arXiv math/00070252000-07-11math.LO

A note on an alleged proof of the relative consistency of P=NP with PA

Ralf Schindler

备注：: 4 pages

英文摘要：

We indicate that an argument of da Costa and Doria in fact proves P=NP. This observation makes their argument appear dubious. We isolate a weak version of one of their lemmas which would already prove P=NP. We point out that even this weak version is probably false. In fact, a generalized form of this weak version is provably false.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

A note on the consistency operator

It is a well known empirical observation that natural axiomatic theories are pre-well-ordered by consistency strength. For any natural theory $T$, the next strongest natural theory is $T+\mathsf{Con}_T$. We formulate and prove a statement to the effect that the consistency operator is the weakest natural way to uniformly extend axiomatic theories.

2019-10-28math.LO

A Note on Iterated Consistency and Infinite Proofs

Schmerl and Beklemishev's work on iterated reflection achieves two aims: It introduces the important notion of $Π^0_1$-ordinal, characterizing the $Π^0_1$-theorems of a theory in terms of transfinite iterations of consistency; and it provides an innovative calculus to compute the $Π^0_1$-ordinals for a range of theories. The present note demonstrates that these achievements are independent: We read off $Π^0_1$-ordinals from a Schütte-style ordinal analysis via infinite proofs, in a direct and transparent way.

2018-07-16math.LO

The problem of artificial precision in theories of vagueness: a note on the role of maximal consistency

The problem of artificial precision is a major objection to any theory of vagueness based on real numbers as degrees of truth. Suppose you are willing to admit that, under sufficiently specified circumstances, a predication of "is red" receives a unique, exact number from the real unit interval [0,1]. You should then be committed to explain what is it that determines that value, settling for instance that my coat is red to degree 0.322 rather than 0.321. In this note I revisit the problem in the important case of Łukasiewicz infinite-valued propositional logic that brings to the foreground the…

2013-06-18math.LO

关于阿普特定理的注记

A Note on a Theorem of Apter

Institute of Mathematics of the Polish Academy of Sciences（波兰科学院数学研究所）

本文探讨了ZF与AD_R条件下Θ的可测性与最小强正则基数关系，揭示了其与可数余数的联系。

2026-10-15math.LO

A note on cardinal preserving embeddings

We prove the consistency of the theory ZFC + there is a strongly compact cardinal from the existence of a cardinal preserving embedding from the universe into an inner model. The proof almost shows that under SCH, every cardinal preserving embedding preserves the cofinality and continuum functions.

2021-02-26math.LO

A note on derivability conditions

We investigate relationships between versions of derivability conditions for provability predicates. We show several implications and non-implications between the conditions, and we discuss unprovability of consistency statements induced by derivability conditions. First, we classify already known versions of the second incompleteness theorem, and exhibit some new sets of conditions which are sufficient for unprovability of Hilbert--Bernays' consistency statement. Secondly, we improve Buchholz's schematic proof of provable $Σ_1$-completeness. Then among other things, we show that Hilbert--Bern…

2020-04-30math.LO

更多相关论文收起更多论文

Consistency of "the ideal of null restricted to some A is kappa-complete not kappa^+-complete, kappa weakly inaccessible and cov(meagre)=aleph_1"

In this note we answer the following question of Grinblat: Is it consistent that for some set X, cov(NULL restriction X)=lambda is a weakly inaccessible cardinal (so X not null of course) while cov(meagre) is small, say it is aleph_1.

2005-04-10math.LO

Consistency Without Cut Elimination

In this note we will show how to get consistency for first order classical logic, in a purely syntactic way, without going through cut elimination. The procedure is very simple and it uses the calculus of structures in an essential way. It also shows how finitaryness (in the sense of finite choice of premises for each rule) is actually a triviality (contrarily to what one would guess from textbooks).

2002-10-24math.LO

A note on some versions of club

This note gives two results on guessing unbounded subsets of lambda^+. The first is a positive result and applies to the situation of lambda regular, while the second is a negative consistency result which applies to the situation of lambda singular. Both results are connected to an earlier result of the same authors in which they showed that a certain version of clubsuit holds at a successor of singular just in ZFC. The first result here shows that that fact can to a certain extent be extended to the successor of singular. The negative result here gives limitations to the extent to which one …

1997-09-12math.LO

Solovay's Relative Consistency Proof for FIM and BI

In 2002 Robert Solovay proved that a subsystem BI of classical second order arithmetic, with bar induction and arithmetical countable choice, can be negatively interpreted in the neutral subsystem BSK of Kleene's intuitionistic analysis FIM using Markov's Principle MP. Combining this result with Kleene's formalized recursive realizability, he established (in primitive recursive arithmetic PRA) that FIM + MP and BI have the same consistency strength. This historical note includes Solovay's original proof, with his permission, and the additional observation that Markov's Principle can be weakene…

2021-01-14math.HOmath.LO

Two remarks on proof theory of first-order arithmetic

In this note let us give two remarks on proof-theory of PA. First a derivability relation is introduced to bound witnesses for provable $Σ_{1}$-formulas in PA. Second Paris-Harrington's proof for their independence result is reformulated to show a `consistency' proof of PA based on a combinatorial principle.

2020-12-31math.LO