Cantor's diagonal argument: Difference between revisions

Revision as of 18:48, 2 May 2023

Revision as of 18:49, 2 May 2023

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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]].

In a set theory, theories of mathematics are [[Model theory|modeled]]. Weaker logical axioms mean less constraints and so allow for a richer class of models. A set may be identified as a [[Construction of the real numbers|model of the field of real numbers]] when it fulfills some [[Tarski's axiomatization of the reals|axioms of real numbers]] or a constructive rephrasing thereof. Various models have been studied, such as the [[Construction_of_the_real_numbers#Construction_from_Cauchy_sequences|Cauchy reals]] or the [[Dedekind cut|Dedekind reals]], among others. The former relate to quotients of sequences while the later are good behaved cuts taken from a powerset, if they exist. In the presence of excluded middle, those are all isomorphic and uncountable. Otherwise, [[Effective_topos#Realizability_topoi|variants]] of the Dedekind reals can be countable or inject into the naturals,<ref>Bauer, A., Hanson, J. A. "The countable reals", 2022</ref> or inject into the naturals, but not jointly. When assuming [[countable choice]], constructive Cauchy reals even without an explicit [[modulus of convergence]] are then [[Cauchy_sequence#Completeness|Cauchy-complete]]<ref>Robert S. Lubarsky, [https://arxiv.org/pdf/1510.00639.pdf ''On the Cauchy Completeness of the Constructive Cauchy Reals''], July 2015</ref> and Dedekind reals simplify so as to become isomorphic to them. Indeed, here choice also aids diagonal constructions and when assuming it, Cauchy-complete models of the reals are uncountable.

===Open questions===