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arxiv: 2312.02540 · v2 · submitted 2023-12-05 · 🧮 math.LO

A Failure of Pi¹_(n+3)-Reduction in the Presence of Sigma¹_(n+3)-Separation

Stefan Hoffelner This is my paper

Pith reviewed 2026-05-24 05:32 UTC · model grok-4.3

classification 🧮 math.LO
keywords forcingprojective setsseparationreductioninner modelsWoodin cardinalsLdescriptive set theory
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The pith

One can force over L that Σ¹₃-separation holds while Π¹₃-reduction fails.

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

The paper constructs a forcing extension of the constructible universe L in which Σ¹₃-separation is true but Π¹₃-reduction is false. This yields the first separation of the two principles at the third projective level. The same forcing lifts to the inner models M_n that contain n Woodin cardinals, producing models where Σ¹_{n+3}-separation holds yet Π¹_{n+3}-reduction fails. A sympathetic reader cares because these principles govern the existence of definable reductions and separations among projective sets of reals, which sit at the boundary between ZFC and stronger axioms such as determinacy.

Core claim

We show that one can force over L that Σ¹₃-separation holds, while Π¹₃-reduction fails, thus separating these two principles for the first time. The construction can be lifted to canonical inner models M_n with n-many Woodin cardinals, yielding that assuming the existence of M_n, Σ¹_{n+3}-separation can hold, yet Π¹_{n+3}-reduction fails.

What carries the argument

A forcing construction over L that preserves the relevant projective properties while destroying reduction.

Load-bearing premise

The forcing construction over L preserves the relevant projective properties and can be lifted to the inner models M_n assuming their existence with n Woodin cardinals.

What would settle it

An explicit verification, inside the generic extension of L by the forcing, that every pair of disjoint Σ¹₃ sets admits a Π¹₃ reduction while some pair of disjoint Π¹₃ sets does not.

read the original abstract

We show that one can force over $L$ that $\Sigma^1_3$-separation holds, while $\Pi^1_3$-reduction fails, thus separating these two principles for the first time. The construction can be lifted to canonical inner models $M_n$ with $n$-many Woodin cardinals, yielding that assuming the existence of $M_n$, $\Sigma^1_{n+3}$-separation can hold, yet $\Pi^1_{n+3}$-reduction fails.

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 manuscript constructs a forcing extension of L in which Σ¹₃-separation holds while Π¹₃-reduction fails, thereby separating the two principles for the first time at this level. The construction is shown to lift to the canonical inner models M_n (n Woodin cardinals), yielding the analogous separation of Σ¹_{n+3}-separation from Π¹_{n+3}-reduction.

Significance. If the forcing construction and its lift are correct, the result supplies the first consistency proof separating these two classical projective principles at the indicated levels. The explicit forcing over L together with the lift to M_n provides a concrete consistency-strength calibration and a template for further separations in the projective hierarchy.

minor comments (2)
  1. [§3] The statement of the main theorem (presumably Theorem 1.1 or the result in §3) would benefit from an explicit list of the preserved projective properties of the forcing poset.
  2. [§4] Notation for the lifted forcing in the M_n case is introduced without a dedicated subsection; a short paragraph comparing the L-case and M_n-case posets would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No circularity; direct forcing construction over L

full rationale

The paper presents a forcing construction over L that forces Σ¹₃-separation while making Π¹₃-reduction fail, with a lift to inner models M_n. No equations, fitted parameters, self-definitional reductions, or load-bearing self-citations appear in the derivation chain. The result is a standard set-theoretic existence proof via forcing, self-contained against external benchmarks in descriptive set theory, with no reduction of the central claim to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on ZFC plus the existence of the inner models M_n; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math ZFC
    Background theory for forcing and inner-model constructions.
  • domain assumption Existence of M_n with n Woodin cardinals
    Required for the lift of the construction in the second part of the result.

pith-pipeline@v0.9.0 · 5614 in / 1100 out tokens · 27563 ms · 2026-05-24T05:32:15.430345+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Forcing $\mathbf{\Sigma}^1_1$-Separation on $\omega_1^{\omega_1}$

    math.LO 2026-05 unverdicted novelty 7.0

    Proves consistency of boldface Σ¹₁-separation on ω₁^ω₁ via forcing from L that preserves CH.

Reference graph

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