Incompleteness in the Finite Domain

Pudlák, Pavel

doi:10.1017/bsl.2017.32

Cited by 40 publications

(30 citation statements)

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“…In fact one can show that for every n ∈ N, IΣ 0 n+1 Con(IΣ 0 n ). 13 For a large class of natural theories U , Pudlák [114] shows that the lengths of the shortest proofs of Con(U ) n for n ∈ ω in the theory U itself are bounded by a polynomial in n. Pudlák conjectures [114] that U does not have polynomial proofs of the finite consistency statements Con(U + Con(U )) n for n ∈ ω.…”

Section: Yong Chengmentioning

confidence: 99%

Current Research on Gödel’s Incompleteness Theorems

Cheng¹

2021

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We give a survey of current research on Gödel's incompleteness theorems from the following three aspects: classifications of different proofs of Gödel's incompleteness theorems, the limit of the applicability of Gödel's first incompleteness theorem, and the limit of the applicability of Gödel's second incompleteness theorem. §1. Introduction. Gödel's first and second incompleteness theorems are some of the most important and profound results in the foundations of mathematics and have had wide influence on the development of logic, philosophy, mathematics, computer science, as well as other fields. Intuitively speaking, Gödel's incompleteness theorems express that any rich enough logical system cannot prove its own consistency, i.e., that no contradiction like 0 = 1 can be derived within this system.Gödel [43] proves his first incompleteness theorem (G1) for a certain formal system P related to Russell-Whitehead's Principia Mathematica based on the simple theory of types over the natural number series and the Dedekind-Peano axioms (see [8, p. 3]). Gödel announces the second incompleteness theorem (G2) in an abstract published in October 1930: no consistency proof of systems such as Principia, Zermelo-Fraenkel set theory, or the systems investigated by Ackermann and von Neumann is possible by methods which can be formulated in these systems (see [147, p. 431]).Gödel comments in a footnote of [43] that G2 is corollary of G1 (and in fact a formalized version of G1): if T is consistent, then the consistency of T is not provable in T where the consistency of T is formulated as the arithmetic formula which says that there exists an unprovable sentence in T. Gödel [43] sketches a proof of G2 and promises to provide full details in a subsequent publication. This promise is not fulfilled, and a detailed proof of G2 for first-order arithmetic only appears in a monograph by Hilbert and Bernays [59]. Abstract logic-free formulations of Gödel's incompleteness theorems have been given by Kleene [76] ("symmetric form"), Smullyan [121] ("representation systems"), and others. The following is a modern reformulation of Gödel's incompleteness theorems.

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Section: Yong Chengmentioning

confidence: 99%

Current Research on Gödel’s Incompleteness Theorems

Cheng¹

2021

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“…We begin this section with some general remarks about the length of proofs: Following Pudlák [Pud86,Pud17] we count the total number of symbols in a proof, rather than just the number of inferences. In other words, the length of a proof is essentially the number of digits in its Gödel code.…”

Section: Preliminariesmentioning

confidence: 99%

“…This is no surprise because the negative answer would imply co -NP = NP and hence P = NP, as pointed out in [Pud86, Proposition 6.2]. In contrast with this there is an interesting new result of Hrubeš (see [Pud17,Theorem 3.4]) which tests the boundaries of Pudlák's conjecture: For any formula ϕ which is consistent with a sequential and finitely axiomatized theory T there is a true Π 1 -formula ψ such that we have T + ϕ ψ but T ⊢ Con(T + ψ) ↾ n with polynomial in n proofs.…”

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confidence: 92%

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Short Proofs for Slow Consistency

Freund¹,

Pakhomov²

2020

Notre Dame J. Formal Logic

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Let Con(T) ↾ x denote the finite consistency statement "there are no proofs of contradiction in T with ≤ x symbols". For a large class of natural theories T, Pudlák has shown that the lengths of the shortest proofs of Con(T) ↾ n in the theory T itself are bounded by a polynomial in n. At the same time he conjectures that T does not have polynomial proofs of the finite consistency statements Con(T + Con(T)) ↾ n. In contrast we show that Peano arithmetic (PA) has polynomial proofs of Con(PA + Con * (PA)) ↾ n, where Con * (PA) is the slow consistency statement for Peano arithmetic, introduced by S.-D. Friedman, Rathjen and Weiermann. We also obtain a new proof of the result that the usual consistency statement Con(PA) is equivalent to ε 0 iterations of slow consistency. Our argument is proof-theoretic, while previous investigations of slow consistency relied on non-standard models of arithmetic.

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“…In fact one can show that for every n ∈ N, IΣ 0 n+1 ⊢ Con(IΣ 0 n ). 15 For a large class of natural theories U , Pudlák [114] shows that the lengths of the shortest proofs of Con(U )↾ n for n ∈ ω in the theory U itself are bounded by a polynomial in n. Pudlák conjectures [114] that U does not have polynomial proofs of the finite consistency statements Con(U + Con(U )) ↾ n for n ∈ ω.…”

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confidence: 99%