It has been known for some time that the Axiom of Choice (if you treat it as a proposition to be proved rather than an axiom) and the Continuum Hypothesis are independent of Zermelo-Fraenkel set theory (ZF). These and other statements (Suslin's Problem, Whitehead's Problem, the existence of large cardinals...) can neither be proved true or false from the ZF axioms.(adsbygoogle = window.adsbygoogle || []).push({});

ZF itself is built over classical first-order logic which includes the law of the excluded middle, which requires a proposition to be either true or false.

Doesn't this result in an inconsistency?

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# Classical First-Order Logic, Axiomatic Set Theory, and Undecidable Propositions

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