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REVIEW 4 major objections 4 minor 4 references

Class of extensions of real field and their topological properties

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For every ordinal, the paper constructs an ordered field from the proper Dedekind cuts over a rational-like tower, and shows the full cut space is connected.

desk verdict The construction is a fresh ordinal-indexed variation on Dedekind cuts over ordered fields, but Theorem 20 is unproven because proper cuts are never shown closed under multiplication, and the paper itself admits products of cuts can fail. read the letter →

arxiv 2506.14838 v1 pith:OKS5EZCL submitted 2025-06-15 math.LO math.FA

E.V. Alexandrov This is my paper · ORCID
classification math.LOmath.FA MSC 28E15
keywords Dedekindcutsorderedfieldsordinalarithmeticnaturaladditionandmultiplicationnon-Archimedeaninfinitesimalsconnectednessinordertopologysurrealnumbers
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

The paper sets out to build an ordinal-indexed tower of extensions of the real numbers, using cuts over rational-like ordered fields. Its central claim is that for every ordinal $\alpha$, the proper Dedekind cuts of the extension $VQ_\alpha$ form a linearly ordered field, while the full space of all cuts is connected and is not closed under addition and multiplication. Each level embeds into the next, so the construction yields a nested tower of ordered fields running through all ordinals. The ordinary reals sit inside every stage, and the new fields contain infinitesimal and infinitely large elements. If the construction is correct, it gives a uniform, explicitly defined hierarchy of ordered field extensions with controlled topological behavior.

What carries the argument

The machinery is the distinction between proper and improper Dedekind cuts on the ordered field $VQ_\alpha$. A cut is improper when the balls that meet both sides of the cut have diameters bounded below by some positive element of $VQ_\alpha$; proper means no such positive gap exists. The field claim is carried by showing that opposite and reciprocal cuts of proper cuts are again proper, and by defining addition and multiplication on cut components. The topology is generated by open balls, and the fact that every bounded set in the full cut space has a supremum inside it is what makes the full space connected.

What would settle it

Take two proper positive cuts on some $VQ_\alpha$, form their product by multiplying the lower sets and upper sets, and check the four Dedekind-cut conditions on the resulting pair. In particular, compute whether the lower product has a largest element or whether the complement of the union of the two products contains more than one point; either outcome would show that the proper cuts are not closed under multiplication and would refute the field claim.

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Extended reading notes

Core claim

The central discovery is Theorem 20: for every ordinal $\alpha$, the set of all proper elements of the Dedekind cuts on $VQ_\alpha$ is a linearly ordered field with the inherited cut operations and order. The construction begins with ordinal-indexed number systems built from natural ordinal arithmetic, then forms integer-like and rational-like quotients, and finally takes Dedekind cuts. The full cut space is a connected order-topological space, contains improper elements once $\alpha>0$, and is not a field; deleting the improper elements leaves the proper subspace, which is disconnected whenever $\alpha>0$. Thus one construction produces two different structures at every level: a connected ambient space and a field inside it.

Load-bearing premise

The load-bearing premise is that the product of two proper Dedekind cuts is again a proper Dedekind cut; the paper proves this closure for sums but does not prove it for products, and without it the proper cuts cannot be certified as a field.

Editorial extensions

If this is right

  • If the paper is right, every ordinal yields a linearly ordered field extending the previous stage, so the construction is an ordinal-indexed tower of ordered fields.
  • At each level the full cut space is connected while the proper field is disconnected for positive ordinals, so topological connectedness is exactly what distinguishes the ambient space from the field inside it.
  • The ordinary real numbers embed into every stage, and each nontrivial stage contains positive elements smaller than every positive real, i.e. infinitesimals.
  • The paper states as a conjecture that the union of all proper stages is isomorphic to the class of surreal numbers; if correct, this connects the construction to a known maximal ordered field.

Reading between the lines

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

  • A direct next step would be to close the product-closure gap: prove that the product of two positive proper cuts satisfies the four cut conditions for its lower and upper sets; the present text proves this for sums but stops short for products.
  • A testable consequence of the tower is that each nontrivial stage is non-Archimedean over the reals, so one can ask whether the order type of each proper stage matches the order type of the corresponding initial segment of surreal numbers; this would operationalize the paper's conjecture.
  • The connected/field split suggests a general recipe: start with a connected space of cuts, carve out exactly those cuts whose boundary is visible at some positive scale, and the remaining points form an ordered field whenever inverses and opposites stay proper.
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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 / 4 minor

Summary. The paper constructs ordinal-indexed extensions VNα, VZα, VQα, VRα and their W-counterparts, using natural operations on ordinals and Dedekind cuts on VQα. The main claims are that the full cut space VRα is connected but not closed under addition and multiplication, while the subspace VRα of 'proper' cuts is a linearly ordered field and is disconnected. The paper also conjectures that the union VR∞ is isomorphic to the surreal numbers. The central algebraic claim is Theorem 20, which asserts that VRα is a linearly ordered field.

Significance. If the construction were correct, it would give a proper-class-indexed tower of field extensions of the reals with controlled topological connectedness, potentially useful for building measures sensitive to Lebesgue null sets. The idea of separating 'proper' Dedekind cuts from 'improper' ones in a non-Archimedean ordered field is original, and the topological dichotomy is appealing. However, the paper is a preliminary draft: the field theorem is not proved, several essential existence assertions are left unjustified, and the topological proofs are too sketchy to verify. The paper does not currently meet the standard for publication.

major comments (4)
  1. [§7.1, Definition 39 and Theorem 20] Theorem 20 asserts that VRα, the set of proper Dedekind cuts, is a linearly ordered field with 'inherited' operations. This requires that addition and multiplication are total binary operations on proper cuts and return proper cuts. Addition is addressed by property 6 of Definition 33, but no analogous closure proof for multiplication is given. Definition 37 defines the product of arbitrary-sign cuts via products of positive cuts, yet item 1 of the 'Properties of multiplication' explicitly states that the product of two Dedekind cuts need not be a Dedekind cut. The paper never proves that for proper (A,B) and (C,D), the pair (A·C, B·D) satisfies Definition 24 and is proper. Theorem 17 only treats x·(1/x) for a single proper x, and its proof relies on an unproved and questionable lemma about reciprocals of sets. Since the field axioms require multiplication to be defined on all pairs of proper cuts, the missing closure proof is load-bearing and Theorem 20 is not established.
  2. [§6.2, Theorem 10] The proof of single-valuedness of the embedding VRα → VRβ assumes the existence of an element δ ∈ VQβ+ such that |δ| < |p| for every nonzero p ∈ VQα (equation (6.12)). This is a substantive assertion about the hierarchy of fields VQα and is not proved anywhere in the paper; it is essentially an extra axiom. Without a proof of the existence of such an infinitesimal relative to VQα, the embedding theorem is unsupported.
  3. [§6.3, Theorem 13] The proof of connectedness of VRα is only a few sentences and is not a valid topological argument. It claims that every open set other than ∅ and VRα has non-empty boundary 'from theorem 12', but Theorem 12 only asserts existence of suprema and infima for bounded sets; it does not show that there are no nontrivial clopen sets. A correct proof must establish that the order topology on the full Dedekind completion is connected, which is a standard but non-obvious fact that needs to be demonstrated in this setting.
  4. [§6.4, Theorems 16 and 17] The proofs of the additive and multiplicative inverse theorems are incomplete. Theorem 16 asserts x + (−x) = 0 for proper x, but the proof does not verify that the constructed sum equals the zero cut; it simply states that 'from definition' the result follows. Theorem 17's proof of the lemma 'if inf(S) is proper, then sup(1/S) is proper' uses the estimate (1/x − 1/y) = (y−x)/(x·y) and claims that x·y is 'limited from above'. This is not justified when x and y are infinitesimal: in that case x·y is an infinitesimal and 1/(x·y) is infinite, so the quotient may not be small. Thus the lemma, which is essential for showing that 1/x is proper, is not established.
minor comments (4)
  1. [§2 and Definition 24] The notation is inconsistent: the symbol Q is used both for the rationals and, in Definition 32, for a set of balls. Also, Definition 24's condition 3 uses inf(B) and sup(A), but it should be stated explicitly in which order these are taken when the ambient order is not complete.
  2. [Definition 32] The definition of improper elements is confusing because the symbol B is used both for the second component of a cut and for a ball in the set Q. This makes the displayed formula B∈Q ⇔ (A∩B≠∅ ∧ B∩B≠∅) hard to read; should be rewritten with distinct letters.
  3. [§7.2, Theorem 21] The proof that VRα is disconnected for α > 0 claims that the set S of infinitesimals has empty border because its border consists of improper elements, but this is not proved. Also, the set S as defined by |x| < |y| for all y∈R presupposes an embedding of R into VRα and that nontrivial infinitesimals exist; both need to be established before this proof goes through.
  4. [Throughout] The manuscript contains many typos and grammatical errors (e.g., 'proove', 'anough', 'lineary', 'Dedeking') that should be corrected before any resubmission.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper derives its claims from its definitions and standard external references, with no fit-to-input or self-citation chain forcing the conclusions.

full rationale

The paper constructs VRα as the full set of Dedekind cuts on VQα and VRα as the subset of proper cuts. The connectedness of VRα (Theorem 13) follows from the order-completeness of the cut space proved in Theorem 12; it is a standard consequence of the Dedekind completion, not an assumption built into the definition. The disconnectedness of VRα (Theorem 21) rests on the existence of improper elements (Theorem 15) and on the claim that the border of the infinitesimal set S consists of improper elements; this follows from the topological definitions, even if the proof is terse. The field claim (Theorem 20) is asserted using inherited operations and the invertibility theorems (Theorems 16 and 17); while the proof leaves closure under multiplication unproven, this is a correctness gap rather than a circular step: the field axioms are not embedded in the definition of 'proper' by construction. The paper relies on external references for ordinal arithmetic and topology; there are no self-citations carrying the argument. The only self-referential statement, the suggestion that VR∞ is isomorphic to the surreals, is explicitly unproven and not load-bearing. No parameter fitting, no prediction from fitted inputs, and no renaming of a known result as a new derivation occurs. Consequently, no specific circularity can be exhibited, and the honest finding is no meaningful circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 2 invented entities

The construction pulls standard ordinal arithmetic and NBG set theory from the literature, plus one ad hoc infinitesimal-existence assumption in the embedding theorem. It introduces no fitted numerical parameters. The main invented entities are the proper/improper cut classification and the conjectured union VR∞; neither has independent evidence outside the paper.

assumptions (3)
  • domain assumption NBG class theory and transfinite recursion over Ord
    Section 3: 'Everywhere in his document will be used axiomatics of Von Neumann-Bernays-Gödel.' The paper uses proper classes and recursively defined ordinal functions throughout.
  • standard math Natural ordinal addition, multiplication, and Cantor normal form
    Definitions 5 and 6 and equation (3.2) import natural operations and Cantor normal form from references [1]-[3]; Theorems 1 and 2 depend on them.
  • ad hoc to paper VQβ contains an infinitesimal δ smaller than every positive element of VQα for α<β
    Equation (6.12) in Theorem 10 states 'Because exists δ...' without proof; the embedding theorem's single-valuedness depends on this.
invented entities (2)
  • Improper Dedekind cuts
    purpose: Distinguish the full cut space VRα (connected) from the subfield VRα of proper cuts (disconnected).
    Defined via the infimum of ball diameters (Definition 32); no externally testable prediction is attached, so the classification is internal to the paper.
  • VR∞ = union of VRα
    purpose: Candidate proper-class extension conjecturally isomorphic to surreal numbers.
    Explicitly unproven ('this claim was not proven'); no falsifiable characterization given.

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Cite this review

Pith. "Pith review of Class of extensions of real field and their topological properties." pith.science (2026). https://pith.science/paper/OKS5EZCL

@misc{pith2026250614838,
  author       = {Pith},
  title        = {Pith review of: Class of extensions of real field and their topological properties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OKS5EZCL}},
  note         = {Machine review of arXiv:2506.14838}
}
read the original abstract

Proper classes of extensions of real field was defined and topological properties of these extensions were studied. These extensions can be connected, in this case such set is not closed under binary operations (addition and multiplication), and not connected, in this case this extension is linearly ordered field. In the future these constructions can be applied to building measure that "feels" set of zero Lebesgue measure.

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Works this paper leans on

4 extracted references · 4 canonical work pages

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    Kuratowski and A

    K. Kuratowski and A. Mostowski.Теория множеств. Russian. Мир, 1970

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    Harry Altman.Intermediate arithmetic operations on ordinal numbers. 2017. arXiv:1501.05747 [math.LO]

    work page Pith review arXiv 2017

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    John. L. Kelley.Общая топология. Russian. 2009. 29

    work page 2009

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