Paradox of Cantor

The paradox of Cantor , or paradox of the largest cardinal , is a Paradoxe of the Set theory discovered by Georg Cantor in the years 1890 (it is mentioned in a letter with David Hilbert gone back to 1897). The paradox shows that the existence of larger cardinal leads to a contradiction. In a too naive set theory, which considers that any property defines a unit, this paradox is indeed a discrepancy, a contradiction deduced from the theory, since one could consider that the cardinal of the class of all the cardinals is then the largest cardinal. But it is not for Cantor. For him, that shows that the largest cardinal, if it can in a certain way of being defined, is not a Ensemble: in modern terms, the class of the cardinals is not a unit.

Cantor gives two way of deducing the paradox. For both it uses that any unit has a cardinal and thus, implicitly, the Axiome of the choice.

It shows that the class of the cardinals is equipotent with the class of the ordinal ones, and thus brings back its paradox to the Paradoxe of Burali-Forti, it states for that a form of the Schéma of axioms of replacement.
It uses its theorem on the cardinality of the Ensemble of the parts: if the largest cardinal is a unit, it thus has a whole of the parts, which then has a cardinal strictly higher than this larger cardinal.

One can eliminate any call with the concept of cardinal, and thus to the axiom of the choice in the second reasoning. That is to say V the class of all the units (of which the cardinal would be naturally the largest cardinal). If V is a unit, its whole of the parts P ( V ) also. Thus P ( V ) ⊂ V , the identity defines an injection of P ( V ) in V and contradicts the theorem of Cantor. In this form, it is very close to the Paradoxe of Russell. Indeed, by adapting the demonstration of the Theorem of Cantor to this particular case, one builds reciprocal on the left a F of the identity of P ( V ) in V , and one considers the unit { X ∈ V | X ∉ F ( X )}, whose intersection with P ( V ) is { X ∈ P ( V ) | X ∉ X }. Bertrand Russell declared besides that it had arrived at his paradox by analyzing the proof of the theorem of Cantor. The paradox of Russell to the advantage of not calling upon the whole of the parts of a unit, and of better isolating the reason from the paradox, which is the nonrestricted comprehension. For the paradox of Cantor one uses it to deduce that V the class of all the units is a unit, which is not possible in the usual set theory, but also to deduce the existence from it from the whole of the parts of a unit, which is licit on the other hand there (it is the Axiome of the whole of the parts), in a way compatible with the analysis of Cantor, which does not have besides this result like a paradox (for more details to see the article on the Paradoxe of Burali-Forti).

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