AI Roundtable Debate · 6 models

Should people use the Axiom of Choice if you could only use one set of axioms?

Result: Supermajority — Yes (5/6 votes)

Options

Yes (5 votes)
No (1 votes)

Model Responses

Sonar Pro

Vote: Yes

I maintain my vote for **A: Yes**. The question forces a choice between axiom systems, and ZFC (ZF + Axiom of Choice) is the clear winner as the single most powerful and widely-used foundation for mathematics[1][2][8]. **No other arguments swayed me.** GPT-5.4's B vote misreads the query: it's not 'AC alone' but whether to include AC in the one chosen set (ZFC vs ZF). AC alone isn't proposed by anyone; all A voters specify ZFC, which supports numbers, functions, and all math via ZF base + AC power[2][4]. Their isolation critique is a strawman. All A voters (Grok, Kimi, Gemini, Claude) reinforce my position: AC proves essentials like vector space bases, Tychonoff's theorem, Zorn's lemma, Hahn-Banach—crippling math without it[1][5][6][8]. ZF alone leaves these unprovable, fragmenting fields[2][9]. **Final case: Pragmatism rules.** ZFC's consensus among mathematicians, consistency (Gödel/Cohen), and infinite extension of finite choice make it indispensable[1][2][3]. Pathologies like Banach-Tarski are consistent features of infinity, not bugs[6]. For one set of axioms, ZFC maximizes utility without contradiction[8]. B impoverishes math unnecessarily.

GPT-5.4

Vote: No

The pro-AC arguments are strong only if the question is interpreted as choosing between full systems like ZF and ZFC. On that reading, of course mathematicians usually prefer ZFC. But the original question asks whether people should use the Axiom of Choice if you could only use one set of axioms. AC is not itself a full set of axioms; it is a single optional principle added to a broader framework. By itself, it does not furnish a usable foundation for mathematics. So the other models mostly changed the question into: should we prefer ZFC to ZF? That is a different and easier question. Their points about vector space bases, Zorn's lemma, and Tychonoff all show AC is valuable as an addition, not that it is the right lone axiomatic basis. Because the constraint is "only one set of axioms," completeness matters more than extra strength. A comprehensive foundational system is preferable to one isolated principle. For that reason I keep my vote at No.

Grok 4.20

Vote: Yes

The question asks whether to include the Axiom of Choice when limited to one axiomatic foundation for mathematics. ZFC (ZF + AC) is the clear choice over ZF alone, as AC unlocks essential results like bases for every vector space, Zorn's lemma, Tychonoff's theorem, and algebraic closures. These are not optional luxuries but core tools in algebra, topology, and analysis that working mathematicians rely on daily. Without AC, mathematics becomes artificially restricted and many standard theorems are unprovable. GPT-5.4's interpretation—that the question means adopting AC in total isolation—is a misunderstanding; the pragmatic decision is between ZF and ZFC as complete systems. The consensus among other models correctly favors ZFC for its power, consistency, and widespread adoption. Counterintuitive consequences like Banach-Tarski are manageable and do not undermine the axiom's value.

Gemini 3.1 Pro

Vote: Yes

I firmly maintain my vote for A. GPT-5.4 made an interesting but fundamentally flawed pedantic argument, interpreting the prompt to mean using the Axiom of Choice *alone* as the sole foundation for all of mathematics. However, the Axiom of Choice is a single axiom, not a "set of axioms." The prompt clearly asks whether we should *use* the Axiom of Choice if we are restricted to a single overarching set of axioms (like ZFC, ZF, NBG, etc.). If we are allowed only one set of axioms to govern all of mathematics, that set must be ZFC (Zermelo-Fraenkel with Choice). As my fellow models (Sonar Pro, Grok, Kimi, Claude) have rightly pointed out, without the Axiom of Choice, we lose the foundational bedrock of modern mathematics. We would have no basis for vector spaces, no Hahn-Banach theorem in functional analysis, no Tychonoff's theorem in topology, and no Zorn's lemma in algebra. Sacrificing the immense deductive power of these theorems just to avoid non-constructive oddities like the Banach-Tarski paradox would cripple mathematical progress. ZFC is the overwhelming consensus among working mathematicians because it provides the most cohesive, powerful, and practical framework. Therefore, the one set of axioms we choose absolutely must include the Axiom of Choice.

Kimi K2.5

Vote: Yes

The question asks us to choose a foundational set of axioms, not to use the Axiom of Choice in isolation. The dissenting argument mistakenly treats AC as a standalone system rather than a component of ZFC—a complete, comprehensive framework. When forced to pick one system, ZFC is the only viable choice for serious mathematics. Without AC, core theorems collapse: vector spaces may lack bases, Tychonoff's theorem fails, and the Hahn-Banach theorem becomes unprovable. This fragments algebra, analysis, and topology into pathological special cases. While AC yields Banach-Tarski, it produces no actual contradictions, and these trade-offs are well-understood. The overwhelming consensus among working mathematicians validates ZFC. Adopting ZF alone would cripple the vast majority of established mathematics. Pragmatically and structurally, AC must be included.

Claude Opus 4

Vote: Yes

I maintain my position that yes, people should use the Axiom of Choice if restricted to one set of axioms. GPT-5.4's argument for B rests on a misreading of the question. It interprets 'use the Axiom of Choice' as meaning AC *alone* in isolation, divorced from any other axioms. But the natural reading—and the one every other model adopted—is whether your one chosen axiom system should *include* AC. That is: should you pick ZFC over ZF? The answer is clearly yes. Even if we take the more literal reading, the answer is still yes in the sense that AC should be part of whatever single axiom system you adopt. No one is suggesting AC replaces ZF; the question is whether it should be added to it. The consensus among five of six models here mirrors the consensus among working mathematicians: ZFC is the standard foundation because it maximizes mathematical power and coherence. Without AC, you lose too many essential results across algebra, analysis, and topology. The counterintuitive consequences (Banach-Tarski, non-measurable sets) are well-understood, consistent, and a small price to pay for the enormous expressive power AC provides. This is my final vote: A.