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text: '[summary: The empty set, $\\emptyset$, is the set with no elements. For every object $x$, $x$ is not in $\\emptyset$.]\n\nThe empty set, $\\emptyset$, is the set with no elements.\nWhy, *a priori*, should this set exist at all?\nWell, if we think of sets as "containers of elements", the idea of an empty container is intuitive: just imagine a box with nothing in it.\n\n# Definitions\n\n## Definition in [ZF]\n\nIn the set theory [ZF], there are exactly two axioms which assert the existence of a set *ex nihilo*; all of the rest of set theory builds sets from those given sets, or else postulates the existence of more sets.\n\nThere is the [-axiom_of_infinity], which asserts the existence of an infinite set, and there is the [-empty_set_axiom], which asserts that an empty set exists.\n\nIn fact, we can deduce the existence of an empty set even without using the empty set axiom, as long as we are allowed to use the [-axiom_of_comprehension] to select a certain specially-chosen subset of an infinite set.\nMore formally: let $X$ be an infinite set (as guaranteed by the axiom of infinity).\nThen select the subset of $X$ consisting of all those members $x$ of $X$ which have the property that $x$ contains $X$ and also does not contain $X$.\n\nThere are no such members, so we must have just constructed an empty set.\n\n## Definition by a universal property\n\nThe empty set [5zr has a definition] in terms of a [-600]: the empty set is the unique set $X$ such that for every set $A$, there is exactly one map from $X$ to $A$.\nMore succinctly, it is the [-initial_object] in the [-category_of_sets] (or in the [-614]).\n\n# Uniqueness of the empty set\n\nThe [-axiom_of_extensionality] states that two sets are the same if and only if they have exactly the same elements.\nIf we had two empty sets $A$ and $B$, then certainly anything in $A$ is also in $B$ ([vacuous_truth vacuously]), and anything in $B$ is also in $A$, so they have the same elements.\nTherefore $A = B$, and we have shown the uniqueness of the empty set.\n\n# A common misconception: $\\emptyset$ vs $\\{ \\emptyset \\}$\n\nIt is very common for people to start out by getting confused between $\\emptyset$ and $\\{\\emptyset\\}$.\nThe first contains no elements; the second is a set containing exactly one element (namely $\\emptyset$).\nThe sets don't [499 biject], because they are finite and have different numbers of elements.\n\n# [vacuous_truth Vacuous truth]: a whistlestop tour\n\nThe idea of vacuous truth can be stated as follows:\n\n> For *any* property $P$, everything in $\\emptyset$ has the property $P$.\n\nIt's a bit unintuitive at first sight: it's true that everything in $\\emptyset$ is the Pope, for instance.\nWhy should this be the case?\n\nIn order for there to be a counterexample to the statement that "everything in $\\emptyset$ is the Pope", we would need to find an element of $\\emptyset$ which was not the Pope. %%note: Strictly, we'd only need to show that such an element must exist, without necessarily finding it.%%\nBut there aren't *any* elements of $\\emptyset$, let alone elements which fail to be the Pope.\nSo there can't be any counterexample to the statement that "everything in $\\emptyset$ is the Pope", so the statement is true.',
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