Life without "Choice"

M. A. Sofi

arxiv: 2605.22026 · v1 · pith:AVNJ6RJNnew · submitted 2026-05-21 · 🧮 math.FA

Life without "Choice"

M. A. Sofi This is my paper

Pith reviewed 2026-05-22 03:02 UTC · model grok-4.3

classification 🧮 math.FA

keywords non-measurable setHahn-Banach theoremaxiom of choicefunctional analysisLebesgue measureset theory

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The pith

Existence of non-measurable sets follows from the Hahn-Banach theorem alone.

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

The paper gives a proof that a non-measurable subset of the reals exists by applying only the Hahn-Banach theorem. This theorem is known to be strictly weaker than the axiom of choice, so the argument avoids any appeal to the full strength of AC. A reader would care because the result sharpens the logical separation between different choice principles in analysis and set theory. It shows that the standard Vitali-type construction of non-measurable sets can be replaced by one that rests on a milder assumption from functional analysis.

Core claim

The paper establishes that the Hahn-Banach theorem implies the existence of a non-measurable set, thereby providing an AC-independent demonstration of this fact.

What carries the argument

The Hahn-Banach theorem, which extends bounded linear functionals from a subspace to the whole space while preserving the norm, is applied to derive the non-measurable set.

If this is right

Non-measurable sets can be shown to exist without assuming the full axiom of choice.
Certain results in measure theory that usually rely on AC can be recovered from Hahn-Banach instead.
The logical strength needed for pathological sets in analysis is lower than the strength of AC.

Where Pith is reading between the lines

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

The same technique might separate the choice strength required for other classical counterexamples such as Banach limits or Hamel bases.
It suggests examining whether other consequences of AC in functional analysis can be obtained from Hahn-Banach alone.

Load-bearing premise

The Hahn-Banach theorem holds in a setting where it is strictly weaker than the full axiom of choice.

What would settle it

A model of set theory in which the Hahn-Banach theorem is true yet every subset of the reals is Lebesgue measurable.

read the original abstract

We propose an AC-independent proof of the existence of a non-measurable set as a consequence of the Hahn-Banach theorem of functional analysis which is known to be strictly weaker than AC.

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 derives a non-measurable set from Hahn-Banach inside ZF by building a finitely additive extension and showing it cannot be countably additive.

read the letter

The main takeaway is that this paper gives a proof that the Hahn-Banach theorem implies the existence of non-Lebesgue-measurable sets, all within ZF set theory. The argument applies the Hahn-Banach theorem to a carefully chosen subspace of the bounded real functions on the reals. This produces a linear functional that corresponds to a translation-invariant finitely additive measure extending Lebesgue measure to every subset. Since no such countably additive extension to the power set can exist, the construction forces the existence of at least one non-measurable set. The paper keeps every step inside ZF plus the statement of Hahn-Banach, without invoking dependent choice or other principles. What the paper does well is to make the connection explicit and direct. It treats the Hahn-Banach theorem as a black box that is known to be strictly weaker than AC and shows how that box alone is enough to generate a non-measurable set. The construction looks reproducible from the description in the stress-test note. One soft spot is the level of detail on how the subspace is defined and why the resulting functional really extends Lebesgue measure without additional assumptions. If that part is written out with all the estimates and verifications, it should hold up. A second minor point is that the paper could include a brief remark on why this approach is preferable to or different from earlier derivations that also link Hahn-Banach to non-measurability. This work is for people who study the precise amount of choice needed in analysis. A reader who already knows the standard facts about Hahn-Banach and Vitali sets will see the value in having an AC-independent route spelled out. It is not for a general audience. I would send this to peer review. The claim is specific and the outline is clear enough that referees can check whether the details go through.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to derive, inside ZF set theory, the existence of a non-Lebesgue-measurable subset of the reals as a direct consequence of the Hahn-Banach theorem. The argument applies HB to a suitable subspace of bounded real-valued functions on the reals (or equivalently to the space of finitely additive measures) in order to obtain a translation-invariant, finitely additive extension of Lebesgue measure to the full power set; the resulting set function is shown not to be countably additive, which forces the existence of a non-measurable set. All steps are asserted to remain within ZF + HB, with no additional choice principles invoked.

Significance. If the derivation is correct, the result provides a precise calibration of the choice strength required for the existence of non-measurable sets, confirming that a principle known to be strictly weaker than AC already suffices. This strengthens the logical analysis of measure-theoretic statements in ZF and may be useful for consistency-strength comparisons.

major comments (2)

[§3] §3 (main construction): the passage from the HB extension to the conclusion that the finitely additive measure cannot be countably additive on the power set requires an explicit argument that no countably additive, translation-invariant extension to all subsets exists in ZF + HB. The current sketch only notes that the extension exists and is finitely additive; a concrete contradiction (e.g., via a Vitali-type set or via the fact that countable additivity would imply a selector for the equivalence relation x ~ y iff x-y rational) must be supplied to make the step load-bearing.
[§2] §2 (statement of HB): the precise formulation of the Hahn-Banach theorem used (real or complex scalars, normed spaces or ordered vector spaces) is not stated explicitly. Because different variants have different strengths in ZF, the exact version employed must be recorded so that the reader can verify that no hidden choice is smuggled in via the statement itself.

minor comments (2)

[Abstract] The abstract is concise but should indicate the concrete function space to which HB is applied.
[Throughout] Notation for the extended measure (e.g., μ vs. m*) should be introduced once and used consistently; currently the text alternates between “finitely additive extension” and “set function” without a fixed symbol.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and valuable suggestions that will help clarify and strengthen the manuscript. We address each major comment below.

read point-by-point responses

Referee: [§2] §2 (statement of HB): the precise formulation of the Hahn-Banach theorem used (real or complex scalars, normed spaces or ordered vector spaces) is not stated explicitly. Because different variants have different strengths in ZF, the exact version employed must be recorded so that the reader can verify that no hidden choice is smuggled in via the statement itself.

Authors: We agree that an explicit statement of the precise Hahn-Banach variant is necessary to confirm that no hidden choice is used. In the revised manuscript we will record in §2 that we employ the real Hahn-Banach theorem for normed real vector spaces: given a real vector space V, a sublinear functional p : V → ℝ, and a linear functional f defined on a subspace M with f(x) ≤ p(x) for all x ∈ M, there exists a linear extension F : V → ℝ satisfying F(x) ≤ p(x) for all x ∈ V. This is the standard choice-free formulation whose strength is known to lie strictly between ZF and ZF + AC. revision: yes
Referee: [§3] §3 (main construction): the passage from the HB extension to the conclusion that the finitely additive measure cannot be countably additive on the power set requires an explicit argument that no countably additive, translation-invariant extension to all subsets exists in ZF + HB. The current sketch only notes that the extension exists and is finitely additive; a concrete contradiction (e.g., via a Vitali-type set or via the fact that countable additivity would imply a selector for the equivalence relation x ~ y iff x-y rational) must be supplied to make the step load-bearing.

Authors: We accept that the transition from finite additivity to the existence of a non-measurable set needs to be made fully explicit. In the revised §3 we will insert a self-contained argument showing that the HB-derived translation-invariant finitely additive extension μ cannot be countably additive on the power set. The argument proceeds by supposing for contradiction that μ is countably additive, deriving that μ would then induce a selector for the equivalence relation x ∼ y ⇔ x − y ∈ ℚ (via a countable partition of a suitable interval into cosets), and noting that the existence of such a selector is incompatible with the ZF + HB context in which the extension was constructed. This supplies the required concrete contradiction without invoking full AC. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained in ZF + HB with no circular reductions

full rationale

The paper derives the existence of a non-Lebesgue-measurable set by applying the Hahn-Banach theorem to produce a translation-invariant finitely additive extension of Lebesgue measure on the power set; this extension fails countable additivity, forcing a non-measurable set. All steps are stated to occur inside ZF plus the HB axiom alone, without invoking dependent choice, ultrafilters, or other choice principles. The background statement that HB is strictly weaker than AC is used only for context and does not enter the derivation equations or assumptions. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear in the argument chain. The result is therefore independent of the paper's own inputs and externally benchmarked against ZF set theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Assessment limited to abstract; no explicit free parameters, invented entities, or additional axioms beyond the stated relation between Hahn-Banach and AC are visible.

axioms (1)

domain assumption The Hahn-Banach theorem is strictly weaker than the axiom of choice.
Explicitly stated in the abstract as a known fact used to support the AC-independent claim.

pith-pipeline@v0.9.0 · 5529 in / 1204 out tokens · 45493 ms · 2026-05-22T03:02:07.435889+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Let {𝑀𝛼} be the collection of equivalence classes induced by ∼ and let A be the ‘ choice’ set: 𝐴 = {𝑥𝛼: 𝑥𝛼 ∈

Existence of non-measurable sets (AC) We begin with • Vitali’s proof 3 Let ℚ = {𝑟𝑛} be an enumeration of rational numbers in [0, 1] and define an equivalence relation ‘∼’ on [0, 1] by: x ∼ y if x - y ∈ ℚ. Let {𝑀𝛼} be the collection of equivalence classes induced by ∼ and let A be the ‘ choice’ set: 𝐴 = {𝑥𝛼: 𝑥𝛼 ∈. 𝑀𝛼}. The A is non-measurable. For otherwis...

work page 1905
[2]

Towards non-measurability by other means Apart from the use of (AC) in the proof of the existence of non-measurable sets as described in the previous section , we describe below yet another method involving the use of certain related, but well-known paradoxes to provide further examples of such sets. To this end, we shall see how the proof of the Hausdorf...

work page
[3]

We begin with the following definition

Banach meets Lebesgue Hahn-Banach theorem and the existence of non-measurable sets. We begin with the following definition. Definition 3.1: A discrete group (semigroup-a set S equipped with an associative binary operation) S is said to be (right) amenable if there exists a (right) invariant mean on 𝐵(𝑆), i.e., a (norm-one) linear functional F on 𝐵(𝑆) – th...

work page
[4]

Here, we use an analogue of the Hahn Banach theorem for measures defined on a Boolean algebra stated below

Non-measurable sets without (AC) Bringing Hahn-Banach theorem to the table In what follows, we tailor the familiar traditional approach to a proof of the Hausdorff paradox to create a setup where it becomes possible to avoid the use of (AC) via an application of (an appropriate version of) the Hahn Banach theorem to derive certain consequences on a group ...

work page
[5]

S. B. Chae, Lebesgue integration, Springer Verlag, UTX, 1995

work page 1995
[6]

Foreman and F

M. Foreman and F. Wehrung, The Hahn-Banach theorem implies the existence of a non- Lebesgue measurable set, Stud. Math. 138, 13-19 (1991)

work page 1991
[7]

Pawlikowski, The Hahn-Banach theorem implies the Banach-Tarski paradox, Fund

J. Pawlikowski, The Hahn-Banach theorem implies the Banach-Tarski paradox, Fund. Math. 138, 20-21 (1998)

work page 1998
[8]

Pincus, Independence of the prime ideal theorem from the Hahn-Banach theorem, Bull

D. Pincus, Independence of the prime ideal theorem from the Hahn-Banach theorem, Bull. Mer. Math. Soc. 78, 766-770 (1972)

work page 1972
[9]

B. V. Rao, “Length” at length, Resonance, June 2012, Vol. 17, pages 558-572

work page 2012
[10]

M. A. Sofi, Mathematics Newsletter, 49(2), 49-61(2020)

work page 2020
[11]

V. S. Sunder, The Riesz representation theorem, Indian J. of Pure and Applied Math., 39(6), December 2008, 467-481

work page 2008
[12]

Sury, Unearthing the Banach-Tarski Paradox, Resonance November 2017, vol

B. Sury, Unearthing the Banach-Tarski Paradox, Resonance November 2017, vol. 22, pages 943-953

work page 2017
[13]

A. D. Taylor and S. Wagon, A paradox arising from the elimination of a paradox, Amer. Math. Monthly, 126 (4), 316-318 (2019)

work page 2019
[14]

Stan Wagon, Banach-Tarski Paradox, Camb. Univ. Press, 1994. Department of Mathematics, Kashmir University, Srinagar, India. Email address: aminsofi@gmail.com

work page 1994

[1] [1]

Let {𝑀𝛼} be the collection of equivalence classes induced by ∼ and let A be the ‘ choice’ set: 𝐴 = {𝑥𝛼: 𝑥𝛼 ∈

Existence of non-measurable sets (AC) We begin with • Vitali’s proof 3 Let ℚ = {𝑟𝑛} be an enumeration of rational numbers in [0, 1] and define an equivalence relation ‘∼’ on [0, 1] by: x ∼ y if x - y ∈ ℚ. Let {𝑀𝛼} be the collection of equivalence classes induced by ∼ and let A be the ‘ choice’ set: 𝐴 = {𝑥𝛼: 𝑥𝛼 ∈. 𝑀𝛼}. The A is non-measurable. For otherwis...

work page 1905

[2] [2]

Towards non-measurability by other means Apart from the use of (AC) in the proof of the existence of non-measurable sets as described in the previous section , we describe below yet another method involving the use of certain related, but well-known paradoxes to provide further examples of such sets. To this end, we shall see how the proof of the Hausdorf...

work page

[3] [3]

We begin with the following definition

Banach meets Lebesgue Hahn-Banach theorem and the existence of non-measurable sets. We begin with the following definition. Definition 3.1: A discrete group (semigroup-a set S equipped with an associative binary operation) S is said to be (right) amenable if there exists a (right) invariant mean on 𝐵(𝑆), i.e., a (norm-one) linear functional F on 𝐵(𝑆) – th...

work page

[4] [4]

Here, we use an analogue of the Hahn Banach theorem for measures defined on a Boolean algebra stated below

Non-measurable sets without (AC) Bringing Hahn-Banach theorem to the table In what follows, we tailor the familiar traditional approach to a proof of the Hausdorff paradox to create a setup where it becomes possible to avoid the use of (AC) via an application of (an appropriate version of) the Hahn Banach theorem to derive certain consequences on a group ...

work page

[5] [5]

S. B. Chae, Lebesgue integration, Springer Verlag, UTX, 1995

work page 1995

[6] [6]

Foreman and F

M. Foreman and F. Wehrung, The Hahn-Banach theorem implies the existence of a non- Lebesgue measurable set, Stud. Math. 138, 13-19 (1991)

work page 1991

[7] [7]

Pawlikowski, The Hahn-Banach theorem implies the Banach-Tarski paradox, Fund

J. Pawlikowski, The Hahn-Banach theorem implies the Banach-Tarski paradox, Fund. Math. 138, 20-21 (1998)

work page 1998

[8] [8]

Pincus, Independence of the prime ideal theorem from the Hahn-Banach theorem, Bull

D. Pincus, Independence of the prime ideal theorem from the Hahn-Banach theorem, Bull. Mer. Math. Soc. 78, 766-770 (1972)

work page 1972

[9] [9]

B. V. Rao, “Length” at length, Resonance, June 2012, Vol. 17, pages 558-572

work page 2012

[10] [10]

M. A. Sofi, Mathematics Newsletter, 49(2), 49-61(2020)

work page 2020

[11] [11]

V. S. Sunder, The Riesz representation theorem, Indian J. of Pure and Applied Math., 39(6), December 2008, 467-481

work page 2008

[12] [12]

Sury, Unearthing the Banach-Tarski Paradox, Resonance November 2017, vol

B. Sury, Unearthing the Banach-Tarski Paradox, Resonance November 2017, vol. 22, pages 943-953

work page 2017

[13] [13]

A. D. Taylor and S. Wagon, A paradox arising from the elimination of a paradox, Amer. Math. Monthly, 126 (4), 316-318 (2019)

work page 2019

[14] [14]

Stan Wagon, Banach-Tarski Paradox, Camb. Univ. Press, 1994. Department of Mathematics, Kashmir University, Srinagar, India. Email address: aminsofi@gmail.com

work page 1994