Boolos‐style proofs of limitative theorems

Serény, György

doi:10.1002/malq.200310091

Cited by 6 publications

(4 citation statements)

References 11 publications

(22 reference statements)

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“…A natural generalization of this theorem is the following (cf. Chapter III of [15], or Corollary 1 of [12]):…”

Section: Semantic Form Of Gödel's Theoremmentioning

confidence: 99%

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Godel-Rosser's Incompleteness Theorems for Non-Recursively Enumerable Theories

Salehi,

Seraji

2015

Preprint

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Gödel's First Incompleteness Theorem is generalized to definable theories, which are not necessarily recursively enumerable, by using a couple of syntactic-semantic notions; one is the consistency of a theory with the set of all true Π n -sentences or equivalently the Σ n -soundness of the theory, and the other is n-consistency the restriction of ω-consistency to the Σ n -formulas. It is also shown that Rosser's Incompleteness Theorem does not generally hold for definable non-recursively enumerable theories; whence Gödel-Rosser's Incompleteness Theorem is optimal in a sense. Though the proof of the incompleteness theorem using the Σ n -soundness assumption is constructive, it is shown that there is no constructive proof for the incompleteness theorem using the n-consistency assumption, for n > 2.

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“…A natural generalization of this theorem is the following (cf. Chapter III of [15], or Corollary 1 of [12]):…”

Section: Semantic Form Of Gödel's Theoremmentioning

confidence: 99%

“…This question has been answered affirmatively in the literature; see e.g. [15] or [12]. Gödel's original first incompleteness theorem did not assume the soundness of the theory in question, and he used the notion of ω-consistency for that purpose.…”

Section: Introductionmentioning

confidence: 96%

Godel-Rosser's Incompleteness Theorems for Non-Recursively Enumerable Theories

Salehi,

Seraji

2015

Preprint

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“…As a matter of fact, different proofs for Tarski's undefinability theorem can lead to different proofs for the semantic version of this lemma. We will review two such proofs (presented in [2,5,6,7,11]) which are supposedly diagonal-free. Having different proofs will, hopefully, shed some new light on the nature of this lemma and will increase our understanding about it.…”

Section: Introductionmentioning

confidence: 99%

Tarski's Undefinability Theorem and Diagonal Lemma

Salehi

2020

Preprint

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We prove the equivalence of the semantic version of Tarski's theorem on the undefinability of truth with a semantic version of the Diagonal Lemma, and also show the equivalence of syntactic Tarski's Undefinability Theorem with a weak syntactic diagonal lemma. We outline two seemingly diagonal-free proofs for these theorems from the literature, and show that syntactic Tarski's theorem can deliver Gödel-Rosser's Incompleteness Theorem.

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“…Later, Boolos [2] gave another proof for the first incompleteness theorem of Gödel, in 1989, which was based on Berry's paradox too; see also [1], [3], and [4,Section 17.3]. Berry's paradox has been used for proving Tarski's undefinability theorem as well, see [6] and [27,Corollary 2]. The research on Berry-based proofs is a live topic, the two most recent publications on which are [17] and [26].…”

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confidence: 99%

On the Diagonal Lemma of Gödel and Carnap

Salehi¹

2020

Bull. symb. log

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A cornerstone of modern mathematical logic is the diagonal lemma of Gödel and Carnap. It is used in, for example, the classical proofs of the theorems of Gödel, Rosser, and Tarski. From its first explication in 1934, just essentially one proof has appeared for the diagonal lemma in the literature; a proof that is so tricky and hard to relate that many authors have tried to avoid the lemma altogether. As a result, some so-called diagonal-free proofs have been given for the above-mentioned fundamental theorems of logic. In this paper, we provide new proofs for the semantic formulation of the diagonal lemma, and for a weak version of the syntactic formulation of it.

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Boolos‐style proofs of limitative theorems

Abstract: Boolos's proof of incompleteness is extended straightforwardly to yield simple "diagonalization-free" proofs of some classical limitative theorems of logic.

Cited by 6 publications

References 11 publications

Godel-Rosser's Incompleteness Theorems for Non-Recursively Enumerable Theories

Godel-Rosser's Incompleteness Theorems for Non-Recursively Enumerable Theories

Tarski's Undefinability Theorem and Diagonal Lemma

On the Diagonal Lemma of Gödel and Carnap

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