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REVIEW 4 major objections 5 minor 49 references

On Conway's Numbers and Games, the Von Neumann Universe, and Pure Set Theory

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper claims that surreal numbers are exactly the sets of ordinals with a maximal element, and that this definition yields the full Conway field inside the von Neumann universe.

desk verdict A clean, honest re-presentation of the surreal numbers as sets of ordinals with a maximum, whose order and tree structure are genuinely proved but whose field arithmetic and game equivalence are imported rather than proven. read the letter →

arxiv 2501.04412 v2 pith:BD7EX3OA submitted 2025-01-08 math.LO

Wolfgang Bertram (IECL) This is my paper · ORCID
classification math.LO MSC 03A0503E0505C0506F2512-0291A4697F5097H50
keywords surrealnumbersConwayvonNeumannuniverseordinalsignexpansionsrealscombinatorialgametheorypureset
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that Conway's surreal numbers have a simple home in pure set theory: define a number to be a set of von Neumann ordinals with a maximal element, and call that maximum its birthday. On this class NO the surreal order, binary tree, and reals arise directly from set inclusion and symmetric difference, with no prior notion of game or sign sequence needed. The author claims this presentation is equivalent to Gonshor's sign expansions, Alling's axiomatic surreal fields, and Conway's original games, and that the Conway reals form an order-isomorphic copy of R inside the von Neumann universe. If right, surreal numbers become an elementary chapter of set theory rather than an exotic side branch, and the von Neumann universe itself carries the absolute arithmetic continuum.

What carries the argument

The load-bearing object is the class $\mathrm{NO}$ of sets of ordinals with a maximal element, equipped with birthday $b(x)=\max x$. The sign-expansion $s_x$ turns each number into a sign string; the discriminant $\delta(x,y)=\min(x\triangle y)$ gives the total order by asking which side contains the discriminant; children $x^+=x\cup\{b(x)+1\}$ and $x^-=x\cup\{b(x)+1\}\setminus\{b(x)\}$ generate the binary tree. The Fundamental Existence Theorem, which gives every Conway cut $\langle L,R\rangle$ a unique number $c$ with minimal birthday and $c\preceq y$ for any $y$ between $L$ and $R$, is what lets Conway's cut formulas for addition, multiplication, and multiplicative inverses be imported, and it also identifies each stage $\mathrm{NO}_{\alpha+1}$ as the Cuesta-Dutari completion of $\mathrm{NO}_\alpha$.

What would settle it

Compute ε · ω_Co where ε = {0, ω} and ω_Co = ω + 1 is the Conway ordinal ω: if the transfer is sound, the product must be 1_Co = {0, 1}. A direct calculation from the cut formulas, or a check that the result depends on the choice of timely cut, that yields any other value would refute the claim that the new NO carries Conway arithmetic.

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Extended reading notes

Core claim

The central claim is that the von Neumann universe already contains the surreal numbers as a definable subclass: NO consists of the sets of ordinals that have a maximum b(x), called the birthday. Every such set encodes a sign expansion (plus for element, minus for hole, zero after the birthday), the total order is read off the least ordinal distinguishing two numbers (the discriminant), and the tree order x ≼ y says x is an initial segment of y below b(x). Starting from nothing, the stages NO_α are the Cuesta-Dutari completions of the previous stage, and the whole tree is connected and complete. The paper argues that Conway's cut-based addition, multiplication, and inverses can be transplanted to this setting, with proofs sketched through the Fundamental Existence Theorem and deferred to detailed presentations, and that the result is the ordered Field NO containing all ordinals and a canonical copy R_Co of the reals.

Load-bearing premise

The proof depends on showing that the usual rules for adding, multiplying, and taking inverses of surreal numbers work when numbers are represented as sets of ordinals with a largest element, and match Conway's original game operations; the paper sketches this and points to other books for details.

Editorial extensions

If this is right

  • If NO as defined here is a field, then surreal numbers need not be introduced through games: the whole ordered Field, with all ordinals and infinitesimals, exists inside the von Neumann hierarchy and can be taught immediately after ordinals.
  • The Conway reals R_Co, short numbers plus long reals of the form X ∪ {ω}, form a field isomorphic to R, with dyadic rationals as the short numbers and rationals exactly the eventually periodic sign expansions.
  • Each stage NO_α+1 is the Cuesta-Dutari completion of NO_α, so the hierarchy of numbers is generated from 0_Co = {0} by completions and limits, giving a purely set-theoretic construction of the full binary number tree.
  • The equivalence with Alling's axioms, Gonshor's sign expansions, and Conway's games means that the same absolute arithmetic continuum is reached from pure sets, from sign strings, and from partizan games.
  • The ordinal operations of Cantor and the Hessenberg operations both reappear inside NO, with the Conway ordinals being exactly the successor von Neumann ordinals and the field operations extending the commutative natural operations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next test is to make the arithmetic fully combinatorial: the paper leaves open a direct sign-sequence formula for x + y and xy, and its Grothendieck-group sketch suggests that such formulas would connect surreal arithmetic to transseries and generalized power series.
  • If the von Neumann-universe realization is accepted, the author's pure set theory program extends beyond numbers: the same cut and tree language could be used to build canonical copies of the surcomplex numbers and possibly p-adic-like completions, though the paper only speculates about these.
  • The philosophical claim that the Conway reals are the only natural construction of R avoiding Kuratowski pairs is stronger than the mathematical equivalence claims; it would require a precise definition of naturality to become testable.
  • The quantum-versus-classical framing of rank and birthday is an interpretive layer rather than a proven result, and its value would have to be judged by whether it produces new structural theorems about the von Neumann universe.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a presentation of surreal numbers as sets of von Neumann ordinals having a maximal element, called the birthday (Definition 2.1). It develops the total order, the descendance tree, limits, and cut-theoretic completeness in Sections 2.4–2.6, and gives an algorithmic bijection between the 'Conway reals' and the usual reals in Section 2.3. The paper then states, but does not fully prove, the transfer of Conway arithmetic to this setting (Theorems 2.67–2.70) and the equivalence with Conway's original games construction (Theorem 3.30). The remaining chapters discuss nimbers, a graded von Neumann universe for partizan games, philosophical interpretations, surcomplex and 'cocomplex' numbers, and open problems.

Significance. If the arithmetic transfer and game-equivalence were fully proved, the paper would give a strikingly simple set-theoretic entry point to surreal numbers and a convincing argument that the ordered surreal tree is a natural structure of the von Neumann universe. The order-theoretic core—total order, tree completeness, and the Fundamental Existence Theorem (Theorems 2.24, 2.41, 2.45, 2.72)—is proved carefully and appears sound. The paper also deserves credit for making the Conway reals explicit through the Berlekamp-style algorithm in Theorem 2.16 and for framing impartial and partizan games in terms of 'pure set theory' and a graded universe in a way that may be pedagogically and conceptually useful. However, the central field-theoretic claims are currently delegated to [Sim], [ONAG], and [S], so the manuscript as written establishes the ordered tree and its completeness, but not the full claimed foundation of the surreal numbers as a field.

major comments (4)
  1. [§2.8, Theorems 2.67–2.69] Theorems 2.67 and 2.68 assert the existence, uniqueness, associativity, commutativity, distributivity, and ordered-field structure of addition and multiplication on NO, but the text gives only an 'Idea of proof' that refers to [Sim] and [ONAG]. The preceding results do not immediately supply the missing content: Theorem 2.72 guarantees a cut number for any Conway cut, yet one still has to prove, by transfinite induction, that the cut defining x+y (and xy) satisfies the required inequalities, that the result is independent of the chosen cut representation, and that the resulting operations satisfy the ring axioms and admit multiplicative inverses. Because the claimed 'firm ground of pure set theory' includes the field structure, this is a load-bearing gap rather than a cosmetic omission.
  2. [§3.3, Theorem 3.30] The equivalence theorem with Conway's game construction is stated without a proof. The text says 'In order to prove the theorem, there are a lot of things to check' and 'In principle, everything is contained in [S]', but no theorem-by-theorem derivation is given. In particular, the injectivity of [G], the characterization of the image by condition (3.7), and the transfer of field operations to the quotient are exactly the assertions needed to substantiate the paper's claim that NO is equivalent to Conway's original numbers-as-games. This external delegation should be stated explicitly in the theorem, or the proof should be included.
  3. [§2.3 and §2.8, Theorem 2.70] Theorem 2.70 claims that RCo is a subfield of NO and that the bijection of Theorem 2.16 is a field isomorphism. The proof of Theorem 2.16 only establishes an order-preserving bijection between R and RCo; compatibility of this bijection with addition and multiplication is not shown in Section 2.3 and is not covered by the 'Idea of proof' in Section 2.8. Since the paper advertises the Conway reals as a canonical copy of R inside the von Neumann universe, the field-isomorphism statement needs a proof or an explicit reference to a proved theorem.
  4. [§0.7 and §4.2.2] The Introduction states that the Fundamental Existence Theorem 'entails' Alling's axioms and 'establishes equivalence' with other approaches, and Section 2.9 repeats this claim. Yet Section 4.2.2 explicitly says that a purely combinatorial definition of Conway arithmetic is still a programme whose missing details are 'remote'. This discrepancy should be reconciled: either the theorems in Section 2.8 are meant as imported known results, or the introduction should describe the contribution as the order-theoretic tree plus a formal translation of previously known arithmetic rather than as a fully self-contained foundation.
minor comments (5)
  1. [§0.1 and §0.3] The manuscript contains several typos, including 'would should' in the abstract and 'exploses' and 'take akes' in the introductory sections; these should be corrected during revision.
  2. [§1.3.3 and §2.8] There are unresolved cross-reference markers such as 'Equation (??)' in the proof sketch of Theorem 2.67; these should be replaced by the intended equation numbers.
  3. [§2.3, Theorem 2.16] The proof of bijectivity in Theorem 2.16 is very concise: it should spell out how the 'long ends' convention removes the binary-expansion ambiguity on the negative side and how the finite/infinite distinction is preserved by the inverse map.
  4. [§2.6, Definition 2.50] The 'topology' of closed sets on the proper class NO is informal; since it is used mainly for motivation, the text should explicitly state that it is not a topological space in the usual sense and that no separation axioms are being claimed.
  5. [§4.3, Definition 4.6] Definition 4.6 and Table 4.1 are labelled 'tentative' and 'speculative', which is honest, but the surrounding text should make even clearer that the cocomplex-number construction is an outlook and not part of the paper's main theorem set.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the core order/tree construction is derived from Definition 2.1, and the arithmetic and equivalence results are imported from external sources rather than presupposed.

full rationale

The paper's central construction (Definition 2.1, numbers as sets of ordinals with maximum) is independent of the results it later compares with Conway's games, Gonshor's sign expansions, and Alling's axiomatization. The order, tree, truncation, limit, canonical-cut, and CD-completion theorems (Theorems 2.24, 2.26, 2.41, 2.45, 2.72, 2.77) are proved in the text from this definition and from standard ordinal facts. The load-bearing field theorems (2.67-2.69) and the game-theoretic equivalence (Theorem 3.30) are not proved in full; the paper explicitly delegates them to [Sim], [ONAG], [S], and [Go]. That is a proof-obligation gap and a conditional claim, not circularity: the cited works are external, do not use the paper's conclusions as premises, and are not self-citations. The only self-citation is the aside in Section 0.10 crediting [Be08] in Ehrlich's overview; it plays no role in the derivation. No fitted parameter is renamed as a prediction, and no definition is circularly expressed in terms of the target theorem. The paper is honest that the arithmetic transfer is imported: 'In principle, everything is contained in [S]' (Section 3.3). Thus the appropriate finding is no significant circularity; the correctness risk about unproved transfer belongs to completeness and rigor, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The central contribution is a new presentation and some proofs; no empirical free parameters are involved. The main external load-bearing inputs are Conway's field arithmetic and game equivalence, plus Alling's uniqueness theorem. The speculative cocomplex numbers and the graded universe are flagged as tentative.

assumptions (5)
  • standard math Zermelo-Fraenkel set theory with transfinite induction and foundation provides the von Neumann universe and the class ON.
    Chapter 1 recalls the von Neumann hierarchy and transfinite induction as background; the whole construction lives in VN.
  • domain assumption Conway's cut-based definitions of addition, multiplication, and inverse on surreal numbers are valid and yield an ordered field.
    Theorems 2.67-2.69 are asserted with proofs delegated to [Sim] and [ONAG]; the paper's transfer to the new NO depends on these results.
  • domain assumption The equivalence between the new NO and Conway's game-theoretic construction (Theorem 3.30) holds as in Siegel's book [S].
    Section 3.3 says 'In principle, everything is contained in [S]' and the proof is sketched, not given.
  • domain assumption The graded von Neumann universe is independent of the chosen ordered-pair encoding.
    Section 3.14 asserts without proof that any ordered-pair definition gives the same intrinsic theory.
  • domain assumption Alling's full surreal number systems of height beta are isomorphic (Theorem 2.80).
    Section 2.9.2 cites [A] for this uniqueness result, used to claim equivalence of approaches.
invented entities (2)
  • Cocomplex numbers (Conway complex numbers) as sets of shuffle ordinals
    purpose: To realize a natural copy of the surcomplex field NO[i] inside VN without using Kuratowski ordered pairs and to guess a hexagonal tree structure.
    Definition 4.6 is labeled 'tentative and ad-hoc'; Table 4.1 is 'purely speculative' and no field operations are defined or tested.
  • Graded von Neumann universe with two membership relations
    purpose: To formalize partizan combinatorial games as a two-sorted set theory, providing a universe for Conway's games.
    Introduced as a definition in Section 3.2; its properties are developed in the paper, and it has no external observable consequences; it is a mathematical construction, not an empirical entity.

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Cite this review

Pith. "Pith review of On Conway's Numbers and Games, the Von Neumann Universe, and Pure Set Theory." pith.science (2026). https://pith.science/paper/BD7EX3OA

@misc{pith2026250104412,
  author       = {Pith},
  title        = {Pith review of: On Conway's Numbers and Games, the Von Neumann Universe, and Pure Set Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BD7EX3OA}},
  note         = {Machine review of arXiv:2501.04412}
}
read the original abstract

We take up Dedekind's question ''Was sind und was sollen die Zahlen?'' (''What are numbers, and would should they be?''), with the aim to describe the place that Conway's (Surreal) Numbers and Games take, or deserve to take, in the whole of mathematics. Rather than just reviewing the work of Conway, and subsequent one by Gonshor, Alling, Ehrlich, and others, we propose a new setting which puts the theory of surreal numbers onto the firm ground of ''pure'' set theory. This approach is closely related to Gonshor's one by ''sign expansions'', but appears to be significantly simpler and clearer, and hopefully may contribute to realizing that ''surreal'' numbers are by no means surrealistic, goofy or wacky. They could, and probably should, play a central role in mathematics. We discuss the interplay between the various approaches to surreal numbers, and analyze the link with Conway's original approach via Combinatorial Game Theory (CGT). To clarify this, we propose to call pure set theory the algebraic theory of pure sets, or in other terms, of the algebraic structures of the von Neumann universe. This topic may be interesting in its own right: it puts CGT into a broad context which has a strong ''quantum flavor'', and where Conway's numbers (as well as their analogue, the nimbers) arise naturally.

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Reference graph

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