of pages.
- Alexei Andreev - Odds
Odds express a relative probability.
- Eliezer Yudkowsky
wiki
- Addition of rational numbers (Math 0)
The simplest operation on rational numbers is addition.
- Patrick Stevens - Advanced agent properties
How smart does a machine intelligence need to be, for its niceness to become an issue? "Advanced" is a broad term to cover cognitive abilities such that we'd need to start considering AI alignment.
- Eliezer Yudkowsky - Algebraic structure tree
When is a monoid a semilattice? What's the difference between a semigroup and a groupoid? Find out here!
- Ryan Hendrickson - Arbital
Arbital is the place for crowdsourced, intuitive math explanations.
- Alexei Andreev - Arbital Blog
Stay up to date on all things Arbital
- Alexei Andreev - Arbital Markdown
All about Arbital's extended Markdown syntax.
- Alexei Andreev - Arbital as single conversational locus
Proposal and outline of Arbital's steps to create a single conversational locus.
- Alexei Andreev - Arbital biographies
As a very strong default (presently an absolute rule), Joe Smith's page only says nice things about Joe. Even if a negative fact is true, it doesn't go on Joe's page.
- Eliezer Yudkowsky - Arbital community input
Do you have ideas about how to improve Arbital which you think the community should discuss?
- Alexei Andreev - Arbital content license
What license does Arbital use for its content?
- Alexei Andreev - Arbital domain
What is a domain? Why is it important?
- Alexei Andreev - Arbital editor
How to use Arbital's page editor.
- Alexei Andreev - Arbital editor: Advanced
Advanced features of Arbital editor.
- Alexei Andreev - Arbital editor: Basics
The basics of how to use the Arbital editor.
- Alexei Andreev - Arbital features
Overview of all Arbital features.
- Alexei Andreev - Arbital greenlink
What happens when you hover over an Arbital link?
- Alexei Andreev - Arbital hidden text
How to hide text in Markdown behind a button.
- Alexei Andreev - Arbital lens
A lens is a page that presents another page's content from a different angle.
- Alexei Andreev - Arbital likes
What are likes? When should I use them? What happens when I like something?
- Alexei Andreev - Arbital mark
What is a mark on Arbital? When is it created? Why is it important?
- Alexei Andreev - Arbital page
The Arbital is a series of pages.
- Alexei Andreev - Arbital page alias
The alias is a short, unique name assigned to each page. For example: "arbital_alias".
The alias u…
- Eric Rogstad - Arbital page clickbait
The text you are reading right now is clickbait.
- Eric Rogstad - Arbital page summaries
Because only one summary is not enough!
- Alexei Andreev - Arbital playpen
Want to test a feature? Feel free to edit this page! asdfasfdasfda
- Eliezer Yudkowsky - Arbital practices
Guidelines and rules for interacting on Arbital.
- Eliezer Yudkowsky - Arbital proposed project
Collecting all project proposals under this page.
- Alexei Andreev - Arbital requisites
To understand a thing you often need to understand some other things.
- Alexei Andreev - Arbital subscriptions
What's a subscription? How do you change it? What to expect?
- Alexei Andreev - Arbital unlisted page
What do you call a page that's not part of any domain?
- Alexei Andreev - Arbital: Google Maps for knowledge
Take your understanding from where it is to where it wants to be.
- Alexei Andreev - Arbital: information hub
How will Arbital help you keep up to date on any given subject?
- Alexei Andreev - Arbital: learning from Wikipedia
How is Arbital different from Wikipedia?
- Alexei Andreev - Arithmetic of rational numbers (Math 0)
How do we combine rational numbers together?
- Patrick Stevens - Associativity: Examples
Yes: [Addition], [multiplication], string concatenation. No: [subtraction], [division], a Function …
- Nate Soares - Associativity: Intuition
Associative functions can be interpreted as families of functions that reduce lists down to a singl…
- Nate Soares - Author's guide to Arbital
How to write intuitive, flexible content on Arbital.
- Alexei Andreev - Axiom of Choice
The most controversial axiom of the 20th century.
- Mark Chimes - Bayes' rule: Definition
Bayes' rule is the mathematics of probability theory governing how to update your beliefs in the lig…
- Nate Soares - Bayes' rule: Functional form
Bayes' rule for to continuous variables.
- Eliezer Yudkowsky - Bayesian view of scientific virtues
Why is it that science relies on bold, precise, and falsifiable predictions? Because of Bayes' rule, of course.
- Eliezer Yudkowsky - Belief revision as probability elimination
Update your beliefs by throwing away large chunks of probability mass.
- Eliezer Yudkowsky - Bit
The term "bit" refers to different concepts in different fields. The common theme across all the us…
- Nate Soares - Boolean
A value in logic that evaluates to either "true" or "false".
- Malcolm McCrimmon - Bézout's theorem
Bézout's theorem is an important link between highest common factors and the integer solutions of a certain equation.
- Patrick Stevens - Cantor-Schröder-Bernstein theorem
This theorem tells us that comparing sizes of sets makes sense: if one set is smaller than another, and the other is smaller than the one, then they are the same size.
- Patrick Stevens - Cardinality
The "size" of a set, or the "number of elements" that it has.
- Joe Zeng - Category (mathematics)
A description of how a collection of mathematical objects are related to one another.
- Mark Chimes - Category theory
How mathematical objects are related to others in the same category.
- Mark Chimes - Church-Turing thesis
A thesis about computational models
- Jaime Sevilla Molina - Church-Turing thesis: Evidence for the Church-Turing thesis
Why do we believe in CT thesis?
- Jaime Sevilla Molina - Commutativity: Intuition
We can think of commutativity either as an artifact of notation, or as a symmetry in the output of a…
- Nate Soares - Creating a /learn/ link
What options are available when creating a /learn/ link?
- Alexei Andreev - Currying
Transforms a function of many arguments into a function into a function of a single argument
- M Yass - Cyclic group
Cyclic groups form one of the most simple classes of groups.
- Patrick Stevens - Decimal notation
The winning architecture for numerals
- Michael Cohen - Decision problem
Formalization of general problems
- Jaime Sevilla Molina - Derivative
How things change
- Michael Cohen - Eigenvalues and eigenvectors
We applied this linear transformation to one of its eigenvectors; you won't believe what happened next!
- Zack M. Davis - Elementary Algebra
How do we describe relations between different things? How can we figure out new true things from tr…
- Adele Lopez - Epistemology
What is truth?
- Eliezer Yudkowsky - Equivalence relation
A relation that allows you to partition a set into equivalence classes.
- Dylan Hendrickson - Expected value
Trying to assign value to an uncertain state? The weighted average of outcomes is probably the tool you need.
- Michael Cohen - Exponential
Any function that constantly gets larger as a proportion of itself.
- Joe Zeng - Extraordinary claims require extraordinary evidence
The people who adamantly claim they were abducted by aliens do provide some evidence for aliens. They just don't provide quantitatively enough evidence.
- Eliezer Yudkowsky - Factorial
The number of ways you can order things. (Alternately subtitled: Is that exclamation point a factorial, or are you just excited to see me?)
- Michael Cohen - Finishing your Bayesian path on Arbital
The page that comes at the end of reading the Arbital Guide to Bayes' rule
- Eliezer Yudkowsky - Free group
The free group is "the purest way to make a group containing a given set".
- Patrick Stevens - Greatest common divisor
The greatest common divisor of two natural numbers is… the largest number which is a divisor of both. The clue is in the name, really.
- Patrick Stevens - Group
The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.
- Nate Soares - Group homomorphism
A group homomorphism is a "function between groups" that "respects the group structure".
- Patrick Stevens - Group orbit
When we have a group acting on a set, we are often interested in how the group acts on a particular …
- Adele Lopez - Group: Examples
Why would anyone have invented groups, anyway? What were the historically motivating examples, and what examples are important today?
- Qiaochu Yuan - Gödel's first incompleteness theorem
The theorem that destroyed Hilbert's program
- Jaime Sevilla Molina - Identity element
An element in a set with a binary operation that leaves every element unchanged when used as the other operand.
- Joe Zeng - In notation
There's a weird E-looking symbol called \in in LaTeX. What does it mean?
- Qiaochu Yuan - Integer
An **integer** is a Number that can be represented as either a Natural number or its [-additive\_inv…
- Michael Cohen - Intentional Communities wiki pages
A categorized, handpicked list of articles relevant to the Accelerator project
- Toon Alfrink - Inverse function
The inverse of a function returns an input of the original function when fed the original's corresponding output.
- Michael Cohen - Irreducible element (ring theory)
This is the appropriate abstraction of the concept of "prime number" to general rings.
- Patrick Stevens - Isomorphism
A morphism between two objects which describes how they are "essentially equivalent" for the purposes of the theory under consideration.
- Mark Chimes - Isomorphism: Intro (Math 0)
Things which are basically the same, except for some stuff you don't care about.
- Mark Chimes - Join and meet
Let $\langle P, \leq \rangle$ be a poset, and let $S \subseteq P$. The **join** of $S$ in $P$, deno…
- Kevin Clancy - Lagrange theorem on subgroup size: Intuitive version
Lagrange's theorem strongly restricts the size a subgroup of a group can be.
- Patrick Stevens - Less Wrong
A community blog devoted to refining the art of human rationality.
- Alexei Andreev - Log base infinity
There is no log base infinity, but if there were, it would send everything to zero
- Nate Soares - Math style guidelines
Stylistic conventions specific to pages about math.
- Dylan Hendrickson - Mathematics
Mathematics is the study of numbers and other ideal objects that can be described by axioms.
- Eliezer Yudkowsky - Modal combat
Modal combat
- Jaime Sevilla Molina - Modal logic
The logic of boxes and bots.
- Jaime Sevilla Molina - Modular arithmetic
Addition as traveling around a circle, instead of along a line.
- Malcolm McCrimmon - Morphism
A morphism is the abstract representation of a relation between mathematical objects.
Usually, it i…
- Jaime Sevilla Molina - Needs exercises
Add this tag to a page which doesn't have enough exercises.
- Alexei Andreev - Needs lenses
This page has only a technical introduction. If you're able to, please help by adding an intuitive explanation!
- Nate Soares - Needs summary
This page does not have a summary which provides an informative overview of the page's primary topic.
- Alexei Andreev - Newcomb's Problem
There are two boxes in front of you, Box A and Box B. You can take both boxes, or only Box B. Box A contains $1000. Box B contains $1,000,000 if and only if Omega predicted you'd take only Box B.
- Eliezer Yudkowsky - Non-standard terminology
A tag for terminology that is Arbital-specific, Arbital-originated, or just not very common outside …
- Nate Soares - Now I am become Life, the protector of worlds
In which Arbital is brought forth into the world.
- Alexei Andreev - Odds
Odds express a relative probability.
- Eliezer Yudkowsky - Ontology identification problem
How do we link an agent's utility function to its model of the world, when we don't know what that model will look like?
- Eliezer Yudkowsky - Operations in Set theory
An operation in set theory is a Function of two sets, that returns a set.
Common set operations inc…
- M Yass - Orbit-Stabiliser theorem: External Resources
External resources on the Orbit-Stabiliser theorem.
- Mark Chimes - Out of date
Meta tag used when the page has a lot of information that's obsolete
- Alexei Andreev - P (Polynomial Time Complexity Class)
P is the class of problems which can be solved by algorithms whose run time is bounded by a polynomial.
- Eric Leese - Partially ordered set
A set endowed with a relation that is reflexive, transitive, and antisymmetric.
- Kevin Clancy - Pi
Pi, usually written $π$, is a number equal to the ratio of a circle's [-circumference] to its [-diam…
- Michael Cohen - Poset: Exercises
Try these exercises to test your poset knowledge.
# Corporate Ladder
Imagine a company with five …
- Kevin Clancy - Prime element of a ring
Despite the name, "prime" in ring theory refers not to elements which are "multiplicatively irreducible" but to those such that if they divide a product then they divide some term of the product.
- Patrick Stevens - Principal ideal domain
A principal ideal domain is a kind of ring, in which all ideals have a certain nice form.
- Patrick Stevens - Prior probability
What we believed before seeing the evidence.
- Eliezer Yudkowsky - Probability
The degree to which someone believes something, measured on a scale from 0 to 1, allowing us to do math to it.
- Eliezer Yudkowsky - Probability notation for Bayes' rule: Intro (Math 1)
How to read, and identify, the probabilities in Bayesian problems.
- Eliezer Yudkowsky - Project outline: Intro to the Universal Property
Outline detailing all the work required for a proposed Arbital Project
- Eric Rogstad - Project proposal: Intro to numbers
Should Arbital's first "project" be a guide to numbers?
- Eric Rogstad - Proof by contradiction
Discover what 'reductio ad absurdum' means!
- Jaime Sevilla Molina - Proof of Bayes' rule: Probability form
Let $\mathbf H$ be a [random\_variable variable] in $\mathbb P$ for the true hypothesis, and let $H_…
- Nate Soares - Quality meta tags
Meta tags which determine the page's quality.
- Alexei Andreev - Real number
A **real number** is any number that can be used to represent a physical quantity.
Intuitively, rea…
- Michael Cohen - Realistic (Math 1)
Real-life examples of Bayesian reasoning
- Eliezer Yudkowsky - Relative likelihood
How relatively likely an observation is, given two or more hypotheses, determines the strength and direction of evidence.
- Eliezer Yudkowsky - Report likelihoods not p-values: FAQ
This page answers frequently asked questions about the Report likelihoods, not p-values proposal for…
- Nate Soares - Report likelihoods, not p-values
If scientists reported likelihood functions instead of p-values, this could help science avoid p-ha…
- Nate Soares - Rescuing the utility function
If your utility function values 'heat', and then you discover to your horror that there's no ontologically basic heat, switch to valuing disordered kinetic energy. Likewise 'free will' or 'people'.
- Eliezer Yudkowsky - Set
An unordered collection of distinct objects.
- Nate Soares - Standard provability predicate
Encoding provability as a statement of arithmetic
- Jaime Sevilla Molina - Start
This page gives a basic overview of the topic, but may be missing important information or have stylistic issues. If you're able to, please help expand or improve it!
- Eliezer Yudkowsky - Strong Church Turing thesis
A strengthening of the Church Turing thesis
- Jaime Sevilla Molina - Stub
This page only gives a very brief overview of the topic. If you're able to, please help expand or improve it!
- Eliezer Yudkowsky - Style guidelines
Various stylistic conventions people should follow on Arbital
- Alexei Andreev - Submitting a page to a domain
How and why to submit a page to a domain
- Alexei Andreev - Transparent Newcomb's Problem
Omega has left behind a transparent Box A containing $1000, and a transparent Box B containing $1,000,000 or nothing. Box B is full iff Omega thinks you one-box on seeing a full Box B.
- Eliezer Yudkowsky - Turing machine
A Turing Machine is a simple mathematical model of computation that is powerful enough to describe any computation a computer can do.
- Eric Leese - Uncomputability
The diagonal function and the halting problem
- Jaime Sevilla Molina - Uncountability
Some infinities are bigger than others. Uncountable infinities are larger than countable infinities.
- Jason Gross - Uncountability (Math 3)
Formal definition of uncountability, and foundational considerations.
- Patrick Stevens - Uncountability: Intro (Math 1)
Not all infinities are created equal. The infinity of real numbers is infinitely larger than the infinity of counting numbers.
- Jason Gross - Uncountability: Intuitive Intro
Are all sizes of infinity the same? What does "the same" even mean here?
- Jason Gross - Universal property of the disjoint union
Just as the empty set may be described by a universal property, so too may the disjoint union of sets.
- Patrick Stevens - Universal property of the empty set
The empty set can be characterised by how it interacts with other sets, rather than by any explicit property of the empty set itself.
- Patrick Stevens - Up to isomorphism
A phrase mathematicians use when saying "we only care about the structure of an object, not about specific implementation details of the object".
- Patrick Stevens - Waterfall diagrams and relative odds
A way to visualize Bayes' rule that yields an easier way to solve some problems
- Eliezer Yudkowsky - What is a logarithm?
Logarithms are a group of functions that take a number as input and produce another number. There i…
- Nate Soares - Work in progress
This page is being actively worked on by an editor. Check with them before making major changes.
- Eliezer Yudkowsky
no-type
wiki
- A googol
A pretty small large number.
- Nate Soares - A googolplex
A moderately large number, as large numbers go.
- Nate Soares - AIXI
How to build an (evil) superintelligent AI using unlimited computing power and one page of Python code.
- Eliezer Yudkowsky - Abelian group
A group where the operation commutes. Named after Niels Henrik Abel.
- Nate Soares - Abstract algebra
The study of groups, fields, vector spaces, arithmetics, algebras, and more.
- Nate Soares - Ackermann function
The slowest-growing fast-growing function.
- Alex Appel - Algebraic structure
Roughly speaking, an algebraic structure is a set $X$, known as the underlying set, paired with a co…
- Nate Soares - Answer to sparking widgets problem
Odds of 1 : 3, probability of 1/4.
- Eliezer Yudkowsky - Antisymmetric relation
A binary relation where no two distinct elements are related in both directions
- M Yass - Arbital
Arbital is the place for crowdsourced, intuitive math explanations.
- Alexei Andreev - Arbital Markdown
All about Arbital's extended Markdown syntax.
- Alexei Andreev - Arbital Markdown questionnaire
How to ask questions in Markdown.
- Alexei Andreev - Arbital content license
What license does Arbital use for its content?
- Alexei Andreev - Arbital external resources
Arbital wants to link users to great content, wherever it is!
- Eric Bruylant - Arbital lens
A lens is a page that presents another page's content from a different angle.
- Alexei Andreev - Arbital mark
What is a mark on Arbital? When is it created? Why is it important?
- Alexei Andreev - Arbital page: Basics
Explaining the basic features of an Arbital page.
- Alexei Andreev - Arbital playpen
Want to test a feature? Feel free to edit this page! asdfasfdasfda
- Eliezer Yudkowsky - Arbital scope
What kind of content is Arbital looking for?
- Eric Bruylant - Arbital unlisted page
What do you call a page that's not part of any domain?
- Alexei Andreev - Arbital: learning from Wikipedia
How is Arbital different from Wikipedia?
- Alexei Andreev - Associative operation
An **associative operation** $\bullet : X \times X \to X$ is a binary operation such that for all $x…
- Nate Soares - Assuming significant overhead in monitoring recipients of a microloan, it's more efficient to let them keep the money.
A claim about microfinance.
- Alexei Andreev - Author's guide to Arbital
How to write intuitive, flexible content on Arbital.
- Alexei Andreev - Bayes' rule: Functional form
Bayes' rule for to continuous variables.
- Eliezer Yudkowsky - Bayes' rule: Guide
The Arbital guide to Bayes' rule
- Eliezer Yudkowsky - Bayes' rule: Log-odds form
A simple transformation of Bayes' rule reveals tools for measuring degree of belief, and strength of evidence.
- Eliezer Yudkowsky - Bayes' rule: Probability form
The original formulation of Bayes' rule.
- Nate Soares - Bayes' rule: Proportional form
The fastest way to say something both convincing and true about belief-updating.
- Eliezer Yudkowsky - Bayesian view of scientific virtues
Why is it that science relies on bold, precise, and falsifiable predictions? Because of Bayes' rule, of course.
- Eliezer Yudkowsky - Big-picture strategic awareness
We start encountering new AI alignment issues at the point where a machine intelligence recognizes the existence of a real world, the existence of programmers, and how these relate to its goals.
- Eliezer Yudkowsky - Binary notation
A way to write down numbers using powers of two.
- Malcolm McCrimmon - Bit (of data)
A bit of data is the amount of data required to single out one message from a set of two. Equivalen…
- Nate Soares - C-Class
This page has substantial content, but may not thoroughly cover the topic, may not meet style and prose standards, or may not explain the concept in a way the target audience will reliably understand.
- Eric Bruylant - Cardinality
The "size" of a set, or the "number of elements" that it has.
- Joe Zeng - Church-Turing thesis
A thesis about computational models
- Jaime Sevilla Molina - Claim-tagging is worth trying more broadly
Source of claim: Improve comments by tagging claims by Benjamin Hoffman
- Stephanie Zolayvar - Codomain vs image
It is useful to distinguish codomain from image both (a) when the type of thing that the function pr…
- Nate Soares - Colon-to notation
Find out what the notation "f : X -> Y" means that everyone keeps using.
- Qiaochu Yuan - Comments are a high-quality, high-sensitivity measure of engagement with little in the way of viable substitutes.
Source of claim: Improve comments by tagging claims by Benjamin Hoffman
- Stephanie Zolayvar - Complexity theory
Study of the computational resources needed to compute something
- Jaime Sevilla Molina - Compressing multiple messages
How many bits of data does it take to encode an $n$-message? Naively, the answer is $\lceil \log_2(n…
- Nate Soares - Contributing to Arbital
Want to help Arbital become awesome?
- Eric Bruylant - Convergent strategies of self-modification
The strategies we'd expect to be employed by an AI that understands the relevance of its code and hardware to achieving its goals, which therefore has subgoals about its code and hardware.
- Eliezer Yudkowsky - Correspondence visualizations for different interpretations of "probability"
Let's say you have a model which says a particular coin is 70% likely to be heads. How should we as…
- Nate Soares - Deep Blue
The chess-playing program, built by IBM, that first won the world chess championship from Garry Kasparov in 1996.
- Eliezer Yudkowsky - Digit wheel
A mechanical device for storing a number from 0 to 9.
 = 3$ bits of data to carry one message from a set of 8 possible messages. Simila…
- Nate Soares - Fractional digits
When $b$ and $x$ are integers, $\log_b(x)$ has a few good interpretations. It's roughly the length o…
- Nate Soares - Free group
The free group is "the purest way to make a group containing a given set".
- Patrick Stevens - Freely reduced word
"Freely reduced" captures the idea of "no cancellation" in a free group.
- Patrick Stevens - Frequency diagrams: A first look at Bayes
The most straightforward visualization of Bayes' rule.
- Nate Soares - Function
Intuitively, a function $f$ is a procedure (or machine) that takes an input and performs some opera…
- Nate Soares - Fundraisers should have a threshold amount which, if not hit, results in a refund.
When starting a fundraiser, a nonprofit should declare a threshold amount. If the nonprofit doesn't …
- Alexei Andreev - Generalized associative law
Given an associative operator $\cdot$ and a list $[a, b, c, \ldots]$ of parameters, all ways of red…
- Nate Soares - Group
The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.
- Nate Soares - Group action
"Groups, as men, will be known by their actions."
- Qiaochu Yuan - Group theory
What kinds of symmetry can an object have?
- Nate Soares - Group theory: Examples
What does thinking in terms of group theory actually look like? And what does it buy you?
- Qiaochu Yuan - Hiring
What's Omniment's protocol for hiring employees?
- Alexei Andreev - How many bits to a trit?
$\log_2(3) \approx 1.585.$ This can be interpreted a few different ways:
1. If you multiply the nu…
- Nate Soares - How to author on Arbital!
Want to contribute pages to Arbital? Here's our current version of the ad-hoc guide to being an author!
- Eliezer Yudkowsky - Improve comments by tagging claims
Comment sections are more important for discourse than I thought. They can be improved by explicitly tagging an article's main claims as anchors for discussion.
- Benjamin Hoffman - Interpretations of "probability"
What does it *mean* to say that a fair coin has a 50% probability of coming up heads?
- Nate Soares - Introduction to Effective Altruism
Effective altruism (EA) means using evidence and reason to take actions that help others as much as …
- Aaron Gertler - Introductory guide to logarithms
Welcome to the Arbital introduction to logarithms! In modern education, logarithms are often mention…
- Nate Soares - Inverse function
The inverse of a function returns an input of the original function when fed the original's corresponding output.
- Michael Cohen - Irrelevant nitpicks are an important problem in comment sections on sites such as LessWrong.
Source of claim: Improve comments by tagging claims by Benjamin Hoffman
- Stephanie Zolayvar - Isomorphism
A morphism between two objects which describes how they are "essentially equivalent" for the purposes of the theory under consideration.
- Mark Chimes - Isomorphism: Intro (Math 0)
Things which are basically the same, except for some stuff you don't care about.
- Mark Chimes - Join and meet: Examples
A union of sets and the least common multiple of a set of natural numbers can both be viewed as join…
- Kevin Clancy - Lattice (Order Theory)
A poset that is closed under binary joins and meets.
- Kevin Clancy - Libya will remain a mess in 2017 - Alexei Andreev
- Life in logspace
The log lattice hints at the reason that engineers, scientists, and AI researchers find logarithms s…
- Nate Soares - Likelihood functions, p-values, and the replication crisis
What's the whole Bayesian-vs.-frequentist debate about?
- Eliezer Yudkowsky - Location on the comments-links continuum is an important aspect of discourse design.
Source of claim: Improve comments by tagging claims by Benjamin Hoffman
- Stephanie Zolayvar - Log as the change in the cost of communicating
When interpreting logarithms as a generalization of the notion of "length" and as digit exchange rat…
- Nate Soares - Logarithm
The logarithm base $b$ of a number $n,$ written $\log_b(n),$ is the answer to the question "how man…
- Nate Soares - Logarithms invert exponentials
The function $\log_b(\cdot)$ inverts the function $b^{(\cdot)}.$ In other words, $\log_b(n) = x$ imp…
- Nate Soares - Logical game
Game's mathematical structure at its purest form.
- Eliezer Yudkowsky - Logical system
Logical systems (a.k.a. formal systems) are mathematical abstractions that aim to capture the notion…
- Jaime Sevilla Molina - Löb's theorem
Löb's theorem
- Jaime Sevilla Molina - Löb's theorem and computer programs
The close relationship between [ logic and computability] allows us to frame Löb's theorem in terms …
- Jaime Sevilla Molina - Mapsto notation
There's an arrow called \mapsto in LaTeX. What does it mean?
- Qiaochu Yuan - Math playpen
Playpen page for Math domain
- Alexei Andreev - Math style guidelines
Stylistic conventions specific to pages about math.
- Dylan Hendrickson - Mindcrime: Introduction
The more predictive accuracy we want from a model, the more detailed the model becomes. A very roug…
- Eliezer Yudkowsky - Natural number
The numbers we use to count: 0, 1, 2, 3, ...
- Jaime Sevilla Molina - Needs summary
This page does not have a summary which provides an informative overview of the page's primary topic.
- Alexei Andreev - No exchange of fire over "tiny stupid islands" in 2017 - Alexei Andreev
- Odds
Odds express a relative probability.
- Eliezer Yudkowsky - On the importance of Less Wrong, or another single conversational locus
In this post, Anna Salamon talks about how Less Wrong used to a locus of discussion, and that it is…
- Alexei Andreev - P vs NP: Arguments against P=NP
Why we believe P and NP are different
- Jaime Sevilla Molina - Partially ordered set
A set endowed with a relation that is reflexive, transitive, and antisymmetric.
- Kevin Clancy - Patch resistance
One does not simply solve the value alignment problem.
- Eliezer Yudkowsky - Perfect rolling sphere
If you don't understand something, start by assuming it's a perfect rolling sphere.
- Eliezer Yudkowsky - Pi
Pi, usually written $π$, is a number equal to the ratio of a circle's [-circumference] to its [-diam…
- Michael Cohen - Pi is irrational
The number pi is famously not rational, in spite of joking attempts at legislation to fix its value at 3 or 22/7.
- Patrick Stevens - Poset: Examples
The standard $\leq$ relation on integers, the $\subseteq$ relation on sets, and the $|$ (divisibilit…
- Kevin Clancy - Posterior probability
What we believe, after seeing the evidence and doing a Bayesian update.
- Eliezer Yudkowsky - Probability
The degree to which someone believes something, measured on a scale from 0 to 1, allowing us to do math to it.
- Eliezer Yudkowsky - Proof by contradiction
Discover what 'reductio ad absurdum' means!
- Jaime Sevilla Molina - Properties of the logarithm
- $\log_b(x \cdot y) = \log_b(x) + \log_b(y)$ for any $b$, this is the defining characteristic of …
- Nate Soares - Provability predicate
A provability predicate of a theory $T$ is a formula $P(x)$ with one free variable $x$ such that:
…
- Jaime Sevilla Molina - Quotient group
Given a group $G$ with operation $\bullet$ and a special kind of subgroup $N \leq G$ called the "no…
- Adele Lopez - Rational number
The rational numbers are "fractions".
- Patrick Stevens - Real number
A **real number** is any number that can be used to represent a physical quantity.
Intuitively, rea…
- Michael Cohen - Relative likelihood
How relatively likely an observation is, given two or more hypotheses, determines the strength and direction of evidence.
- Eliezer Yudkowsky - Report likelihoods, not p-values
If scientists reported likelihood functions instead of p-values, this could help science avoid p-ha…
- Nate Soares - Rescuing the utility function
If your utility function values 'heat', and then you discover to your horror that there's no ontologically basic heat, switch to valuing disordered kinetic energy. Likewise 'free will' or 'people'.
- Eliezer Yudkowsky - Set
An unordered collection of distinct objects.
- Nate Soares - Shannon
The shannon (Sh) is a unit of Information. One shannon is the difference in [info\_entropy entropy] …
- Nate Soares - Square visualization of probabilities on two events
$$
\newcommand{\true}{\text{True}}
\newcommand{\false}{\text{False}}
\newcommand{\bP}{\mathbb{P}}
…
- Tsvi BT - Standard provability predicate
Encoding provability as a statement of arithmetic
- Jaime Sevilla Molina - Strictly factual question
A "question of strict fact" is one which is true or false about the material universe (and maybe some math) without introducing any issues of values, perspectives, etcetera.
- Eliezer Yudkowsky - Strong Church Turing thesis
A strengthening of the Church Turing thesis
- Jaime Sevilla Molina - Submitting a page to a domain
How and why to submit a page to a domain
- Alexei Andreev - Superintelligent
A "superintelligence" is strongly superhuman (strictly higher-performing than any and all humans) on every cognitive problem.
- Eliezer Yudkowsky - The characteristic of the logarithm
Any time you find an output that adds whenever the input multiplies, you're probably looking at a (…
- Nate Soares - The composition of two group homomorphisms is a homomorphism
The collection of group homomorphisms is closed under composition.
- Patrick Stevens - The log lattice
Log as the change in the cost of communicating and other pages give physical interpretations of what…
- Nate Soares - There is only one logarithm
All logarithm functions are the same, up to a multiplicative constant.
- Nate Soares - To math explanations and beyond!
In which Arbital doubles down on math explanations.
- Alexei Andreev - Totally ordered set
A set where all the elements can be compared as greater than or less than.
- Joe Zeng - Two independent events
What do [a pair of dice], [a pair of coins], and [a pair of people on opposite sides of the planet] all have in common?
- Tsvi BT - Two independent events: Square visualization
$$
\newcommand{\true}{\text{True}}
\newcommand{\false}{\text{False}}
\newcommand{\bP}{\mathbb{P}}
…
- Tsvi BT - Uncomputability
The diagonal function and the halting problem
- Jaime Sevilla Molina - Uncountability: Intuitive Intro
Are all sizes of infinity the same? What does "the same" even mean here?
- Jason Gross - Underlying set
What do a Group, a Partially ordered set, and a [ topological space] have in common? Each is a Set …
- Nate Soares - Unforeseen maximum
When you tell AI to produce world peace and it kills everyone. (Okay, some SF writers saw that one coming.)
- Eliezer Yudkowsky - Waterfall diagram
Visualizing Bayes' rule as the mixing of probability streams.
- Eliezer Yudkowsky - What is a logarithm?
Logarithms are a group of functions that take a number as input and produce another number. There i…
- Nate Soares - Why is log like length?
If a number $x$ is $n$ digits long (in Decimal notation), then its logarithm (base 10) is between $n…
- Nate Soares - You can't get the coffee if you're dead
An AI given the goal of 'get the coffee' can't achieve that goal if it has been turned off; so even an AI whose goal is just to fetch the coffee may try to avert a shutdown button being pressed.
- Eliezer Yudkowsky - Zero-Shot Translation with Google’s Multilingual Neural Machine Translation System
Sorry, old news. But I'll leave it up.
- Andrea Gallagher - n-message
A message singling out one thing from a set of $n$ is sometimes called an $n$-message. For example,…
- Nate Soares
group
no-type
- Arbital playpen
Want to test a feature? Feel free to edit this page! asdfasfdasfda
- Eliezer Yudkowsky
- Bayes' rule: Vector form
For when you want to apply Bayes' rule to lots of evidence and lots of variables, all in one go. (This is more or less how spam filters work.)
- Eliezer Yudkowsky
- Mathematics
Mathematics is the study of numbers and other ideal objects that can be described by axioms.
- Eliezer Yudkowsky
- Coherent decisions imply consistent utilities
Why do we all use the 'expected utility' formalism? Because any behavior that can't be viewed from that perspective, must be qualitatively self-defeating (in various mathy ways).
- Eliezer Yudkowsky
- Possible math pages
A list of things which we may want math pages on
- Eric Bruylant
- Bayes' rule: Vector form
For when you want to apply Bayes' rule to lots of evidence and lots of variables, all in one go. (This is more or less how spam filters work.)
- Eliezer Yudkowsky - Mutually exclusive and exhaustive
The condition needed for probabilities to sum to 1
- Eliezer Yudkowsky
- Bayes' rule: Vector form
For when you want to apply Bayes' rule to lots of evidence and lots of variables, all in one go. (This is more or less how spam filters work.)
- Eliezer Yudkowsky
- Elementary Algebra
How do we describe relations between different things? How can we figure out new true things from tr…
- Adele Lopez
- Extraordinary claims require extraordinary evidence
The people who adamantly claim they were abducted by aliens do provide some evidence for aliens. They just don't provide quantitatively enough evidence.
- Eliezer Yudkowsky
- Odds
Odds express a relative probability.
- Eliezer Yudkowsky
- Solomonoff induction: Intro Dialogue (Math 2)
An introduction to Solomonoff induction for the unfamiliar reader who isn't bad at math
- Eliezer Yudkowsky - Zermelo-Fraenkel provability oracle
We might be able to build a system that can safely inform us that a theorem has a proof in set theory, but we can't see how to use that capability to save the world.
- Eliezer Yudkowsky
- Bayes' rule: Proportional form
The fastest way to say something both convincing and true about belief-updating.
- Eliezer Yudkowsky
- Empty set
The empty set does what it says on the tin: it is the set which is empty.
- Patrick Stevens - Set
An unordered collection of distinct objects.
- Nate Soares
- Colon-to notation
Find out what the notation "f : X -> Y" means that everyone keeps using.
- Qiaochu Yuan
- Causal decision theories
On CDT, to choose rationally, you should imagine the world where your physical act changes, then imagine running that world forward in time. (Therefore, it's irrational to vote in elections.)
- Eliezer Yudkowsky
- Arbital "parent" relationship
Parent-child relationship between pages implies a strong, inseparable connection.
- Alexei Andreev
- Algorithmic complexity
When you compress the information, what you are left with determines the complexity.
- Eliezer Yudkowsky - Axiom
An **axiom** of a [theory\_mathematics theory] $T$ is a [well\_formed well-formed] [sentence\_mathem…
- Eric Bruylant - Axiom of Choice
The most controversial axiom of the 20th century.
- Mark Chimes - Category theory
How mathematical objects are related to others in the same category.
- Mark Chimes - Proof of Bayes' rule: Intro
Proof of Bayes' rule, assuming you know the rule itself, and the notations for the quantities involved.
- Eliezer Yudkowsky - Two independent events: Square visualization
$$
\newcommand{\true}{\text{True}}
\newcommand{\false}{\text{False}}
\newcommand{\bP}{\mathbb{P}}
…
- Tsvi BT - Uncountability
Some infinities are bigger than others. Uncountable infinities are larger than countable infinities.
- Jason Gross
- Laplace's Rule of Succession
Suppose you flip a coin with an unknown bias 30 times, and see 4 heads and 26 tails. The Rule of Succession says the next flip has a 5/32 chance of showing heads.
- Eliezer Yudkowsky
- Coordinative AI development hypothetical
What would safe AI development look like if we didn't have to worry about anything else?
- Eliezer Yudkowsky
- Mindcrime
Might a machine intelligence contain vast numbers of unhappy conscious subprocesses?
- Eliezer Yudkowsky
- What is a logarithm?
Logarithms are a group of functions that take a number as input and produce another number. There i…
- Nate Soares
- Addition of rational numbers exercises
Test and cement your understanding of how we add rational numbers!
- Patrick Stevens - Arbital math levels
How mathy do you like your pages?
- Eric Bruylant - Bayes' rule: Guide
The Arbital guide to Bayes' rule
- Eliezer Yudkowsky - Bayes' rule: Log-odds form
A simple transformation of Bayes' rule reveals tools for measuring degree of belief, and strength of evidence.
- Eliezer Yudkowsky - Equivalence relation
A relation that allows you to partition a set into equivalence classes.
- Dylan Hendrickson - Group
The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.
- Nate Soares - Integer
An **integer** is a Number that can be represented as either a Natural number or its [-additive\_inv…
- Michael Cohen - Logarithms invert exponentials
The function $\log_b(\cdot)$ inverts the function $b^{(\cdot)}.$ In other words, $\log_b(n) = x$ imp…
- Nate Soares - Math 0
Are you not actively bad at math, nor traumatized about math?
- Eliezer Yudkowsky - Math 1
Is math sometimes fun for you, and are you not anxious if you see a math puzzle you don't know how to solve?
- Eliezer Yudkowsky - Math 3
Can you read the sort of things that professional mathematicians read, aka LaTeX formulas with a minimum of explanation?
- Eliezer Yudkowsky - Natural number
The numbers we use to count: 0, 1, 2, 3, ...
- Jaime Sevilla Molina - Ordered ring
A ring with a total ordering compatible with its ring structure.
- Dylan Hendrickson - Ordering of rational numbers (Math 0)
How do we know if one lot of apples is "more apples" than another lot?
- Patrick Stevens - Partially ordered set
A set endowed with a relation that is reflexive, transitive, and antisymmetric.
- Kevin Clancy - Poset: Examples
The standard $\leq$ relation on integers, the $\subseteq$ relation on sets, and the $|$ (divisibilit…
- Kevin Clancy - Project proposal: Intro to numbers
Should Arbital's first "project" be a guide to numbers?
- Eric Rogstad - Rational number
The rational numbers are "fractions".
- Patrick Stevens - Rational numbers: Intro (Math 0)
The rational numbers are "fractions". While the natural numbers measure the answer to the question …
- Patrick Stevens - Real number
A **real number** is any number that can be used to represent a physical quantity.
Intuitively, rea…
- Michael Cohen - Real number (as Cauchy sequence)
There are several ways to construct real numbers; this is the most natural way to use them in computations.
- Patrick Stevens - Relation
A **relation** is a set of [tuple\_mathematics tuples], all of which have the same [tuple\_arity ar…
- Kevin Clancy - Subtraction of rational numbers (Math 0)
In which we meet anti-apples.
- Patrick Stevens - The rationals form a field
The set $\mathbb{Q}$ of rational numbers is a field.
# Proof
$\mathbb{Q}$ is a (commutative) ring …
- Patrick Stevens - The reals (constructed as classes of Cauchy sequences of rationals) form a field
The reals are an archetypal example of a field, but if we are to construct them from simpler objects, we need to show that our construction does indeed have the right properties.
- Patrick Stevens - Transcendental number
A transcendental number is one which is not the root of any integer-coefficient polynomial.
- Patrick Stevens - Uncountability: Intuitive Intro
Are all sizes of infinity the same? What does "the same" even mean here?
- Jason Gross - Union
The union of two sets is the set of elements which are in one or the other, or both
- M Yass - Well-ordered set
An ordered set with an order that always has a "next element".
- Dylan Hendrickson - What is a logarithm?
Logarithms are a group of functions that take a number as input and produce another number. There i…
- Nate Soares
- Frequency diagrams: A first look at Bayes
The most straightforward visualization of Bayes' rule.
- Nate Soares
- Bayes' rule: Log-odds form
A simple transformation of Bayes' rule reveals tools for measuring degree of belief, and strength of evidence.
- Eliezer Yudkowsky
- Löbstacle
Imagine you have an artificial intelligence $D1$ who uses a logical [ knowledge base] and a certain …
- Jaime Sevilla Molina
- Introduction to Bayes' rule: Odds form
Bayes' rule is simple, if you think in terms of relative odds.
- Eliezer Yudkowsky
- Introduction to Logical Decision Theory for Computer Scientists
'Logical decision theory' from a math/programming standpoint, including how two agents with mutual knowledge of each other's code can cooperate on the Prisoner's Dilemma.
- Eliezer Yudkowsky
- Examination through isomorphism
Isomorphism is the correct notion of equality between objects in a category. From the category-theor…
- Luke Sciarappa - In notation
There's a weird E-looking symbol called \in in LaTeX. What does it mean?
- Qiaochu Yuan - Least common multiple
The **least common multiple (LCM)** of two positive natural numbers a, b is the smallest natural …
- Johannes Schmitt - Mapsto notation
There's an arrow called \mapsto in LaTeX. What does it mean?
- Qiaochu Yuan - Math style guidelines
Stylistic conventions specific to pages about math.
- Dylan Hendrickson - Mathematical induction
Proving a statement about all positive integers by knocking them down like dominoes.
- Douglas Weathers - Metric
A metric is a function that defines a distance between elements in a set and follows some basic rules.
- Bryce Woodworth - Real number
A **real number** is any number that can be used to represent a physical quantity.
Intuitively, rea…
- Michael Cohen - Real number (as Dedekind cut)
A way to construct the real numbers that follows the intuition of filling in the gaps.
- Joe Zeng - Set
An unordered collection of distinct objects.
- Nate Soares - Totally ordered set
A set where all the elements can be compared as greater than or less than.
- Joe Zeng
- Waterfall diagrams and relative odds
A way to visualize Bayes' rule that yields an easier way to solve some problems
- Eliezer Yudkowsky
- Introduction to Bayes' rule: Odds form
Bayes' rule is simple, if you think in terms of relative odds.
- Eliezer Yudkowsky
- Geneva Macro Labs
Leveraging the collective intelligence of Geneva and surroundings to solve the world's biggest problems.
- Solutions Paper - Solving Employment in the Light of Automation
GeM Labs solution space to tackle labour market challenges that arise from the rise of automation and artificial intelligence.
- Geneva Macro Labs
- Bayes' rule: Log-odds form
A simple transformation of Bayes' rule reveals tools for measuring degree of belief, and strength of evidence.
- Eliezer Yudkowsky
- Predictions For 2017
Scott Alexander made 105 predictions for 2017. Most of them are not personal and are listed below. …
- Alexei Andreev
- Belief revision as probability elimination
Update your beliefs by throwing away large chunks of probability mass.
- Eliezer Yudkowsky - The rocket alignment problem
If people talked about the problem of space travel the way they talked about AI...
- Eliezer Yudkowsky
- Group
The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.
- Nate Soares - Monoid
A monoid $M$ is a pair $(X, \diamond)$ where $X$ is a [set\_theory\_set set] and $\diamond$ is an [a…
- Nate Soares
- Empty set
The empty set does what it says on the tin: it is the set which is empty.
- Patrick Stevens - Universal property of the product
The product can be defined in a very general way, applicable to the natural numbers, to sets, to algebraic structures, and so on.
- Patrick Stevens
- Iff
If and only if...
- Alexei Andreev - Proof of Löb's theorem
Proving that I am Santa Claus
- Jaime Sevilla Molina - Real number
A **real number** is any number that can be used to represent a physical quantity.
Intuitively, rea…
- Michael Cohen
- Fractional digits
When $b$ and $x$ are integers, $\log_b(x)$ has a few good interpretations. It's roughly the length o…
- Nate Soares - Life in logspace
The log lattice hints at the reason that engineers, scientists, and AI researchers find logarithms s…
- Nate Soares - Löb's theorem
Löb's theorem
- Jaime Sevilla Molina - Löb's theorem and computer programs
The close relationship between [ logic and computability] allows us to frame Löb's theorem in terms …
- Jaime Sevilla Molina - Proof of Löb's theorem
Proving that I am Santa Claus
- Jaime Sevilla Molina - The log lattice
Log as the change in the cost of communicating and other pages give physical interpretations of what…
- Nate Soares - Uncountability
Some infinities are bigger than others. Uncountable infinities are larger than countable infinities.
- Jason Gross - Uncountability: Intuitive Intro
Are all sizes of infinity the same? What does "the same" even mean here?
- Jason Gross
- Task-directed AGI
An advanced AI that's meant to pursue a series of limited-scope goals given it by the user. In Bostrom's terminology, a Genie.
- Eliezer Yudkowsky
wiki
- Researchers in value alignment theory
Who's working full-time in value alignment theory?
- Eliezer Yudkowsky - Waterfall diagrams and relative odds
A way to visualize Bayes' rule that yields an easier way to solve some problems
- Eliezer Yudkowsky
no-type
- What is a logarithm?
Logarithms are a group of functions that take a number as input and produce another number. There i…
- Nate Soares
wiki
- An introductory guide to modern logic
Logic, provability, Löb, Gödel and more!
- Jaime Sevilla Molina - Cyclic group
Cyclic groups form one of the most simple classes of groups.
- Patrick Stevens - Group
The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.
- Nate Soares - Group: Examples
Why would anyone have invented groups, anyway? What were the historically motivating examples, and what examples are important today?
- Qiaochu Yuan - Group: Exercises
Test your understanding of the definition of a group with these exercises.
- Qiaochu Yuan - Morphism
A morphism is the abstract representation of a relation between mathematical objects.
Usually, it i…
- Jaime Sevilla Molina - Orbit-stabiliser theorem
The Orbit-Stabiliser theorem tells us a lot about how a group acts on a given element.
- Patrick Stevens - P vs NP
Is creativity purely mechanical?
- Jaime Sevilla Molina - Poset: Exercises
Try these exercises to test your poset knowledge.
# Corporate Ladder
Imagine a company with five …
- Kevin Clancy - Proof by contradiction
Discover what 'reductio ad absurdum' means!
- Jaime Sevilla Molina - Quotient group
Given a group $G$ with operation $\bullet$ and a special kind of subgroup $N \leq G$ called the "no…
- Adele Lopez - Stabiliser (of a group action)
If a group acts on a set, it is useful to consider which elements of the group don't move a certain element of the set.
- Patrick Stevens - Stabiliser is a subgroup
Given a group acting on a set, each element of the set induces a subgroup of the group.
- Patrick Stevens - Uncountability: Intro (Math 1)
Not all infinities are created equal. The infinity of real numbers is infinitely larger than the infinity of counting numbers.
- Jason Gross
no-type
- A whirlwind tour
A rapid tour of Eric's thoughts on the accelerator project.
- Eric Bruylant - The missing step between Zero and Hero
Creating a space for high potential people grow and improve the world at scale.
- Eric Bruylant
- Natural number
The numbers we use to count: 0, 1, 2, 3, ...
- Jaime Sevilla Molina - Transitive relation
If a is related to b and b is related to c, then a is related to c.
- Dylan Hendrickson
- Advanced agent properties
How smart does a machine intelligence need to be, for its niceness to become an issue? "Advanced" is a broad term to cover cognitive abilities such that we'd need to start considering AI alignment.
- Eliezer Yudkowsky - Corrigibility
"I can't let you do that, Dave."
- Nate Soares - List: value-alignment subjects
Bullet point list of core VAT subjects.
- Eliezer Yudkowsky - Methodology of foreseeable difficulties
Building a nice AI is likely to be hard enough, and contain enough gotchas that won't show up in the AI's early days, that we need to foresee problems coming in advance.
- Eliezer Yudkowsky
- Log as generalized length
To estimate the log (base 10) of a number, count how many digits it has.
- Nate Soares - Solomonoff induction: Intro Dialogue (Math 2)
An introduction to Solomonoff induction for the unfamiliar reader who isn't bad at math
- Eliezer Yudkowsky
- High-speed intro to Bayes's rule
A high-speed introduction to Bayes's Rule on one page, for the impatient and mathematically adept.
- Eliezer Yudkowsky
- Bayes' rule: Probability form
The original formulation of Bayes' rule.
- Nate Soares
wiki
- AI alignment
The great civilizational problem of creating artificially intelligent computer systems such that running them is a good idea.
- Eliezer Yudkowsky - AI arms races
AI arms races are bad
- Eliezer Yudkowsky - Ackermann function
The slowest-growing fast-growing function.
- Alex Appel - Arbital
Arbital is the place for crowdsourced, intuitive math explanations.
- Alexei Andreev - Arbital Markdown
All about Arbital's extended Markdown syntax.
- Alexei Andreev - Arbital: Solving online explanations
An explanation of Arbital's mid-term goals
- Eliezer Yudkowsky - Author's guide to Arbital
How to write intuitive, flexible content on Arbital.
- Alexei Andreev - Bayes' rule
Bayes' rule is the core theorem of probability theory saying how to revise our beliefs when we make a new observation.
- Eliezer Yudkowsky - Bayes' rule examples
Interesting problems solvable by Bayes' rule
- Eliezer Yudkowsky - Bayes' rule: Functional form
Bayes' rule for to continuous variables.
- Eliezer Yudkowsky - Bayes' rule: Guide
The Arbital guide to Bayes' rule
- Eliezer Yudkowsky - Bayes' rule: Log-odds form
A simple transformation of Bayes' rule reveals tools for measuring degree of belief, and strength of evidence.
- Eliezer Yudkowsky - Bayes' rule: Odds form
The simplest and most easily understandable form of Bayes' rule uses relative odds.
- Eliezer Yudkowsky - Bayes' rule: Odds form (Intro, Math 1)
Introduction to the odds form of Bayes' rule
- Eliezer Yudkowsky - Bayes' rule: Proportional form
The fastest way to say something both convincing and true about belief-updating.
- Eliezer Yudkowsky - Bayes' rule: Vector form
For when you want to apply Bayes' rule to lots of evidence and lots of variables, all in one go. (This is more or less how spam filters work.)
- Eliezer Yudkowsky - Bayesian view of scientific virtues
Why is it that science relies on bold, precise, and falsifiable predictions? Because of Bayes' rule, of course.
- Eliezer Yudkowsky - Behaviorist genie
An advanced agent that's forbidden to model minds in too much detail.
- Eliezer Yudkowsky - Belief revision as probability elimination
Update your beliefs by throwing away large chunks of probability mass.
- Eliezer Yudkowsky - Category theory
How mathematical objects are related to others in the same category.
- Mark Chimes - Central examples
List of central examples in Value Alignment Theory domain.
- Eliezer Yudkowsky - Conditional probability
The notation for writing "The probability that someone has green eyes, if we know that they have red hair."
- Eliezer Yudkowsky - Cosmic endowment
The 'cosmic endowment' consists of all the stars that could be reached from probes originating on Earth; the sum of all matter and energy potentially available to be transformed into life and fun.
- Eliezer Yudkowsky - Decimal notation
The winning architecture for numerals
- Michael Cohen - Diamond maximizer
How would you build an agent that made as much diamond material as possible, given vast computing power but an otherwise rich and complicated environment?
- Eliezer Yudkowsky - Diseasitis
20% of patients have Diseasitis. 90% of sick patients and 30% of healthy patients turn a tongue depressor black. You turn a tongue depressor black. What's the chance you have Diseasitis?
- Eliezer Yudkowsky - Edge instantiation
When you ask the AI to make people happy, and it tiles the universe with the smallest objects that can be happy.
- Eliezer Yudkowsky - Eliezer's vision for Arbital
Why are we building this? What's the goal?
- Eric Bruylant - Exponential
Any function that constantly gets larger as a proportion of itself.
- Joe Zeng - Extraordinary claims require extraordinary evidence
The people who adamantly claim they were abducted by aliens do provide some evidence for aliens. They just don't provide quantitatively enough evidence.
- Eliezer Yudkowsky - Finishing your Bayesian path on Arbital
The page that comes at the end of reading the Arbital Guide to Bayes' rule
- Eliezer Yudkowsky - Inductive prior
Some states of pre-observation belief can learn quickly; others never learn anything. An "inductive prior" is of the former type.
- Eliezer Yudkowsky - Integer
An **integer** is a Number that can be represented as either a Natural number or its [-additive\_inv…
- Michael Cohen - Introduction to Bayes' rule: Odds form
Bayes' rule is simple, if you think in terms of relative odds.
- Eliezer Yudkowsky - Introductory Bayesian problems
Bayesian problems to try to solve yourself, before beginning to learn about Bayes' rule.
- Eliezer Yudkowsky - Likelihood functions, p-values, and the replication crisis
What's the whole Bayesian-vs.-frequentist debate about?
- Eliezer Yudkowsky - List: value-alignment subjects
Bullet point list of core VAT subjects.
- Eliezer Yudkowsky - Mathematics
Mathematics is the study of numbers and other ideal objects that can be described by axioms.
- Eliezer Yudkowsky - Methodology of unbounded analysis
What we do and don't understand how to do, using unlimited computing power, is a critical distinction and important frontier.
- Eliezer Yudkowsky - Mind projection fallacy
Uncertainty is in the mind, not in the environment; a blank map does not correspond to a blank territory. In general, the territory may have a different ontology from the map.
- Eliezer Yudkowsky - Mindcrime
Might a machine intelligence contain vast numbers of unhappy conscious subprocesses?
- Eliezer Yudkowsky - Natural number
The numbers we use to count: 0, 1, 2, 3, ...
- Jaime Sevilla Molina - Odds
Odds express a relative probability.
- Eliezer Yudkowsky - Ontology identification problem
How do we link an agent's utility function to its model of the world, when we don't know what that model will look like?
- Eliezer Yudkowsky - P vs NP: Arguments against P=NP
Why we believe P and NP are different
- Jaime Sevilla Molina - Partially ordered set
A set endowed with a relation that is reflexive, transitive, and antisymmetric.
- Kevin Clancy - Path: Multiple angles on Bayes's Rule
A learning-path placeholder page for learning multiple angles on Bayes's Rule.
- Eliezer Yudkowsky - Pivotal event
Which types of AIs, if they work, can do things that drastically change the nature of the further game?
- Eliezer Yudkowsky - Posterior probability
What we believe, after seeing the evidence and doing a Bayesian update.
- Eliezer Yudkowsky - Prior probability
What we believed before seeing the evidence.
- Eliezer Yudkowsky - Probability notation for Bayes' rule
The probability notation used in Bayesian reasoning
- Eliezer Yudkowsky - Probability notation for Bayes' rule: Intro (Math 1)
How to read, and identify, the probabilities in Bayesian problems.
- Eliezer Yudkowsky - Product (Category Theory)
How a product is characterized rather than how it's constructed
- Mark Chimes - Proof of Bayes' rule
Proofs of Bayes' rule, with graphics
- Eliezer Yudkowsky - Proof of Bayes' rule: Intro
Proof of Bayes' rule, assuming you know the rule itself, and the notations for the quantities involved.
- Eliezer Yudkowsky - Quine
A computer program that prints (or does other computations to) its own source code, using indirect self-reference.
- Patrick LaVictoire - Rational number
The rational numbers are "fractions".
- Patrick Stevens - Real number
A **real number** is any number that can be used to represent a physical quantity.
Intuitively, rea…
- Michael Cohen - Relation
A **relation** is a set of [tuple\_mathematics tuples], all of which have the same [tuple\_arity ar…
- Kevin Clancy - Relative likelihood
How relatively likely an observation is, given two or more hypotheses, determines the strength and direction of evidence.
- Eliezer Yudkowsky - Solomonoff induction: Intro Dialogue (Math 2)
An introduction to Solomonoff induction for the unfamiliar reader who isn't bad at math
- Eliezer Yudkowsky - Some men just want to watch the world learn
In which Arbital gets paths.
- Alexei Andreev - Standard agent properties
What's a Standard Agent, and what can it do?
- Eliezer Yudkowsky - Strength of Bayesian evidence
From a Bayesian standpoint, the strength of evidence can be identified with its likelihood ratio.
- Eliezer Yudkowsky - Strictly confused
A hypothesis is strictly confused by the raw data, if the hypothesis did much worse in predicting it than the hypothesis itself expected.
- Eliezer Yudkowsky - Subjective probability
Probability is in the mind, not in the environment. If you don't know whether a coin came up heads or tails, that's a fact about you, not a fact about the coin.
- Eliezer Yudkowsky - Task-directed AGI
An advanced AI that's meant to pursue a series of limited-scope goals given it by the user. In Bostrom's terminology, a Genie.
- Eliezer Yudkowsky - The rocket alignment problem
If people talked about the problem of space travel the way they talked about AI...
- Eliezer Yudkowsky - The square root of 2 is irrational
The number whose square is 2 can't be written is a quotient of natural numbers
- Dylan Hendrickson - Vinge's Law
You can't predict exactly what someone smarter than you would do, because if you could, you'd be that smart yourself.
- Eliezer Yudkowsky - Vingean uncertainty
You can't predict the exact actions of an agent smarter than you - so is there anything you _can_ say about them?
- Eliezer Yudkowsky - Waterfall diagram
Visualizing Bayes' rule as the mixing of probability streams.
- Eliezer Yudkowsky - Waterfall diagrams and relative odds
A way to visualize Bayes' rule that yields an easier way to solve some problems
- Eliezer Yudkowsky
no-type
- Bayes' rule: Vector form
For when you want to apply Bayes' rule to lots of evidence and lots of variables, all in one go. (This is more or less how spam filters work.)
- Eliezer Yudkowsky
- Bayes' rule: Vector form
For when you want to apply Bayes' rule to lots of evidence and lots of variables, all in one go. (This is more or less how spam filters work.)
- Eliezer Yudkowsky
- Arithmetical hierarchy: If you don't read logic
The arithmetical hierarchy is a way of stratifying statements by how many "for every number" and "th…
- Eliezer Yudkowsky
- Effective altruism outreach
Effective Altruism Outreach (EAO) is part of the Centre for Effective Altruism and is an organizatio…
- Alexei Andreev
- Approval-based agents
An alternative to goal-directed behavior
- Paul Christiano - Consequentialist cognition
The cognitive ability to foresee the consequences of actions, prefer some outcomes to others, and output actions leading to the preferred outcomes.
- Eliezer Yudkowsky - Omnipotence test for AI safety
Would your AI produce disastrous outcomes if it suddenly gained omnipotence and omniscience? If so, why did you program something that *wants* to hurt you and is held back only by lacking the power?
- Eliezer Yudkowsky - Scalable AI Control
By AI control, I mean the problem of getting AI systems to do what we want them to do, to the best o…
- Paul Christiano
- Bayesian view of scientific virtues
Why is it that science relies on bold, precise, and falsifiable predictions? Because of Bayes' rule, of course.
- Eliezer Yudkowsky
- Laplace's Rule of Succession
Suppose you flip a coin with an unknown bias 30 times, and see 4 heads and 26 tails. The Rule of Succession says the next flip has a 5/32 chance of showing heads.
- Eliezer Yudkowsky
- Arithmetical hierarchy: If you don't read logic
The arithmetical hierarchy is a way of stratifying statements by how many "for every number" and "th…
- Eliezer Yudkowsky - Bayes' rule: Vector form
For when you want to apply Bayes' rule to lots of evidence and lots of variables, all in one go. (This is more or less how spam filters work.)
- Eliezer Yudkowsky - Bayesian view of scientific virtues
Why is it that science relies on bold, precise, and falsifiable predictions? Because of Bayes' rule, of course.
- Eliezer Yudkowsky - Löb's theorem
Löb's theorem
- Jaime Sevilla Molina - Löb's theorem and computer programs
The close relationship between [ logic and computability] allows us to frame Löb's theorem in terms …
- Jaime Sevilla Molina - Mathematical induction
Proving a statement about all positive integers by knocking them down like dominoes.
- Douglas Weathers - Mild optimization
An AGI which, if you ask it to paint one car pink, just paints one car pink and doesn't tile the universe with pink-painted cars, because it's not trying *that* hard to max out its car-painting score.
- Eliezer Yudkowsky - Modal combat
Modal combat
- Jaime Sevilla Molina - Modal logic
The logic of boxes and bots.
- Jaime Sevilla Molina
- Arbital features
Overview of all Arbital features.
- Alexei Andreev - Cardinality
The "size" of a set, or the "number of elements" that it has.
- Joe Zeng - Category (mathematics)
A description of how a collection of mathematical objects are related to one another.
- Mark Chimes - Complex number
A complex number is a number of the form $z = a + b\textrm{i}$, where $\textrm{i}$ is the imaginary …
- Eliana Ruby - Group
The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.
- Nate Soares - Group action
"Groups, as men, will be known by their actions."
- Qiaochu Yuan - Group orbit
When we have a group acting on a set, we are often interested in how the group acts on a particular …
- Adele Lopez - Group theory
What kinds of symmetry can an object have?
- Nate Soares - Group: Examples
Why would anyone have invented groups, anyway? What were the historically motivating examples, and what examples are important today?
- Qiaochu Yuan - Group: Exercises
Test your understanding of the definition of a group with these exercises.
- Qiaochu Yuan - Integer
An **integer** is a Number that can be represented as either a Natural number or its [-additive\_inv…
- Michael Cohen - Isomorphism
A morphism between two objects which describes how they are "essentially equivalent" for the purposes of the theory under consideration.
- Mark Chimes - Least common multiple
The **least common multiple (LCM)** of two positive natural numbers a, b is the smallest natural …
- Johannes Schmitt - Likelihood functions, p-values, and the replication crisis
What's the whole Bayesian-vs.-frequentist debate about?
- Eliezer Yudkowsky - Logarithm tutorial overview
The logarithm tutorial covers the following six subjects:
1. What are logarithms?
2. Logarithms as…
- Nate Soares - Math 2 example statements
If you can read these formulas, you're in Math 2!
- Joe Zeng - Math playpen
Playpen page for Math domain
- Alexei Andreev - Monoid
A monoid $M$ is a pair $(X, \diamond)$ where $X$ is a [set\_theory\_set set] and $\diamond$ is an [a…
- Nate Soares - Morphism
A morphism is the abstract representation of a relation between mathematical objects.
Usually, it i…
- Jaime Sevilla Molina - Natural number
The numbers we use to count: 0, 1, 2, 3, ...
- Jaime Sevilla Molina - Operator
An operation $f$ on a set $S$ is a function that takes some values from $S$ and produces a new value…
- Nate Soares - Orbit-Stabiliser theorem: External Resources
External resources on the Orbit-Stabiliser theorem.
- Mark Chimes - Order of a group
The order $|G|$ of a group $G$ is the size of its underlying set. For example, if $G=(X,\bullet)$ an…
- Nate Soares - Order relation
A way of determining which elements of a set come "before" or "after" other elements.
- Joe Zeng - Project outline: Intro to the Universal Property
Outline detailing all the work required for a proposed Arbital Project
- Eric Rogstad - Proof that there are infinitely many primes
Suppose there were finitely many primes. Then consider the product of all the primes plus 1...
- Joe Zeng - Real number
A **real number** is any number that can be used to represent a physical quantity.
Intuitively, rea…
- Michael Cohen - Real number (as Dedekind cut)
A way to construct the real numbers that follows the intuition of filling in the gaps.
- Joe Zeng - Ring
A ring is a kind of Algebraic structure which we obtain by considering groups as being "things with…
- Nate Soares - Set
An unordered collection of distinct objects.
- Nate Soares - Strong Church Turing thesis
A strengthening of the Church Turing thesis
- Jaime Sevilla Molina - There is only one logarithm
All logarithm functions are the same, up to a multiplicative constant.
- Nate Soares - Turing machine
A Turing Machine is a simple mathematical model of computation that is powerful enough to describe any computation a computer can do.
- Eric Leese - Uncountability: Intro (Math 1)
Not all infinities are created equal. The infinity of real numbers is infinitely larger than the infinity of counting numbers.
- Jason Gross - Underlying set
What do a Group, a Partially ordered set, and a [ topological space] have in common? Each is a Set …
- Nate Soares - Union
The union of two sets is the set of elements which are in one or the other, or both
- M Yass - Vector space
A vector space is a field $F$ paired with a Group $V$ and a function $\cdot : F \times V \to V$ (cal…
- Nate Soares
- Contributing to Arbital
Want to help Arbital become awesome?
- Eric Bruylant
- Neutral genie metaphor
Definition. A neutral-genie metaphor is an attempt to illustrate a possible formal problem via an in…
- Alexei Andreev - Researchers in value alignment theory
Who's working full-time in value alignment theory?
- Eliezer Yudkowsky
- Bayes' rule: Log-odds form
A simple transformation of Bayes' rule reveals tools for measuring degree of belief, and strength of evidence.
- Eliezer Yudkowsky
- Abelian group
A group where the operation commutes. Named after Niels Henrik Abel.
- Nate Soares - Eigenvalues and eigenvectors
We applied this linear transformation to one of its eigenvectors; you won't believe what happened next!
- Zack M. Davis - Group
The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.
- Nate Soares - Group theory
What kinds of symmetry can an object have?
- Nate Soares - Order of a group
The order $|G|$ of a group $G$ is the size of its underlying set. For example, if $G=(X,\bullet)$ an…
- Nate Soares
- A reply to Francois Chollet on intelligence explosion
A quick run-through of what I'd consider the standard replies to the arguments in Keras inventor Francois Chollet's essay "The impossibility of intelligence explosion".
- Eliezer Yudkowsky - Coherent extrapolated volition (alignment target)
A proposed direction for an extremely well-aligned autonomous superintelligence - do what humans would want, if we knew what the AI knew, thought that fast, and understood ourselves.
- Eliezer Yudkowsky - Cosmic endowment
The 'cosmic endowment' consists of all the stars that could be reached from probes originating on Earth; the sum of all matter and energy potentially available to be transformed into life and fun.
- Eliezer Yudkowsky - The rocket alignment problem
If people talked about the problem of space travel the way they talked about AI...
- Eliezer Yudkowsky
- AI safety mindset
Asking how AI designs could go wrong, instead of imagining them going right.
- Eliezer Yudkowsky - Goodhart's Curse
The Optimizer's Curse meets Goodhart's Law. For example, if our values are V, and an AI's utility function U is a proxy for V, optimizing for high U seeks out 'errors'--that is, high values of U - V.
- Eliezer Yudkowsky - Problem of fully updated deference
Why moral uncertainty doesn't stop an AI from defending its off-switch.
- Eliezer Yudkowsky - Relevant limited AI
Can we have a limited AI, that's nonetheless relevant?
- Eliezer Yudkowsky - Understandability principle
The more you understand what the heck is going on inside your AI, the safer you are.
- Eliezer Yudkowsky - Why waiting to donate harms charities
A blog post explaining the potential reasons why someone would choose to wait to donate and how that leads to suboptimal outcomes for the charity.
- Alexei Andreev - Zermelo-Fraenkel provability oracle
We might be able to build a system that can safely inform us that a theorem has a proof in set theory, but we can't see how to use that capability to save the world.
- Eliezer Yudkowsky
- Solomonoff induction: Intro Dialogue (Math 2)
An introduction to Solomonoff induction for the unfamiliar reader who isn't bad at math
- Eliezer Yudkowsky
- Waterfall diagrams and relative odds
A way to visualize Bayes' rule that yields an easier way to solve some problems
- Eliezer Yudkowsky
wiki
- Arbital playpen
Want to test a feature? Feel free to edit this page! asdfasfdasfda
- Eliezer Yudkowsky
no-type
- Introduction to Bayes' rule: Odds form
Bayes' rule is simple, if you think in terms of relative odds.
- Eliezer Yudkowsky
- Arbital Markdown
All about Arbital's extended Markdown syntax.
- Alexei Andreev - Creating a /learn/ link
What options are available when creating a /learn/ link?
- Alexei Andreev
- Eigenvalues and eigenvectors
We applied this linear transformation to one of its eigenvectors; you won't believe what happened next!
- Zack M. Davis
- Bayes' rule: Vector form
For when you want to apply Bayes' rule to lots of evidence and lots of variables, all in one go. (This is more or less how spam filters work.)
- Eliezer Yudkowsky
- Frequency diagrams: A first look at Bayes
The most straightforward visualization of Bayes' rule.
- Nate Soares
- Waterfall diagram
Visualizing Bayes' rule as the mixing of probability streams.
- Eliezer Yudkowsky
- Bayes' rule
Bayes' rule is the core theorem of probability theory saying how to revise our beliefs when we make a new observation.
- Eliezer Yudkowsky
- Introduction to Logical Decision Theory for Computer Scientists
'Logical decision theory' from a math/programming standpoint, including how two agents with mutual knowledge of each other's code can cooperate on the Prisoner's Dilemma.
- Eliezer Yudkowsky
- The n-th root of m is either an integer or irrational
In other words, no power of a rational number that is not an integer is ever an integer.
- Joe Zeng
- Welcome to Arbital
Front page explaining what Arbital is all about.
- Alexei Andreev
wiki
- Arbital playpen
Want to test a feature? Feel free to edit this page! asdfasfdasfda
- Eliezer Yudkowsky - Eliezer office hours 2016-04-19
### Steph's notes from the actual meeting:
- "good question" -> people upvoting your question is a g…
- Eric Rogstad - Eliezer office hours 2016-05-07
##Questions for Eliezer##
###Strategy###
Q: What does a discussion of which EA cause to donate to …
- Eric Rogstad - Group theory
What kinds of symmetry can an object have?
- Nate Soares - Odds
Odds express a relative probability.
- Eliezer Yudkowsky - Our community should relocate to a country other than the US
Claims with specific country proposals:
* [claim([6v8])]
* [claim([6v9])]
Relevant claims:
* [cla…
- Alexei Andreev - Project proposal: Intro to numbers
Should Arbital's first "project" be a guide to numbers?
- Eric Rogstad - Sample space
The set of possible things that could happen in a part of the world that you are uncertain about.
- Tsvi BT
group
no-type
- AI safety mindset
Asking how AI designs could go wrong, instead of imagining them going right.
- Eliezer Yudkowsky
- Bayes' rule: Log-odds form
A simple transformation of Bayes' rule reveals tools for measuring degree of belief, and strength of evidence.
- Eliezer Yudkowsky
- Axiom of Choice: History and Controversy
Really? The *most* controversial axiom of the 20th century? Yes.
- Mark Chimes - Logical Induction (incomplete)
The theoretically ideal algorithm for bounded reasoning with lots of computational resources
- Alex Appel
- Possible math pages
A list of things which we may want math pages on
- Eric Bruylant
- Bayes' rule: Guide
The Arbital guide to Bayes' rule
- Eliezer Yudkowsky
- Arbital
Arbital is the place for crowdsourced, intuitive math explanations.
- Alexei Andreev
- Introduction to Bayes' rule: Odds form
Bayes' rule is simple, if you think in terms of relative odds.
- Eliezer Yudkowsky
- Big-O Notation
This notation describes asymptotic behavior of functions.
# O(x)
A function f is O(g(x)) if, for la…
- Aeneas Mackenzie - Finite set
A finite set is one which is not infinite. Some of these are the least complicated sets.
- Patrick Stevens - Report likelihoods not p-values: FAQ
This page answers frequently asked questions about the Report likelihoods, not p-values proposal for…
- Nate Soares - Set
An unordered collection of distinct objects.
- Nate Soares - Solomonoff induction: Intro Dialogue (Math 2)
An introduction to Solomonoff induction for the unfamiliar reader who isn't bad at math
- Eliezer Yudkowsky
- Arbital: Google Maps for knowledge
Take your understanding from where it is to where it wants to be.
- Alexei Andreev - Arbital: learning from Wikipedia
How is Arbital different from Wikipedia?
- Alexei Andreev - Corrigibility
"I can't let you do that, Dave."
- Nate Soares - Group theory
What kinds of symmetry can an object have?
- Nate Soares - Joint probability
The notation for writing the chance that both X and Y are true.
- Eliezer Yudkowsky
- Bayes' rule: Proportional form
The fastest way to say something both convincing and true about belief-updating.
- Eliezer Yudkowsky
- Bayes' rule: Log-odds form
A simple transformation of Bayes' rule reveals tools for measuring degree of belief, and strength of evidence.
- Eliezer Yudkowsky
- Bayes' rule: Proportional form
The fastest way to say something both convincing and true about belief-updating.
- Eliezer Yudkowsky
- Introductory Bayesian problems
Bayesian problems to try to solve yourself, before beginning to learn about Bayes' rule.
- Eliezer Yudkowsky
- Axiom of Choice
The most controversial axiom of the 20th century.
- Mark Chimes
- Report likelihoods not p-values: FAQ
This page answers frequently asked questions about the Report likelihoods, not p-values proposal for…
- Nate Soares
- Bayes' rule: Vector form
For when you want to apply Bayes' rule to lots of evidence and lots of variables, all in one go. (This is more or less how spam filters work.)
- Eliezer Yudkowsky
- Information theory
The study (and quantificaiton) of information, and its communication and storage.
- Nate Soares
- Group
The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.
- Nate Soares
- Extraordinary claims require extraordinary evidence
The people who adamantly claim they were abducted by aliens do provide some evidence for aliens. They just don't provide quantitatively enough evidence.
- Eliezer Yudkowsky
- Logic
Logic is the study of correct arguments.
- Erik Istre
- Introduction to Bayes' rule: Odds form
Bayes' rule is simple, if you think in terms of relative odds.
- Eliezer Yudkowsky