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text: '[summary: [3jz Collections of things] which are the same [4w5 size] as or smaller than the collection of all [-45h natural numbers] are called *countable*, while larger collections (like the set of all [-4bc real numbers]) are called *uncountable*.\n\nAll uncountable collections (and some countable collections) are [infinity infinite]. There is a meaningful and [5ss well-defined] way to compare the sizes of different infinite collections of things, and some infinite collections are larger than others.]\n\n[3jz Collections of things] which are the same [4w5 size] as or smaller than the collection of all [-45h natural numbers] are called *countable*, while larger collections (like the set of all [-4bc real numbers]) are called *uncountable*.\n\nAll uncountable collections (and some countable collections) are [infinity infinite]. There is a meaningful and [5ss well-defined] way to compare the sizes of different infinite collections of things %%note: Specifically, mathematical systems which use the [69b], see the [4zp technical] page for details.%%. To demonstrate this, we'll use a 2d grid.\n\n[toc:]\n\n## Real and Rational numbers\n\n[4bc Real numbers] are numbers with a [4sl decimal expansion], for example 1, 2, 3.5, $\\pi$ = 3.14159265... Every real number has an infinite decimal expansion, for example 1 = 1.0000000000..., 2 = 2.0000000000..., 3.5 = 3.5000000000... Recall that the rational numbers are [fraction fractions] of [48l integers], for example $1 = \\frac11$, $\\frac32$, $\\frac{100}{101}$, $\\frac{22}{7}$. The positive integers are the integers greater than zero (i.e. 1, 2, 3, 4, ..). \n\nThere is a [-theorem] in math that states that the rational numbers are *countable* %%note: You can see the theorem [511 here].%%, that is, that the set of rational numbers is the same size as the set of positive integers, and another theorem which states that the real numbers are *uncountable*, that is, that the set of real numbers is strictly bigger. By "same size" and "strictly bigger", we mean that it is possible to match every rational number with some positive integer in a way so that there are no rational numbers, nor positive integers, left unmatched, but that any matching you make between real numbers and positive integers leaves some real numbers not matched with anything. \n\n## Rational grid\n\nIf you imagine laying the rational numbers out on a two-dimensional grid, so that the number $p / q$ falls at $(p, q)$, then we may match the positive integers with the rational numbers by walking in a spiral pattern out from zero, skipping over numbers that we have already counted (or that are undefined, such as zero divided by any number). The beginning of this sequence is $\\frac01$, $\\frac11$, $\\frac12$, $\\frac{-1}{2}$, $\\frac{-1}{1}$, ... Graphically, this is:\n\n\n\nThis shows that the rational numbers are countable.\n\n## Reals are uncountable\n\nThe real numbers, however, cannot be matched with the positive integers. I show this by [46z contradiction]. %%note:That is to say, I show that if there is such a matching, then we can conclude nonsensical statements (and if making a new assumption allows us to conclude nonsense, then the assumption itself must be nonsense.%%\n\nSuppose we have such a matching. We can construct a new real number that differs in its $n^\\text{th}$ decimal digit from the real number matched with $n$.\n\nFor example, if we were given a matching that matched 1 with 1.8, 2 with 1.26, 3 with 5.758, 4 with 1, and 5 with $\\pi$, then our new number could be 0.11111, which differs from 1.8 in the first decimal place (the 0.1 place), 1.26 in the second decimal place (the 0.01 place), and so on. It is clear that this number cannot be matched with any number under the matching we are given, because, if it were matched with $n$, then it would differ from itself in the $n^\\text{th}$ decimal digit, which is nonsense. Thus, there is no matching between the real numbers and the positive integers.\n\n## See also\n\nIf you enjoyed this explanation, consider exploring some of [3d Arbital's] other [6gg featured content]!\n\nArbital is made by people like you, if you think you can explain a mathematical concept then consider [-4d6]!',
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