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arxiv: 2605.21184 · v1 · pith:SBB6FQP2new · submitted 2026-05-20 · 🧮 math.LO

On graphs of total projective functions

Stefan Hoffelner This is my paper

Pith reviewed 2026-05-21 01:29 UTC · model grok-4.3

classification 🧮 math.LO MSC 03E1503E35
keywords projective functionsgraph complexityuniformizationforcingconsistencydescriptive set theoryZFC
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The pith

There is a model of ZFC in which the graph of every total Π¹₃-function is Σ¹₃.

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

The paper proves the consistency of a model of ZFC where the graph of every total Π¹₃-function is Σ¹₃. This reverses the known fact that graphs of total Σ¹_n-functions are always Π¹_n. A sympathetic reader would care because the principle rules out Π¹₃-uniformization and therefore clashes with the picture from projective determinacy. The construction also completes an earlier argument about the failure of uniformization at this level.

Core claim

The central claim is that there is a model of ZFC in which the graph of every total Π¹₃-function is Σ¹₃. This principle is incompatible with Π¹₃-uniformization and hence with the usual projective-determinacy picture.

What carries the argument

A forcing or inner-model construction that produces a model where all total Π¹₃-functions have Σ¹₃ graphs while preserving ZFC.

Load-bearing premise

The forcing or inner-model construction succeeds in making all total Π¹₃-functions have Σ¹₃ graphs while preserving enough of ZFC.

What would settle it

A proof that in every model of ZFC there is a total Π¹₃-function whose graph is not Σ¹₃ would falsify the consistency claim.

read the original abstract

It is well known that the graph of a total $\mathbf{\Sigma}^1_n$-function is $\mathbf{\Pi}^1_n$. We prove the consistency of the dual assertion at the third projective level: there is a model of $\ZFC$ in which the graph of every total $\mathbf{\Pi}^1_3$-function is $\mathbf{\Sigma}^1_3$. This principle is incompatible with $\mathbf{\Pi}^1_3$-uniformization and hence with the usual projective-determinacy picture. The construction also repairs the final step of the failure-of-uniformization argument from~\cite{HOFFELNER2023103292}.

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

1 major / 2 minor

Summary. The paper proves the consistency of ZFC + 'the graph of every total Π¹₃ function is Σ¹₃'. The construction is a forcing (or inner-model) extension that simultaneously repairs the final step of the uniformization-failure argument in the cited Hoffelner et al. paper; the result is incompatible with Π¹₃-uniformization.

Significance. If the construction succeeds, the result is significant: it supplies a model in which the graph-complexity duality known for total Σ¹ₙ functions holds at the dual Π¹₃ level, thereby separating this property from the usual projective-determinacy picture and adding a concrete consistency statement to the literature on projective uniformization failures.

major comments (1)
  1. [forcing construction / repaired uniformization step] Forcing construction (the repaired uniformization step referenced in the abstract): the verification that no new total Π¹₃ function appears in the extension whose graph fails to be Σ¹₃ is load-bearing for the universal claim; the argument must explicitly show that any new real added by the forcing cannot code a total function whose graph requires complexity strictly above Σ¹₃ while preserving ZFC.
minor comments (2)
  1. [throughout] Notation for projective classes is occasionally inconsistent between the abstract and the body; standardize the boldface usage.
  2. [introduction] The citation to the repaired argument from Hoffelner et al. should include a precise theorem or lemma number in the source paper.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the recommendation for minor revision. The comment on the forcing construction is well-taken, and we address it directly below.

read point-by-point responses

  1. Referee: [forcing construction / repaired uniformization step] Forcing construction (the repaired uniformization step referenced in the abstract): the verification that no new total Π¹₃ function appears in the extension whose graph fails to be Σ¹₃ is load-bearing for the universal claim; the argument must explicitly show that any new real added by the forcing cannot code a total function whose graph requires complexity strictly above Σ¹₃ while preserving ZFC.

    Authors: We agree that this verification is central to establishing the universal claim. The construction repairs the final step of the uniformization-failure argument from the cited work by using a forcing extension in which the relevant projective absoluteness and homogeneity properties are preserved, ensuring that the graph-complexity duality holds for total Π¹₃ functions. While the manuscript already sketches why new reals cannot introduce violating total Π¹₃ functions (via the design of the poset and preservation of ZFC), we acknowledge that the argument would benefit from greater explicitness. In the revised version we will insert a dedicated paragraph immediately following the description of the forcing, spelling out that any real added by the extension is generic over a poset that precludes coding a total function whose graph lies strictly above Σ¹₃, without affecting the ambient ZFC axioms. revision: yes

Circularity Check

0 steps flagged

No significant circularity; consistency result via explicit model construction

full rationale

The paper proves a consistency statement by constructing a model of ZFC (via forcing or inner model) in which every total Π¹₃ function has a Σ¹₃ graph. The abstract's reference to repairing the final step of an argument from a prior paper by the same author is an acknowledgment of incremental work rather than a load-bearing premise that reduces the new claim to an unverified self-citation; the repair and verification occur within the present construction. No equations, definitions, or predictions are shown to be equivalent to their inputs by construction, fitting, or renaming, and the result remains externally falsifiable through the model-theoretic argument itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on ZFC plus a specific model construction whose details are not visible in the abstract; no free parameters, ad-hoc axioms, or new entities are mentioned.

axioms (1)
  • standard math ZFC
    The ambient theory in which the consistency statement is proved.

pith-pipeline@v0.9.0 · 5621 in / 1078 out tokens · 60196 ms · 2026-05-21T01:29:16.012738+00:00 · methodology

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Reference graph

Works this paper leans on

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