Theorem of Cantor

The theorem of Cantor is a mathematical theorem, in the field of the Set theory, which owes its name with the Mathématicien Georg Cantor.

Cantor shows that, for any unit E , the cardinal of E is always strictly lower than the cardinal of $\backslash \; mathfrak\; P\; (E)$ together of the parts of E .

When E is a finished unit, the result is obvious because the cardinal of E is the number of elements in E and, if E contains N elements, it is shown that the whole of the parts of E contains $2^n$ elements. It is then easy to check that, for entire N , .

When E is an infinite unit, it is necessary to set out again on the comparison of the cardinals.

$\backslash \; mathrm\; \{card\}\; (A)\; \backslash \; Leq\; \backslash \; mathrm\; \{card\}\; (B)$ if and only if, there exists a injection of towards B.

has Cantor shows that by a Reasoning by the absurdity: it supposes that $\backslash \; mathrm\; \{card\}\; (E)\; \backslash \; geq\; \backslash \; mathrm\; \{card\}\; (\backslash \; mathfrak\; P\; (E))$, therefore that there exists an injection of $\backslash \; mathfrak\; P\; (E)$ towards E and arrives at a contradiction.

One calls F this injection. One then builds a subset B of E in the following way:

is X an element of E ,

* if X then does not have an antecedent by F X is not in B

* if X has an antecedent by F , it is single because F is injective. This antecedent $A\_x$ is noted. If X belongs to $A\_x$ then X is not in B, if X does not belong to $A\_x$ then X is in B .

B is part of E , and thus has an image by F , which one names there. The question which installation is: “ there is it or not an element of B ? ”. has there as an antecedent B .

if is there in B then, by construction of B , does not belong there to its antecedent thus does not belong there to… B

if is not there in B , always according to the construction of B , must there belong to its antecedent, therefore belongs there to… B

the two assumptions thus lead to a contradiction it cannot exist of injection of $\backslash \; mathfrak\; P\; (E)$ towards E

This type of reasoning, which one calls diagonal Argument, was used by Russell (and Zermelo) for the paradox of the whole of the units which are not belonged.

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