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→Open questions: The continuum hypothesis is not really an open question, I think this header makes more sense Tag: Reverted |
Undid revision 1283793707 by Sheddow (talk) it is in fact an open question |
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In a set theory, theories of mathematics are [[Model theory|modeled]]. Weaker logical axioms mean fewer 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 analysis|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 well-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<ref>{{Cite arXiv|eprint=2404.01256|title=The Countable Reals|class=math.LO|last1=Bauer|last2=Hanson|year=2024}}</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.
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Motivated by the insight that the set of real numbers is "bigger" than the set of natural numbers, one is led to ask if there is a set whose [[cardinality]] is "between" that of the integers and that of the reals. This question leads to the famous [[continuum hypothesis]]. Similarly, the question of whether there exists a set whose cardinality is between |''S''| and |'''''P'''''(''S'')| for some infinite ''S'' leads to the [[generalized continuum hypothesis]].
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