Can You Cut a Ball Apart and Reassemble It into Two? How Axioms Split Mathematics into Parallel Universes

The article explains how Gödel's and Cohen's proofs of the independence of the continuum hypothesis showed that mathematics is not a monolithic edifice with a single set of truths, but a branching landscape of parallel universes contingent on the choice of axioms. It describes the philosophical dispute between pluralists (Hamkins) and proponents of a single canonical universe (Woodin), and examines whether AI systems such as AlphaProof or Lean 4 can systematically explore these axiomatic landscapes.

In April 1963, the twenty-nine-year-old mathematician Paul Cohen sent a letter to Kurt Gödel in Princeton. It contained a proof of something that for eighty-five years had seemed impossible — that one of the oldest questions in mathematics has no answer. Gödel invited Cohen to his home, checked the proof, and wrote to him that he had given "the best possible proof." A year later, in a published text, Gödel called Cohen's work "undoubtedly the greatest advance in the foundations of set theory since its axiomatization."

Cohen's result — together with Gödel's own contribution of 1940 — showed that mathematics is not a single monolithic edifice with one set of truths. It is a branching landscape, where the choice of starting assumptions opens up fundamentally different worlds. In some, a sphere can be cut into five pieces and reassembled into two spheres. In others, this is impossible. In some there are "intermediate levels" of infinity, in others there are not. All these worlds are internally consistent.

This discovery gave birth to a philosophical dispute that continues to this day: is there a single "correct" mathematical universe, or are there infinitely many? And in an era when artificial intelligence solves olympiad problems at gold-medal level and formalizes proofs in the Lean system, a new question arises — can AI systematically explore these parallel worlds?

The story begins in 1873, when Georg Cantor made a shocking discovery: there are "more" real numbers than natural numbers. He showed that the natural numbers (1, 2, 3, …) have cardinality ℵ₀ (aleph-null), while the real numbers — the entire number line — have cardinality 2^ℵ₀, which cannot be identified with ℵ₀. Cantor's original 1873 proof used the method of nested intervals. The famous diagonal argument, which is better known today, came only in 1891: we assume that all the real numbers in the interval (0, 1) can be arranged into a list, and then from this list we construct a number that differs from the first in the first decimal position, from the second in the second, and so on. This number is missing from the list — a contradiction.

A question naturally arose: is there some "intermediate set" whose size would fall exactly between these two infinities?

Cantor's continuum hypothesis (CH) asserts that there is not — that 2^ℵ₀ = ℵ₁, that is, that the cardinality of the continuum is the immediate "next" infinity after ℵ₀ with no intermediate step whatsoever. Cantor formulated this hypothesis in 1878 and regarded it as the key question of set theory.

David Hilbert considered this question so fundamental that in 1900 he placed it as problem number 1 on his famous list of 23 unsolved problems. A resolution came, but in a completely different way than Hilbert had imagined: it turned out that CH is independent of the standard axioms.

In a deep irony of fate, on 8 September 1930 in Königsberg Hilbert delivered his famous Wir müssen wissen, wir werden wissen — we must know, we will know. The day before, at a linked conference on the epistemology of the exact sciences, the young Kurt Gödel announced a preliminary version of his first incompleteness theorem, which showed that Hilbert's optimism had fundamental limits.

Kurt Gödel approached the problem by constructing an inner model of set theory called L — the constructible universe. The standard set-theoretic universe V is built up in stages: at each level one takes the subsets of the previous level. Gödel tightened the procedure — into L he admitted only those subsets that can be defined by a formula with parameters from the previous level. The result is a "leaner" universe, in which every set is explicitly describable.

Figuratively speaking: when you enter the "room" L and close the door, you see only definable objects. About the undefinable sets that may exist "outside" you know nothing — and from your point of view CH simply holds.

Gödel's result formally states: if ZF is consistent, then so is ZFC + CH. In other words, CH cannot be refuted. Gödel himself, however, believed that CH is false. In his 1947 article he wrote that he found it suspicious that, against the many arguments opposing CH, there was no plausible argument in its favor. In 1972 he even extended an argument that the true cardinality of the continuum is ℵ₂, but the proof contained an error and was withdrawn.

The second step was taken by Paul Cohen — an analyst, not a logician, which makes his achievement all the more remarkable. Cohen invented a method called forcing, which makes it possible to extend an existing model of set theory by adding a new generic object without disturbing consistency.

How does it work? We start with a model M in which CH holds. We then add to M ℵ₂ new real numbers — but we do so carefully, so that the cardinalities do not collapse. He achieved this with a condition called the countable chain condition, which ensures that the cardinalities of existing sets do not change. The result is a new model in which 2^ℵ₀ ≥ ℵ₂ > ℵ₁, and so CH fails.

Cohen announced his result in 1963 and in 1966 received the Fields Medal — to this day the only one awarded for work in mathematical logic. Scott Aaronson compares forcing to the construction of oracles in computational complexity theory — we insert a new function into the universe and, step by step, ensure that it satisfies the required properties. Timothy Chow, in his text "A Beginner's Guide to Forcing," calls forcing an open exposition problem, because for most mathematicians the method remains difficult to access.

The results of Gödel and Cohen together state: CH is independent of ZFC. There are consistent mathematical worlds in which CH holds, and worlds in which it fails.

This is an analogy with Euclid's parallel postulate. For two thousand years mathematicians tried to prove the fifth postulate from the other axioms. Euclid himself formulated it as a postulate, not a theorem, perhaps because he sensed its different character. In the 19th century Lobachevsky, Bolyai, and Gauss showed that there are consistent geometries in which it does not hold. Today no one regards Euclidean geometry as "the only true one" — the hyperbolic and elliptic geometries are equally legitimate.

It is essential to distinguish this independence from Gödel's incompleteness theorems. Gödel incompleteness produces sentences that are true in the standard model of arithmetic but unprovable. Adding their negation creates non-standard, "pathological" models. With CH the situation is different: both directions of extension are mathematically fully fledged and have their proponents. As Aaronson notes, incompleteness gives two consistent models but only one healthy one, whereas the independence of CH offers two plausible ones.

The continuum hypothesis is not an isolated case. Modern mathematics is full of crossroads where the choice of an axiom leads to fundamentally different theories.

The axiom of choice (AC), formulated by Ernst Zermelo in 1904, asserts that for every collection of non-empty sets there is a function that selects exactly one element from each. Bertrand Russell captured it with a famous analogy about socks: to choose one from infinitely many pairs of socks you need the axiom of choice, but for shoes you do not — you always take the left one.

AC is equivalent to a number of other statements: Zorn's lemma, the well-ordering theorem, and Tychonoff's theorem on compactness. Its most provocative consequence is the Banach–Tarski paradox of 1924, in which a ball in three-dimensional space can be cut into a finite number of pieces and reassembled into two balls of the same size.

The apparent violation of the law of conservation of volume is explained by the fact that the "cuts" produce sets so pathologically complex that they have no defined volume — they are not Lebesgue measurable. The axiom of choice is independent of ZF: Fraenkel, Mostowski, and Gödel showed in various models that mathematics can function both with it and without it.

Robert Solovay showed in 1970 that there is a model of ZF in which dependent choice holds (sufficient for ordinary analysis) but every set of real numbers is measurable. This is a dramatic contrast: in one universe you double the ball, in the other it is impossible.

Large cardinals form a hierarchy of infinities so enormous that their existence cannot be proved in ZFC. The hierarchy ranges from inaccessible cardinals through measurable ones up to Woodin and supercompact cardinals. Remarkably, this hierarchy is empirically linearly ordered by consistency strength — a fact that is an observation without a counterexample, not a provable theorem.

Dana Scott proved in 1961 that in the constructible universe L no measurable cardinals exist. This means that accepting the existence of measurable cardinals automatically implies V ≠ L — the real universe of sets is "richer" than Gödel's construction.

Surprisingly, these gigantic cardinals decide questions about "small" objects. Martin and Steel proved in 1989 that the existence of infinitely many Woodin cardinals implies projective determinacy. This gives an elegant and complete structural theory for the projective sets of real numbers, one that ZFC alone does not provide.

The axiom of determinacy (AD), proposed by Mycielski and Steinhaus in 1962, states that in every infinite game between two players one of them has a winning strategy. AD is directly incompatible with the axiom of choice — AC makes it possible to construct a game in which neither player has a strategy. Under AD every set of real numbers is measurable, every one has the Baire property, and ω₁ is a measurable cardinal.

Interesting is the role of forcing axioms such as PFA (Proper Forcing Axiom), which imply 2^ℵ₀ = ℵ₂ — that is, a specific negation of CH.

Saharon Shelah showed in 1974 that a purely algebraic question — "Is every Whitehead group free?" — is independent of ZFC. In the constructible universe L the answer is yes; under Martin's axiom with the negation of CH there exist non-free Whitehead groups. It was one of the first cases in which independence from the axioms appeared directly in ordinary algebra, outside logic and set theory.

Remarkable is a recent turn: Clausen and Scholze showed that within condensed mathematics the analogous version of the Whitehead problem is decidable in ZFC — the independence is tied to the classical notion of a group, not to the problem itself.

Joel David Hamkins, since 2022 the O'Hara Professor of Logic at the University of Notre Dame, is one of the most influential proponents of set-theoretic pluralism. In his article "The set-theoretic multiverse" (Review of Symbolic Logic, 2012) he formulated a radical thesis: there are many legitimate set-theoretic universes and none of them is privileged.

In the follow-up text "More than a Decade in the Set-Theoretic Multiverse" of 2025 he summarizes the development of his position. He argues that the most prominent phenomenon of modern set theory has been precisely the surprising diversity of set-theoretic possibilities.

Hamkins develops the analogy with geometry: just as the alternative geometries were at first curiosities for independence proofs and gradually gained full legitimacy, so too the alternative set-theoretic universes are to be regarded as fully real. On this perspective CH is "solved" — not by finding an answer, but by a deep understanding of how it behaves across the entire multiverse.

At the opposite pole stands W. Hugh Woodin of Harvard, whose intellectual trajectory is itself fascinating. Around 2001 he developed a sophisticated argument that CH is false, and proposed Ω-logic as an extension of the ZFC axioms. Since 2010, however, he has radically changed his position and presented the Ultimate L program — an attempt to construct a canonical inner model compatible with all large cardinals, in which CH would hold.

As Rittberg documents in the article "How Woodin Changed His Mind" (2015), this is an extraordinary case in which a leading mathematician publicly reconsidered his stance on a fundamental question. Woodin claims that the axiom V = Ultimate-L would decide all the questions undecidable by Cohen's method, but he admits that everything depends on the truth of the Ultimate L conjecture.

Saharon Shelah of the Hebrew University of Jerusalem, one of the most prolific living mathematicians, holds a pragmatic stance. In his text "Logical Dreams" he writes that his interest in logic was always mathematical, never philosophical. Against the axiom V = L he is firmly opposed — he considers the constructible universe too special a case. His pcf theory shows that meaningful mathematics about cardinal arithmetic is possible even without deciding CH.

Penelope Maddy (UC Irvine) approaches the axioms naturalistically: mathematical criteria — fruitfulness, unifying power, depth — should decide the adoption of new axioms. She advocates the principle "Maximize!" — the universe of sets should be as rich as possible. The dissertation of her student Jeffrey Schatz (2022) formally proved that Martin's Maximum, in this sense, "maximizes" over V = Ultimate-L.

In July 2024 DeepMind's AlphaProof system solved 4 of the 6 problems of the International Mathematical Olympiad and earned 28 of 42 points — silver-medal level. The system combines reinforcement learning with the Lean 4 formal prover — every proof is machine-verified, so hallucinations are ruled out. Moreover, it solved the hardest problem, P6, on which only a minimum of human contestants succeeded.

A year later, at IMO 2025, DeepMind's Gemini Deep Think reached 35 points out of 42, thereby crossing the gold-medal threshold — and that end-to-end in natural language, without the need for translation into a formal system.

At the same time we are seeing major advances in the autoformalization of mathematics in Lean 4, where AI systems automatically translate informal proofs into a machine-verifiable form.

Terence Tao, in March 2026 at the IPAM conference, declared that AI in mathematics and theoretical physics is "ready for primetime." He argues that AI tools will save more time than it takes to learn them and integrate them into one's workflow. In the same month he announced the "Mathematics Distillation Challenge" — a challenge for AI systems to generate humanly comprehensible explanations of formal proofs from the Equational Theories project. This project systematically decided over 22 million implications among thousands of equational laws — a direct demonstration of AI-assisted exploration of axiomatic systems.

Christian Szegedy, formerly of Google Brain and a co-founder of xAI, today the founder of Math Inc. and chief scientist at Morph Labs, champions autoformalization — the automatic translation of informal mathematics into a machine-verifiable language. His system Gauss completed the formalization of the strong prime number theorem in Lean within a matter of weeks — a task on which Fields Medalist Terence Tao and Alex Kontorovich, with other collaborators, had worked for over a year and a half. Szegedy is convinced that the road to superintelligence runs through formal mathematics.

The Equational Theories Project is so far the closest realization of a systematic exploration of axiomatic landscapes — the project achieved nearly complete coverage in 19 days. Systems such as Axplorer and AI-Hilbert generate hypotheses inductively from data and verify them deductively.

So far, however, no system has autonomously invented a fundamentally new axiomatic framework in the way a mathematician invents non-Euclidean geometry. Current tools explore the consequences of given axioms but do not reflect on the meta-question of which axioms to adopt. It is precisely here that a fascinating research frontier opens up.

Gödel's incompleteness theorems state that any sufficiently strong consistent formal system contains statements that are true but unprovable — and this holds for AI just as it does for humans.

The Lucas–Penrose argument — that the human mind transcends formal systems because it "sees" the truth of Gödel's theorems — is today rejected by the majority. Critics such as Feferman point out that to recognize the truth of Gödel's theorem you must first know that the system is consistent — and that not even a human can do.

The practical relevance of incompleteness for AI is limited — most working mathematics does not run into these limits. The real limit is not incompleteness, but the ability to propose new definitions, to choose a perspective, and to formulate productive questions.

It would be convenient to tell the story as a triumphal march from incompleteness to AI explorers of the multiverse. But the reality is more nuanced.

The multiverse versus practice. Most working mathematicians ignore Hamkins's multiverse — they work in ZFC and encounter independence only exceptionally. Shelah's pragmatism, not philosophical pluralism, is the field's de facto standard.

AI and creativity. Current AI systems are extraordinarily strong at exploring the consequences of given axioms, but so far they have demonstrated nothing comparable to Cohen's invention of forcing — that is, a fundamentally new method that changes the rules of the game. AlphaProof solves olympiad problems, but it does not propose new definitions. The Equational Theories Project maps the landscape, but it does not invent new landscapes. The gulf between "solving a problem" and "asking a question that changes the field" remains open.

Historical precedent. The analogy with geometry is instructive, but not perfect. The non-Euclidean geometries gained legitimacy thanks to physical reality — general relativity showed that spacetime is curved. The alternative set-theoretic universes have no such physical support. The question of whether CH "really" holds may be categorically different from the question of whether the fifth postulate holds.

The story of the branching of mathematics according to its axioms reveals something deeper than a technical detail of logic. It shows that mathematical truth depends on starting points — and that this dependence is not a weakness, but a source of richness.

The philosophical dispute between Hamkins's multiverse and Woodin's search for a canonical model remains open — and it is precisely this openness that is productive. Woodin's change of mind shows that even leading mathematicians revise their positions when new arguments appear.

The entrance of AI onto the scene dramatically accelerates this story. In two years we have gone from a silver medal at the IMO to gold, from hand translation into Lean to end-to-end solutions in natural language. The Equational Theories Project showed that the systematic exploration of axiomatic landscapes is, with AI, not only possible but surprisingly efficient.

The key question for the coming decade is not whether AI can prove theorems — it already can — but whether it can pose questions that would not occur to a human, and propose axioms that open up hitherto unsuspected mathematical worlds. Gödel's theorems say that no system will be complete. But it is precisely this incompleteness that is an invitation to exploration — and today we have tools for this exploration that Cantor, Gödel, and Cohen could not have dreamed of.

Key sources: Stanford Encyclopedia of Philosophy: Continuum Hypothesis, Gödel's Incompleteness Theorems, Independence and Large Cardinals, Large Cardinals and Determinacy · Hamkins, J. D.: "The set-theoretic multiverse" (Review of Symbolic Logic, 2012) · Rittberg, C. J.: "How Woodin Changed His Mind" (Archive for History of Exact Sciences, 2015) · Shelah, S.: "Logical Dreams" (2002) · DeepMind: AlphaProof, Gemini Deep Think IMO 2025 · Tao, T.: Equational Theories Project, Mathematics Distillation Challenge · Chow, T.: "A Beginner's Guide to Forcing" (arXiv:0712.1320) · Math Inc.: Gauss

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