Set Theory and Logic

    

The Conflict Between Non-definable Reals and the Axiom of Choice in ZFC

Authors: Kim Altair

We examine the consequences of applying the Axiom of Choice (AC) within ZFC to the set D := R C, where C denotes the set of computable real numbers. We argue that any choice function defined on D inevitably introduces a structure in which its elements become effectively referable or indexable, thereby contradicting the definition of D as unnameable and unindexable. This leads to a logical inconsis- tency, suggesting that AC is incompatible with the existence of D. To resolve this conflict, we propose a modified axiomatic system, ZFC D/AC, in which the Axiom of Choice is explicitly restricted from applying to sets composed of non-definable real numbers. This result prompts broader meta-mathematical reflection on the in- terplay between definability, choice, and the foundational assumptions underlying mathematical existence.

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[v1] 2025-07-20 20:43:06

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