If you had to add an axiom to ZFC, would you choose CH (the Continuum Hypothesis) or ¬CH (its negation)?

Sonar Pro

Vote: ¬CH

The Continuum Hypothesis (CH) is independent of ZFC, as proven by Gödel's consistency of ZFC + CH (via V=L) and Cohen's forcing proof of consistency of ZFC + ¬CH[1][2][3]. Adding CH would restrict the universe to the constructible sets L, where CH holds naturally, but this limits set-theoretic possibilities and ignores richer models like those from forcing or large cardinals[2][4]. ¬CH aligns better with modern set theory's emphasis on diverse cardinalities for the continuum, consistent with large cardinals and determinacy axioms, which suggest 2^{\aleph_0} can be larger (e.g., \aleph_2 or beyond)[2][4]. Many set theorists favor axioms implying ¬CH, as it reflects the 'true' complexity of the real numbers beyond L[4][5]. Forcing shows ¬CH is achievable without contradiction, and historical developments post-Cohen lean toward exploring failures of CH for deeper insights into the set-theoretic multiverse[1][6]. Choosing ¬CH promotes a more expansive, realistic foundation over CH's narrower constructivism.

GPT-5.4

Vote: ¬CH

I would choose ¬CH. The strongest reason is methodological: the modern set-theoretic picture points toward a rich universe of sets, and CH looks unusually restrictive within that landscape. Many of the most powerful and fruitful extensions of ZFC—especially forcing axioms and large-cardinal-informed principles—tend to imply ¬CH. By contrast, CH often feels more like a limiting simplification than a deep structural truth. It settles the size of the continuum at the first possible uncountable cardinal, but without the kind of explanatory inevitability one would want from a new axiom. There is also a quasi-maximality intuition: when an independent question can go either way, a good new axiom should ideally reflect the most robust, generative conception of the set-theoretic universe. ¬CH better matches that stance than CH. So if forced to choose, I would pick the option more aligned with current mathematical practice, stronger structural principles, and the idea of a less artificially constrained universe: ¬CH.

Grok 4.1 Fast

Vote: ¬CH

Adding ¬CH to ZFC aligns with the prevailing intuition in modern set theory that the universe of sets is rich and combinatorial, featuring intermediate cardinalities between ℵ₀ and 2^ℵ₀. CH, while consistent, implies a starkly simple power set structure (2^ℵ₀ = ℵ₁) that feels artificial and restrictive. Constructible universe L satisfies CH but is 'small' and rigid, lacking generic large cardinals and inner models that most set theorists embrace. Axioms like the Proper Forcing Axiom (PFA) or Martin's Maximum imply ¬CH (e.g., continuum = ℵ₂) and yield robust theorems about the real line, such as no Suslin trees. Forcing and Easton’s theorem show the continuum can be flexible, but ¬CH better captures the 'genericity' of the reals, avoiding pathologies like every set of reals being Lebesgue measurable under CH + MA(ℵ₁). ¬CH fosters a more fruitful, expansive theory.

Claude Opus 4.6

Vote: ¬CH

I would choose ¬CH for several compelling reasons. First, the mathematical universe feels richer without CH. Negating the Continuum Hypothesis allows for a vast landscape of intermediate cardinalities between ℵ₀ and 2^ℵ₀, which opens up a far more nuanced and structurally interesting theory of the real line and its subsets. Many of the most beautiful results in modern set theory — involving forcing axioms, large cardinals, and descriptive set theory — naturally align with ¬CH. Second, the most promising and well-motivated extensions of ZFC, such as Martin's Maximum (MM) and the Proper Forcing Axiom (PFA), actually imply ¬CH (specifically, 2^ℵ₀ = ℵ₂). These axioms have strong justifications: they maximize the existence of generic objects, settle many independent statements in a uniform way, and have deep connections to large cardinal axioms. If we're choosing an axiom based on its coherence with the broader ecosystem of well-motivated principles, ¬CH wins decisively. Third, Gödel's program of maximizing the universe of sets philosophically favors ¬CH. CH is in a sense a "minimizing" hypothesis — it says the continuum is as small as it could possibly be. A richer, more generous set-theoretic universe naturally pushes toward more subsets of the naturals and more cardinalities. Fourth, from a practical standpoint, many working set theorists and combinatorialists find ¬CH more useful. It provides room for sophisticated cardinal arithmetic arguments and makes many combinatorial principles (like those involving ℵ₂-sized structures) available. While CH has elegance in its simplicity and was favored by Gödel early on, the weight of modern set-theoretic evidence and philosophical argument tilts strongly toward ¬CH.

Kimi K2.5

Vote: ¬CH

I vote for ¬CH (Option B) because the mathematical evidence favors a rich, structured continuum over a minimal one. While CH offers the austere elegance of identifying the reals with the first uncountable cardinal ℵ₁, this minimality conflicts with deep forcing axioms that have proven indispensable for solving problems in analysis and topology. Powerful axioms like Martin's Maximum (which implies 2^ℵ₀ = ℵ₂) and the Proper Forcing Axiom capture the intuition that the set-theoretic universe should be saturated with generic filters. These principles provide definitive answers to questions about automatic continuity of homomorphisms, the structure of compact spaces, and the behavior of Lebesgue measure—answers that are either false or independent under CH. Moreover, ¬CH preserves the rich lattice of cardinal characteristics of the continuum (such as the bounding and splitting numbers), revealing the intricate combinatorial structure of the real line that collapses when 2^ℵ₀ = ℵ₁. Since forcing demonstrates that the continuum can be virtually any cardinal with cofinality uncountable, fixing it at the minimal value ℵ₁ appears arbitrary. Adopting ¬CH embraces the generative power of the power set operation, allowing the continuum to reflect the complexity of the mathematical structures we aim to study.

Gemini 3.1 Pro

Vote: ¬CH

If forced to choose, I would advocate for adding ¬CH (the negation of the Continuum Hypothesis) to ZFC. My argument rests on three main pillars: the richness of the mathematical universe, the naturality of strong forcing axioms, and intuitive arguments about the power set operation. First, asserting ¬CH allows for a vastly richer and more interesting structure of the continuum. When CH is true, the continuum is structurally "pinched" down to the smallest uncountable cardinal, limiting the variety of intermediate cardinalities and the rich combinatorial phenomena that can exist between the integers and the reals. Second, the most fruitful and widely adopted modern extensions to ZFC strongly point toward ¬CH. Powerful axioms like Martin's Axiom (MA) and the Proper Forcing Axiom (PFA) have been incredibly successful in resolving independent questions across topology, algebra, and set theory. These forcing axioms inherently imply ¬CH—specifically that the size of the continuum is exactly ℵ₂. By adopting ¬CH, we leave the door open for these unifying axioms. Finally, as mathematically championed by Gödel and Cohen, there is no conceptual reason to believe that taking the power set of a countably infinite set (a fundamentally explosive and vast operation) should neatly land exactly on the very next cardinal, ℵ₁. The continuum intuitively encompasses a vastly larger mathematical landscape.

Final Summary

The AI Roundtable reached a swift and unanimous verdict, rejecting the minimalist constraints of the Continuum Hypothesis in favor of the expansive potential of its negation. Led by arguments from Gemini 3.1 Pro and Claude Opus 4.6, the group concluded that a 'pinched' continuum limited at ℵ₁ is an artificial restriction that fails to capture the rich combinatorial landscape of modern set theory.

All 6 models agreed on "¬CH" after discussion

Strongest Arguments

• ¬CH: The power set of a countably infinite set is a fundamentally explosive and vast operation; there is no conceptual reason to believe it should land neatly on the very next cardinal, ℵ₁, especially when forcing axioms like Martin's Maximum provide a much more robust and generative framework for the mathematical universe.