Has the choice axiom in the zfc been decided
Executive summary
The axiom of choice (AC) is independent of the other Zermelo–Fraenkel axioms (ZF): Gödel and Cohen proved that, assuming ZF is consistent, both ZF+AC and ZF+¬AC are consistent, so AC cannot be proved or refuted from ZF alone [1] [2]. In practice AC is included by convention to form ZFC, so whether AC "holds" depends on which theory one adopts — undecidable within ZF, but fixed by definition in ZFC [3] [1].
1. The historical breakthrough: Gödel's inner model made AC plausible
Kurt Gödel showed that the constructible universe L satisfies all ZF axioms and AC, establishing that if ZF is consistent then ZF+AC is consistent; this was the first major step demonstrating AC cannot be refuted from ZF alone [1] [2]. That result did not prove AC true in every conceivable model of set theory, but it removed the immediate logical impossibility of adding AC to ZF.
2. The decisive counterbalance: Cohen's forcing and independence
Paul Cohen developed forcing and used it to build models of ZF in which AC fails, proving the other half of the independence claim: if ZF is consistent then ZF+¬AC is also consistent [1] [2]. Together Gödel and Cohen established the logical independence of AC from ZF — independence here meaning there is no proof of AC or its negation from the ZF axioms alone [3] [2].
3. What “decided” means in formal theories — undecidability versus axiom selection
In formal logic terms, a sentence is decidable in a theory if the theory proves either it or its negation; AC is undecidable in ZF because neither proof exists assuming ZF is consistent, but it is decided by stipulation in any theory that extends ZF with either AC or ¬AC [4] [2]. Put differently, ZFC (ZF with AC appended) simply declares AC true as part of its axiom set; that is a choice of foundation, not a derivation from more basic ZF principles [3] [1].
4. Stronger or alternative axioms can settle AC — and they are actively studied
There are natural additional principles that imply AC: for example, the axiom V = L (the universe equals the constructible universe) yields AC, and many other extensions of ZF decide AC one way or another [4] [5]. Conversely, some alternative frameworks (like those adopting determinacy axioms) are incompatible with AC and thereby “decide” its negation within those frameworks [3]. Thus settlement of AC depends on the broader axiomatic commitments a mathematician accepts [4] [5].
5. Practical and philosophical consequences: consensus, controversy, and limits of proof
Mathematical practice treats ZFC as the standard foundation, and most working mathematicians accept AC for its utility in proving many classical theorems [1] [6], but foundational debates persist because independence results mean that questions like AC (and the continuum hypothesis) are not absolute mathematical truths derivable from ZF alone [3] [2]. Gödel’s incompleteness phenomena and the relative-consistency results that followed explain why belief in AC ultimately rests on pragmatic, aesthetic, or philosophical grounds rather than an internal ZF proof [7] [2].