Structural-Abstraction Principles

Leach-Krouse, Graham

doi:10.1093/philmat/nkv033

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“…Cieśliński and Urbaniak [27] conjecture that any two distinct instances Y R (m) and Y R (n) of a Rosser-type Yablo formula Y R (x) based on a standard proof predicate are not provably equivalent. Leach-Krouse [88] and Kurahashi [89] construct a standard proof predicate, and a Rosser-type Yablo formula Y R (x) based on this proof predicate such that T ∀x∀y(Y R (x) ↔ Y R (y)) (see [88,Theorem 9] and [89,Corollary 21]).…”

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

Bull. symb. log

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show abstract

Structural-Abstraction Principles

Cited by 1 publication

References 18 publications

Current Research on Gödel’s Incompleteness Theorems

Current Research on Gödel’s Incompleteness Theorems

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