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arxiv: 2604.19360 · v1 · submitted 2026-04-21 · 🧮 math.LO

Recognition: unknown

On boldsymbol{Sigma}¹₃- and Sigma¹₄-uniformization

Stefan Hoffelner

Pith reviewed 2026-05-10 00:56 UTC · model grok-4.3

classification 🧮 math.LO
keywords uniformizationdescriptive set theoryprojective hierarchyboldface propertieslightface propertiesforcingconsistencyinner models
0
0 comments X

The pith

There is a model of set theory where boldface Σ¹₃-uniformization holds but lightface Σ¹₄-uniformization fails.

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

The paper establishes that the boldface Σ¹₃-uniformization property and the lightface Σ¹₄-uniformization property are independent by constructing a model in which one holds and the other does not. A sympathetic reader would care because uniformization principles describe how definable relations on the reals can be selected as functions while controlling their definability complexity. The separation shows that these properties do not automatically transfer across the boldface-lightface distinction or across consecutive levels of the projective hierarchy. The result follows from the consistency of ZFC and uses either a forcing construction or inner models with large cardinals to achieve the separation.

Core claim

Assuming the consistency of ZFC, we construct a model of set theory in which the boldface Σ¹₃-uniformization property holds, yet the lightface Σ¹₄-uniformization property fails, separating these two principles for the first time. We also indicate how to create a universe where Σ¹₃-uniformization holds, but Σ¹₄-uniformization fails using inner models with large cardinals.

What carries the argument

A forcing extension or inner model with large cardinals that preserves boldface Σ¹₃-uniformization while violating lightface Σ¹₄-uniformization.

If this is right

  • Boldface Σ¹₃-uniformization does not entail lightface Σ¹₄-uniformization.
  • The uniformization principles at these levels can be separated while maintaining the consistency of ZFC.
  • Inner models with large cardinals can be used to obtain a similar separation between Σ¹₃-uniformization and Σ¹₄-uniformization.

Where Pith is reading between the lines

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

  • The separation indicates that uniformization properties may be calibrated more finely across boldface and lightface versions than previously expected.
  • Similar constructions could potentially distinguish uniformization at other projective levels.
  • The result opens the possibility of determining the exact consistency strengths required for each uniformization principle independently.

Load-bearing premise

A model of ZFC exists in which boldface Σ¹₃-uniformization can be maintained while lightface Σ¹₄-uniformization is destroyed.

What would settle it

A proof in ZFC that boldface Σ¹₃-uniformization always implies lightface Σ¹₄-uniformization would show that no such separating model can exist.

read the original abstract

Assuming the consistency of $\mathsf{ZFC}$, we construct a model of set theory in which the boldface $\mathbf{\Sigma}^1_3$-uniformization property holds, yet the lightface $\Sigma^1_4$-uniformization property fails, separating these two principles for the first time. We also indicate how to create a universe where $\Sigma^1_3$-uniformization holds, but $\Sigma^1_4$-uniformization fails using inner models with large cardinals.

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.

Referee Report

0 major / 2 minor

Summary. The paper shows that, assuming the consistency of ZFC, there is a model of set theory satisfying the boldface boldface Σ¹₃-uniformization property but not the lightface Σ¹₄-uniformization property. Additionally, it indicates a construction using inner models with large cardinals in which Σ¹₃-uniformization holds while Σ¹₄-uniformization fails.

Significance. This result is significant because it provides the first known separation between boldface Σ¹₃-uniformization and lightface Σ¹₄-uniformization. The explicit forcing construction that preserves Σ¹₃ relations while adding a non-uniformizable Σ¹₄ relation, along with the inner model approach, offers new tools for investigating the projective hierarchy and uniformization principles in descriptive set theory.

minor comments (2)
  1. [Introduction] The discussion of previous results on uniformization at lower levels could benefit from more explicit citations to key papers on Σ¹₁ and Σ¹₂ uniformization.
  2. Ensure consistent use of boldface notation for projective classes throughout the manuscript.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, accurate summary of our results, and recommendation of minor revision. The manuscript establishes the first separation between boldface Σ¹₃-uniformization and lightface Σ¹₄-uniformization via an explicit forcing construction (assuming Con(ZFC)), together with an inner-model construction using large cardinals.

Circularity Check

0 steps flagged

No significant circularity in the consistency construction

full rationale

The paper establishes a consistency result by constructing a forcing extension from Con(ZFC) that preserves boldface Σ¹₃-uniformization while violating lightface Σ¹₄-uniformization, with an additional indication of an inner-model construction using large cardinals for a related separation. These steps rely on standard set-theoretic techniques (forcing and inner models) whose correctness is independent of the target claim and can be verified externally. No equations or definitions reduce the conclusion to its inputs by construction, no parameters are fitted and renamed as predictions, and no load-bearing self-citations or uniqueness theorems are invoked that collapse the argument. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the consistency of ZFC together with standard large-cardinal or forcing machinery whose precise axioms are not detailed in the abstract.

axioms (1)
  • domain assumption Consistency of ZFC
    The entire construction begins from the assumption that ZFC is consistent.

pith-pipeline@v0.9.0 · 5372 in / 1087 out tokens · 40428 ms · 2026-05-10T00:56:46.503730+00:00 · methodology

discussion (0)

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Reference graph

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