arxiv: 2605.13102 · v1 · submitted 2026-05-13 · 🧮 math.HO

Recognition: 2 theorem links

· Lean Theorem

A philosophical history of infinitesimals

Karl Kuhlemann, Mikhail G. Katz, Taras Kudryk, Vladimir Kanovei

Pith reviewed 2026-05-14 01:44 UTC · model grok-4.3

classification 🧮 math.HO

keywords infinitesimalsLeibnizringinalsnonstandard analysisZF set theoryaxiom of choicephilosophy of mathematicshistory of analysis

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The pith

Leibnizian infinitesimals can be formalized rigorously in a choice-free conservative extension of ZF set theory.

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

The paper proposes a framework for infinitesimal analysis based on a new concept called ringinals. These are infinite numbers treated arithmetically, distinct from Cantorian ordinals and cardinals. It shows that such a theory can be developed within a conservative extension of Zermelo-Fraenkel set theory, formalizing Leibniz's definitions and principles without relying on the axiom of choice or ultrafilters. This approach challenges traditional philosophical perspectives that view infinitesimals as requiring nonstandard models dependent on advanced set-theoretic tools. A reader might care because it offers a potentially more elementary and historically faithful foundation for nonstandard analysis.

Core claim

The paper claims that a recent theory of infinitesimal analysis formalizes Leibnizian definitions and heuristic principles in a conservative extension of ZF set theory, using ringinals as arithmetic infinite numbers, while avoiding the axiom of choice and ultrafilters, thereby challenging received views on the nature of infinitesimals.

What carries the argument

Ringinals, which are infinite numbers of an arithmetic nature different from ordinals and cardinals, serving as the basis for a trichotomy with ordinals and cardinals to analyze the continuum and infinitesimals.

If this is right

The continuum need not be identified exclusively with the real numbers R, allowing room for infinitesimals.
Analysis involving unlimited numbers can be carried out using the 'standard' predicate in a conservative extension of ZF.
Philosophical objections to infinitesimals based on the need for choice or ultrafilters are undermined.
Leibnizian heuristic principles receive a rigorous formalization in this setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This framework could enable the development of infinitesimal methods in educational contexts without advanced set theory prerequisites.
Future work might explore whether specific calculus theorems can be proved using ringinals in a choice-free manner.
Connections to other conservative extensions or predicative approaches in mathematics could be investigated to test compatibility.

Load-bearing premise

That the introduced ringinals provide a coherent and consistent arithmetic framework for infinitesimals within a conservative extension of ZF set theory.

What would settle it

Demonstrating an inconsistency when attempting to formalize a specific Leibnizian principle, such as the definition of continuity or derivative using ringinals, in the proposed theory.

Figures

Figures reproduced from arXiv: 2605.13102 by Karl Kuhlemann, Mikhail G. Katz, Taras Kudryk, Vladimir Kanovei.

**Figure 1.** Figure 1: Sonar’s analysis of hornangles Knobloch’s interpretation of Leibniz ultimately leads him to attribute contraditions to Leibniz where there may be none. As analyzed in [88], there is no contradiction between Theorems 11 and 45 in Leibniz’s De Quadratura Arithmetica, as they deal with different notions of infinity: Theorem 11 deals with the infinitum terminatum, whereas Theorem 45 deals with an ideal persp… view at source ↗

read the original abstract

We explore the issue of providing a foundational framework for Leibnizian infinitesimals in the light of modern standard and nonstandard approaches. We outline a trichotomy of ordinals, cardinals and ringinals as a historiographic tool. A ringinal is a concept of infinite number, arithmetic in nature, different from Cantor's transfinite ordinals and cardinals. The continuum is not necessarily identifiable with R; even if one seeks such an identification, infinitesimals are not ruled out. Analysis with unlimited numbers (via the predicate standard) is possible in a conservative extension of Zermelo-Fraenkel set theory and in this sense is epistemologically 'safe'. We sketch a recent theory of infinitesimal analysis that formalizes Leibnizian definitions and heuristic principles while eschewing both the axiom of choice and ultrafilters, thus challenging received philosophical views on the nature of infinitesimals.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper sketches ringinals as an arithmetic infinite number to formalize Leibnizian infinitesimals in a conservative ZF extension without choice or ultrafilters, but the details to check coherence are thin.

read the letter

The main point is that the authors introduce ringinals as a third category of infinite numbers, arithmetic in character and separate from ordinals and cardinals, to give a foundation for Leibniz-style infinitesimals. They argue this can be done in a conservative extension of ZF that uses a standard predicate for unlimited numbers, keeps the continuum flexible, and avoids both the axiom of choice and ultrafilters. This is presented as a way to challenge views that modern set theory rules out such infinitesimals on philosophical grounds. The trichotomy is offered as a historiographic device to separate these notions cleanly. The sketch claims to capture Leibnizian definitions and heuristics while remaining epistemologically safe through conservativity. That is the core new element. The paper does a reasonable job keeping the proposal lightweight and tied to historical principles rather than heavy machinery. It correctly notes that identifying the continuum with R does not automatically exclude infinitesimals if the right extension is used. The approach stays within standard ZF plus a predicate, which is a clear choice. The soft spot is that the abstract and sketch give no explicit axioms for ringinals, no sample operations, and no outline of the conservativity argument. Without those steps it is hard to see whether the construction is consistent or whether it quietly assumes something about the continuum that would undermine the claim. This is a moderate gap for a paper that positions itself as a sketch, but it does leave the central challenge to received views resting on an unshown piece. The work is aimed at philosophers of mathematics and historians of analysis who care about alternatives to standard nonstandard methods. A reader wanting concrete derivations or applications will find it limited, but the engagement with foundations and Leibniz is direct enough to merit a serious referee who can ask for the missing details.

Referee Report

2 major / 2 minor

Summary. The paper explores foundational frameworks for Leibnizian infinitesimals by outlining a trichotomy of ordinals, cardinals, and ringinals as a historiographic tool. It defines ringinals as an arithmetic concept of infinite numbers distinct from Cantor's transfinites, argues that the continuum need not be identified with the reals (and that infinitesimals are not thereby ruled out), and sketches a theory of infinitesimal analysis formalizing Leibnizian definitions and heuristics in a conservative extension of ZF set theory without the axiom of choice or ultrafilters, thereby challenging received philosophical views on the nature of infinitesimals.

Significance. If the sketched construction of ringinals and the associated conservative extension hold, the work would supply an epistemologically 'safe' (conservative) route to Leibnizian-style infinitesimals that avoids ultrafilters and choice, potentially reframing philosophical debates on the legitimacy of non-Archimedean analysis. The trichotomy offers a novel historiographic lens for distinguishing arithmetic from set-theoretic infinities. However, the absence of explicit axioms, operations, or a conservativity argument for ringinals substantially reduces the immediate significance of these claims.

major comments (2)

[Abstract and section introducing ringinals] The central claim that ringinals furnish a coherent arithmetic framework for infinitesimals (distinct from ordinals and cardinals) and integrate with a conservative ZF extension is load-bearing for the challenge to received views, yet the manuscript provides only a sketch mentioning the trichotomy and 'standard' predicate without explicit axioms, arithmetic operations, or a conservativity proof.
[Section on conservative extension and the 'standard' predicate] The assertion that analysis with unlimited numbers via the 'standard' predicate is possible in a conservative extension of ZF (and is therefore epistemologically safe) rests on an unverified step; no derivation, model construction, or error analysis is supplied to support this.

minor comments (2)

[Abstract] The phrase 'we sketch a recent theory' appears without a specific citation or reference to the source work being summarized.
[Introduction] Notation for the 'standard' predicate and the distinction between ringinals and other infinities would benefit from a brief formal definition or example early in the text to aid readability for historians and philosophers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We appreciate the acknowledgment of the potential significance of the trichotomy as a historiographic tool and the conservative extension for Leibnizian infinitesimals. We address each major comment below, indicating planned revisions to strengthen the manuscript while preserving its philosophical and historical focus.

read point-by-point responses

Referee: [Abstract and section introducing ringinals] The central claim that ringinals furnish a coherent arithmetic framework for infinitesimals (distinct from ordinals and cardinals) and integrate with a conservative ZF extension is load-bearing for the challenge to received views, yet the manuscript provides only a sketch mentioning the trichotomy and 'standard' predicate without explicit axioms, arithmetic operations, or a conservativity proof.

Authors: The paper is framed as a philosophical and historiographic exploration that outlines the trichotomy to distinguish arithmetic conceptions of infinity (ringinals) from set-theoretic ones (ordinals and cardinals), rather than a complete technical treatise. Ringinals are presented as an arithmetic notion extending the integers in a ring structure, with the 'standard' predicate enabling Leibnizian heuristics. We agree that additional explicitness would better support the central claims. In revision, we will expand the relevant section to include basic axioms for the ringinal structure (e.g., closure under addition and multiplication extending Z) and a concise description of key operations, while retaining references to the full construction in the cited recent theory for complete details. This addresses the concern without shifting the paper's primary focus. revision: partial
Referee: [Section on conservative extension and the 'standard' predicate] The assertion that analysis with unlimited numbers via the 'standard' predicate is possible in a conservative extension of ZF (and is therefore epistemologically safe) rests on an unverified step; no derivation, model construction, or error analysis is supplied to support this.

Authors: The conservativity claim rests on the extension being formulated so that the 'standard' predicate introduces no new theorems about standard sets, consistent with the avoidance of choice and ultrafilters in the sketched theory. We acknowledge that the manuscript provides only a high-level indication rather than a self-contained derivation. In the revised version, we will add a brief subsection outlining the model construction at a high level (via a suitable non-ultrafilter-based enlargement) and the key steps establishing conservativity over ZF, thereby making the epistemological safety more explicit while keeping the treatment accessible to the paper's intended audience. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained in conservative ZF extension

full rationale

The paper sketches a historiographic trichotomy of ordinals, cardinals and ringinals and outlines infinitesimal analysis in a conservative extension of ZF without AC or ultrafilters. No load-bearing step reduces by construction to its own inputs: the conservativity claim is asserted as an independent property of the extension rather than derived from the target Leibnizian results, and no equations, fitted parameters, or self-citations are shown to force the central conclusions. The framework is presented as epistemologically safe precisely because it avoids self-referential definitions or unverified uniqueness theorems from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard ZF set theory plus the new ringinal concept and the conservative extension property, which are outlined but not independently evidenced beyond the paper's framework.

axioms (2)

standard math Zermelo-Fraenkel set theory (ZF)
Base theory for the conservative extension mentioned in the abstract.
domain assumption Existence of a 'standard' predicate for distinguishing limited and unlimited numbers
Invoked to enable analysis with unlimited numbers in the sketched theory.

invented entities (1)

ringinal no independent evidence
purpose: Arithmetic concept of infinite number distinct from ordinals and cardinals for modeling Leibnizian infinitesimals
Newly postulated as part of the trichotomy to provide a historiographic and foundational tool.

pith-pipeline@v0.9.0 · 5446 in / 1426 out tokens · 43897 ms · 2026-05-14T01:44:27.038346+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
A ringinal is a concept of infinite number, arithmetic in nature, different from Cantor's transfinite ordinals and cardinals... Analysis with unlimited numbers (via the predicate standard) is possible in a conservative extension of Zermelo-Fraenkel set theory
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
SPOT has the following three axioms... T(Transfer)... O(Nontriviality)... SP(Standard Part)

Reference graph

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