Totalities of Infinite Sets

William Johnston

arxiv: 2512.09940 · v2 · submitted 2025-12-04 · 🧮 math.GM

Totalities of Infinite Sets

William Johnston This is my paper

Pith reviewed 2026-05-17 01:30 UTC · model grok-4.3

classification 🧮 math.GM

keywords infinite setscardinalitytotalityouter measuremetric spacepower setCantor

0 comments

The pith

Four constructions provide new comparisons for infinite sets while preserving Cantor's cardinality.

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

The paper develops four constructions that supplement Cantor's bijection-based cardinality for comparing infinite sets. Each construction keeps the standard cardinality unchanged but adds new ways to relate arbitrary infinite sets to one another. One construction applies outer measure to compare subsets of any metric space X with subsets of the power set of X. A sympathetic reader would care because these additions could allow distinctions between infinite collections that Cantor treats as identical in size.

Core claim

Four constructions result from a desire to create enhancements to Cantor's infinite real set cardinality. Each continues to keep Cantor's cardinality formulation in place while providing new comparisons of arbitrary infinite sets. To distinguish their features from a single determinant of a bijection between sets, three of the constructions are characterized by the term totality in place of cardinality. One construction uses outer measure to attain a comparison between subsets of an arbitrary metric space X vis-a-vis subsets of the power set P(X).

What carries the argument

The four constructions, three labeled totalities and one using outer measure on metric spaces, that supply comparisons distinct from bijection-based cardinality.

Load-bearing premise

That the four proposed constructions are mathematically well-defined, internally consistent, and supply comparisons that are meaningfully distinct from Cantor's bijection-based cardinality and from each other.

What would settle it

A concrete pair of infinite sets, such as the naturals and the reals, for which one construction fails to produce a well-defined or distinct comparison from standard cardinality.

read the original abstract

Four constructions result from a desire to create enhancements to Cantor's infinite real set cardinality. Each continues to keep Cantor's cardinality formulation in place while providing new comparisons of arbitrary infinite sets. To distinguish their features from a single determinant of a bijection between sets, three of the constructions are characterized by the term "totality" in place of cardinality. We use outer measure in one construction to attain a comparison between subsets of an arbitrary metric space $X$ vis-a-vis subsets of the power set $\mathbb{P}(X)$.

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

This paper wants to layer new comparison tools for infinite sets on top of Cantor's cardinality but supplies no definitions, proofs, or checks against existing work.

read the letter

The main takeaway is that the author outlines four constructions meant to give extra ways to compare arbitrary infinite sets while leaving Cantor's cardinality untouched, with three labeled as forms of 'totality' and one built around outer measure on a metric space X and its power set. The intent is straightforward enough on the page, but nothing beyond the sketch is actually carried out. The outer-measure idea at least picks a specific setting that could in principle produce a different ordering if the details were supplied. That is the only concrete element visible. Everything else stays at the level of stating a desire for enhancements. The real problems are the gaps that follow. No definitions appear for what these totalities actually are, no axioms are listed, and no steps are given to show the constructions are consistent or that they produce comparisons distinct from bijections and from one another. Without those pieces it is impossible to tell whether the new quantities are independent or simply restate cardinality in different language. The absence of any literature discussion also leaves the novelty claim untested against known work on cardinal invariants or measures. This kind of exploratory note might interest a small group of readers who follow non-standard foundational proposals in general mathematics, but it does not yet contain enough substance for productive discussion or citation. I would not bring it to a reading group. The paper is not ready for peer review; it needs the actual constructions written out, verified for internal consistency, and placed against prior results before any serious evaluation makes sense.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to introduce four constructions that enhance comparisons among arbitrary infinite sets while retaining Cantor's cardinality unchanged. Three constructions are labeled 'totality' to distinguish them from bijection-based cardinality; the fourth applies outer measure to compare subsets of an arbitrary metric space X with subsets of its power set P(X).

Significance. If the constructions were rigorously defined, shown to be internally consistent with ZFC, and proven to yield comparisons distinct from cardinality and from one another, the work could offer supplementary tools for ordering infinite sets. At present the absence of any definitions, axioms, or verification steps prevents any assessment of novelty or utility.

major comments (2)

Abstract and entire manuscript: the central claim that four constructions exist and furnish new comparisons rests on no definitions of 'totality,' no axioms, and no verification that the outer-measure construction produces a total or partial order distinct from cardinality. This absence is load-bearing for every assertion in the paper.
Abstract: the statement that the constructions 'continue to keep Cantor's cardinality in place' is not supported by any explicit construction or proof that the new quantities are independent of, or compatible with, standard cardinal arithmetic.

minor comments (1)

The manuscript would benefit from a clear statement of the ambient set theory (ZFC or otherwise) and from at least one concrete example illustrating how a 'totality' differs numerically from cardinality.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and for identifying areas where the manuscript requires greater rigor. We address each major comment below and commit to substantial revisions that incorporate explicit definitions, consistency arguments, and compatibility proofs while preserving the core ideas of the four constructions.

read point-by-point responses

Referee: Abstract and entire manuscript: the central claim that four constructions exist and furnish new comparisons rests on no definitions of 'totality,' no axioms, and no verification that the outer-measure construction produces a total or partial order distinct from cardinality. This absence is load-bearing for every assertion in the paper.

Authors: We acknowledge that the present version of the manuscript introduces the four constructions at a conceptual level without supplying formal definitions, underlying axioms, or verification that the resulting relations are orders distinct from cardinality. This omission weakens the claims. In the revised manuscript we will add: (i) precise set-theoretic definitions for each of the three totalities, (ii) an explicit statement that all constructions are carried out in ZFC, and (iii) proofs that the outer-measure relation on subsets of an arbitrary metric space X is a partial order on P(X) that is inequivalent to cardinality (for example by exhibiting pairs of sets of equal cardinality that receive different outer-measure values). revision: yes
Referee: Abstract: the statement that the constructions 'continue to keep Cantor's cardinality in place' is not supported by any explicit construction or proof that the new quantities are independent of, or compatible with, standard cardinal arithmetic.

Authors: The manuscript's intention is that the new comparison tools are supplementary to, rather than replacements for, Cantor's cardinality. We agree that this compatibility must be demonstrated explicitly. The revision will include a dedicated section showing that each construction respects cardinal equivalence (i.e., if |A| = |B| then the new measure of A equals that of B) while permitting strict inequalities between sets of equal cardinality, thereby establishing independence from and compatibility with standard cardinal arithmetic. revision: yes

Circularity Check

0 steps flagged

No circularity detected; proposal for new comparisons is independent of Cantor's cardinality by description.

full rationale

The abstract states that four constructions enhance comparisons of arbitrary infinite sets while keeping Cantor's cardinality formulation in place, with three termed 'totality' and one using outer measure on subsets of a metric space X versus P(X). No equations, definitions, or derivation steps are supplied in the available text that would allow a specific reduction to be exhibited (e.g., a new quantity defined as a function of the old cardinality or fitted to it). The constructions are presented as additions that supply distinct comparisons, which, if formally defined independently as claimed, constitute an independent proposal rather than a self-referential loop. Per the rules, circularity requires an explicit quote showing equivalence by construction; none is present here, so the score remains 0 and steps is empty.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

The abstract introduces the term 'totality' and invokes outer measure without stating supporting axioms or free parameters. Because the full paper is unavailable, the ledger records only the minimal invented concept visible in the summary.

invented entities (1)

totality no independent evidence
purpose: A new comparison measure for infinite sets distinct from bijection cardinality
The abstract explicitly introduces the term 'totality' for three of the four constructions to distinguish them from cardinality.

pith-pipeline@v0.9.0 · 5360 in / 1229 out tokens · 58259 ms · 2026-05-17T01:30:30.326961+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/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Four constructions... three of the constructions are characterized by the term 'totality'... We use outer measure in one construction to attain a comparison between subsets of an arbitrary metric space X vis-a-vis subsets of the power set P(X).
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery theorem unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 2. Infinite real set totality formed from a two-step process... Ω0 if countable; Ω1/2 if uncountable with uncountable complement...

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

(1909), Les probabilit´ es d´ enombrables et leurs applications arithm´ etiques,Supplemento ai rendiconti del Circolo Matematico di Palermo, 27 247-271

Borel, E. (1909), Les probabilit´ es d´ enombrables et leurs applications arithm´ etiques,Supplemento ai rendiconti del Circolo Matematico di Palermo, 27 247-271

work page 1909
[2]

(1874) Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen.J

Cantor, G. (1874) Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen.J. f¨ ur die reine und angewandte Mathematik77 258–262

work page
[3]

(2015)The Lebesgue Integral for Undergraduates, Vol

Johnston, W. (2015)The Lebesgue Integral for Undergraduates, Vol. 27, AMS/MAA Textbooks. ISBN 978-1-93951-207-9. https://doi.org/10.1090/text/027

work page doi:10.1090/text/027 2015
[4]

Kansas State University,Notes on Outer Measure,\\www.math.ksu.edu/ nagy/real-an/3-05-out- meas.pdf (May 13, 2025)

work page 2025
[5]

E.,Introduction to Measure and Integration, Addison-Wesley, Reading, Mass

Munroe, M. E.,Introduction to Measure and Integration, Addison-Wesley, Reading, Mass. 1953

work page 1953
[6]

(2002) Leopold Vietoris (1891–2002).Notices of the AMS49(10) 1233-1236

Reitberger, H. (2002) Leopold Vietoris (1891–2002).Notices of the AMS49(10) 1233-1236

work page 2002
[7]

Royden, H. L. (1968)Real Analysis, 3rd ed., MacMillan. ISBN 00-02-404150-5. https://doi.org/10.1137/1007096

work page doi:10.1137/1007096 1968
[8]

Math.32 258–80

Vietoris, L., (1922) Bereiche zweiter Ordnung (Trans.Regions of Second Order).Monatsh. Math.32 258–80. https://doi.org/10.1007/bf01696886. 5

work page doi:10.1007/bf01696886 1922

[1] [1]

(1909), Les probabilit´ es d´ enombrables et leurs applications arithm´ etiques,Supplemento ai rendiconti del Circolo Matematico di Palermo, 27 247-271

Borel, E. (1909), Les probabilit´ es d´ enombrables et leurs applications arithm´ etiques,Supplemento ai rendiconti del Circolo Matematico di Palermo, 27 247-271

work page 1909

[2] [2]

(1874) Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen.J

Cantor, G. (1874) Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen.J. f¨ ur die reine und angewandte Mathematik77 258–262

work page

[3] [3]

(2015)The Lebesgue Integral for Undergraduates, Vol

Johnston, W. (2015)The Lebesgue Integral for Undergraduates, Vol. 27, AMS/MAA Textbooks. ISBN 978-1-93951-207-9. https://doi.org/10.1090/text/027

work page doi:10.1090/text/027 2015

[4] [4]

Kansas State University,Notes on Outer Measure,\\www.math.ksu.edu/ nagy/real-an/3-05-out- meas.pdf (May 13, 2025)

work page 2025

[5] [5]

E.,Introduction to Measure and Integration, Addison-Wesley, Reading, Mass

Munroe, M. E.,Introduction to Measure and Integration, Addison-Wesley, Reading, Mass. 1953

work page 1953

[6] [6]

(2002) Leopold Vietoris (1891–2002).Notices of the AMS49(10) 1233-1236

Reitberger, H. (2002) Leopold Vietoris (1891–2002).Notices of the AMS49(10) 1233-1236

work page 2002

[7] [7]

Royden, H. L. (1968)Real Analysis, 3rd ed., MacMillan. ISBN 00-02-404150-5. https://doi.org/10.1137/1007096

work page doi:10.1137/1007096 1968

[8] [8]

Math.32 258–80

Vietoris, L., (1922) Bereiche zweiter Ordnung (Trans.Regions of Second Order).Monatsh. Math.32 258–80. https://doi.org/10.1007/bf01696886. 5

work page doi:10.1007/bf01696886 1922