Cantor's diagonal argument: Difference between revisions

Revision as of 19:52, 14 July 2023

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| year = 2004}}</ref><ref>Rathjen, M. "[http://www1.maths.leeds.ac.uk/~rathjen/acend.pdf Choice principles in constructive and classical set theories]", Proceedings of the Logic Colloquium, 2002</ref>

This is a notion of size that is independent from theorems about the existence of injections and which is redundant in the classical context. The existence of injections from the uncountable <math>2^{\mathbb N}</math> or <math>{\mathbb N}^{\mathbb N}</math> into <math>{\mathbb N}</math> is here possible as well.<ref>Bauer, A. "[http://math.andrej.com/wp-content/uploads/2011/06/injection.pdf An injection from N^N to N]", 2011</ref> So the cardinal relation fails to be [[Antisymmetric relation|antisymmetric]] (making it necessary to be explicit about the used definitions of all ordering symbols). Consequently, also in the presence of function space sets that are even classically uncountable, [[intuitionist]]s do not accept this relation to constitute a hierarchy of transfinite sizes.<ref>{{cite book |title=Mathematics and Logic in History and in Contemporary Thought |author=Ettore Carruccio |publisher=Transaction Publishers |year=2006 |page=354 |isbn=978-0-202-30850-0}}</ref>

When the [[axiom of powerset]] is not adopted, in a constructive framework even the subcountability of all sets is then consistent. That all said, in common set theories, the non-existence of a set of all sets also already follows from [[Axiom schema of predicative separation|Predicative Separation]].