Cantor's theorem: Difference between revisions

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{{math theorem|name=Theorem (Cantor)|math_statement=Let <math>f</math> be a map from set <math>A</math> to its power set <math>\mathcal{P}(A)</math>. Then <math>f : A \to \mathcal{P}(A)</math> is not [[surjective]]. As a consequence, <math>\operatorname{card}(A) < \operatorname{card}(\mathcal{P}(A))</math> holds for any set <math>A</math>.}}

{{math proof| The set <math> B=\{x \in A \mid x \notin f(x)\}</math> exists according tovia the [[axiom schema of specification]], and <math> B \in \mathcal{P}(A) </math> because <math> B \subseteq A </math>. Aiming<br> to derive a contradiction, supposeAssume <math> f </math> is surjective. <br> Then there exists an elementa <math>\xi \in A</math> such that <math>f(\xi)=B</math>. Moreover<br> From &nbsp;''for all'' <math>x</math> ''in'' <math>A \ [ x \in B \iff x \notin f(x) ]</math> , we deduce &nbsp;<math>\xi \in B \iff \xi \notin f(\xi) </math> &nbsp;via [[universal instantiation]]. <br> The previous deduction yields a contradiction of the form <math>\varphi \Leftrightarrow \lnot \varphi</math>, since <math>f(\xi)=B</math>. <br> Therefore, <math>f</math> is not surjective, via [[reductio ad absurdum]]. <br> We know [[injective function|injective maps]] from <math>A</math> to <math>\mathcal{P}(A)</math> exist. For example, a function <math>g : A \to \mathcal{P}(A)</math> such that <math>g(x) = \{x\}</math>. <br> Consequently, <math>\operatorname{card}(A) < \operatorname{card}(\mathcal{P}(A))</math>. ∎}}

<math display=block>\forall x\in A [x\in B \iff x\notin f(x)].</math>

From this we deduce

<math display=block>\xi \in B \iff \xi \notin f(\xi), </math>

due to [[universal instantiation]]. Recall that <math>\xi</math> is the element of <math>B</math> such that <math>\xi =B</math>, hence the previous deduction yields a contradiction–this can be shown in a straightforward way by method of truth tables. Therefore, using [[reductio ad absurdum]], <math>f</math> is not surjective. We know [[injective function|injective maps]] from <math>A</math> to <math>\mathcal{P}(A)</math> exist. For example, a map <math>g : A \to \mathcal{P}(A)</math> defined by <math>g(x) = \{x\}</math> is surjective. Consequently, <math>\operatorname{card}(A) < \operatorname{card}(\mathcal{P}(A))</math>. ∎}}

By definition of cardinality, we have <math>\operatorname{card}(X) < \operatorname{card}(Y)</math> for any two sets <math>X</math> and <math>Y</math> if and only if there is an [[injective function]] but no [[Bijective Function|bijective function]] from <math>X</math> {{nowrap|to <math>Y</math>.}} It suffices to show that there is no surjection from <math>X</math> {{nowrap|to <math>Y</math>}}. This is the heart of Cantor's theorem: there is no surjective function from any set <math>A</math> to its power set. To establish this, it is enough to show that no function <math>f</math> (that maps elements in <math>A</math> to subsets of <math>A</math>) can reach every possible subset, i.e., we just need to demonstrate the existence of a subset of <math>A</math> that is not equal to <math>f(x)</math> for any <math>x \in A</math>. Recalling that each <math>f(x)</math> is a subset of <math>A</math>, such a subset is given by the following construction, sometimes called the [[Cantor's diagonal argument|Cantor diagonal set]] of <math>f</math>:<ref name="Dasgupta2013">{{cite book|author=Abhijit Dasgupta|title=Set Theory: With an Introduction to Real Point Sets|year=2013|publisher=[[Springer Science & Business Media]]|isbn=978-1-4614-8854-5|pages=362–363}}</ref><ref name="Paulson1992">{{cite book|author=Lawrence Paulson|title=Set Theory as a Computational Logic |url=https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-271.pdf|year=1992|publisher=University of Cambridge Computer Laboratory|page=14}}</ref>