Finding the Limit of Incompleteness I

Cheng, Yong

doi:10.1017/bsl.2020.9

Cited by 5 publications

(8 citation statements)

References 30 publications

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“…Arithmetical completeness theorem for GL: For any modal formula φ in L(GL), GL ⊢ φ iff PA ⊢ φ f for every arithmetic interpretation f . 14 Arithmetical completeness theorem for GLS: For any modal formula φ, GLS ⊢ φ iff N |= φ f for every arithmetic interpretation f . 15 Gödel's incompleteness theorems are closely related to paradox.…”

Section: Some Popular Misinterpretations Of Gödel's First Incompleten...mentioning

confidence: 99%

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On the Depth of Gödel’s Incompleteness Theorems

Cheng

2022

Philosophia Mathematica

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We use Gödel’s incompleteness theorems as a case study for investigating mathematical depth. We examine the philosophical question of what the depth of Gödel’s incompleteness theorems consists in. We focus on the methodological study of the depth of Gödel’s incompleteness theorems, and propose three criteria to account for the depth of the incompleteness theorems: influence, fruitfulness, and unity. Finally, we give some explanations for our account of the depth of Gödel’s incompleteness theorems.

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Section: Some Popular Misinterpretations Of Gödel's First Incompleten...mentioning

confidence: 99%

“…13 For details of the properties of Truth and Prov, we refer to [73] and [97]. 14 GL is a modal system consisting of the following schemes of axiom: (1) all tautologies;…”

Section: Some Popular Misinterpretations Of Gödel's First Incompleten...mentioning

confidence: 99%

On the Depth of Gödel’s Incompleteness Theorems

Cheng

2022

Philosophia Mathematica

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“…T means S is interpretable in T but it is unknown whether S interprets T follows from the idea behind the proof of Theorem 4.7 of [2] that the interpretability degrees of computably enumerable essentially undecidable theories have infima. Hence, if there exists a minimal essentially undecidable theory, then that theory is also the minimum essentially undecidable theory.…”

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confidence: 99%

“…In [7], Jeřábek shows that there is an essentially undecidable theory that is interpretable in R but that does not interpret R. The theory Jeřábek gives is such that all partially recursive functions are representable. In Section 3 of [2], Yong Cheng uses results in [7] to give many examples of essentially undecidable theories that are interpretable in R but that do not interpret R. One of the referees observed that the existence of theories strictly below R w.r.t. interpretability also follows from the work of Shoenfield [14].…”

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Weak essentially undecidable theories of concatenation

Murwanashyaka

2022

Arch. Math. Logic

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In the language $$\lbrace 0, 1, \circ , \preceq \rbrace $$ { 0 , 1 , ∘ , ⪯ } , where 0 and 1 are constant symbols, $$\circ $$ ∘ is a binary function symbol and $$\preceq $$ ⪯ is a binary relation symbol, we formulate two theories, $$ \textsf {WD} $$ WD and $$ {\textsf {D}}$$ D , that are mutually interpretable with the theory of arithmetic $$ {\textsf {R}} $$ R and Robinson arithmetic $${\textsf {Q}} $$ Q , respectively. The intended model of $$ \textsf {WD} $$ WD and $$ {\textsf {D}}$$ D is the free semigroup generated by $$\lbrace {\varvec{0}}, {\varvec{1}} \rbrace $$ { 0 , 1 } under string concatenation extended with the prefix relation. The theories $$ \textsf {WD} $$ WD and $$ {\textsf {D}}$$ D are purely universally axiomatised, in contrast to $$ {\textsf {Q}} $$ Q which has the $$\varPi _2$$ Π 2 -axiom $$\forall x \; [ \ x = 0 \vee \exists y \; [ \ x = Sy \ ] \ ] $$ ∀ x [ x = 0 ∨ ∃ y [ x = S y ] ] .

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A new perspective on completeness and finitist consistency

Santos

Sieg

Kahle

2023

Journal of Logic and Computation

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In this paper, we study the metamathematics of consistent arithmetical theories $T$ (containing $\textsf {I}\varSigma _{1}$); we investigate numerical properties based on proof predicates that depend on numerations of the axioms. Numeral Completeness. For every true (in $\mathbb {N}$) sentence $\vec {Q}\vec {x}.\varphi (\vec {x})$, with $\varphi (\vec {x})$ a $\varSigma _{1}(\textsf {I}\varSigma _1)$-formula, there is a numeration $\tau $ of the axioms of $T$ such that $\textsf {I}\varSigma _1\vdash \vec {Q}\vec {x}. \texttt {Pr}_{\tau }(\ulcorner \varphi (\overset {\text{.} }{\vec {x}})\urcorner )$, where $\texttt {Pr}_{\tau }$ is the provability predicate for the numeration $\tau $. Numeral Consistency. If $T$ is consistent, there is a $\varSigma _{1}(\textsf {I}\varSigma _1)$-numeration $\tau $ of the axioms of $\textsf {I}\varSigma _{1}$ such that $\textsf {I}\varSigma _1\vdash \forall\, x. \texttt {Pr}_{\tau }(\ulcorner \neg \textit {Prf}(\ulcorner \perp \urcorner , \overset {\text{.}}{x})\urcorner )$, where $\textit {Prf}(x,y)$ denotes a $\varDelta _{1}(\textsf {I}\varSigma _1)$-definition of ‘$y$ is a $T$-proof of $x$’. Finitist consistency is addressed by generalizing a result of Artemov: Partial finitism. If $T$ is consistent, there is a primitive recursive function $f$ such that, for all $n\in \mathbb {N}$, $f(n)$ is the code of an $\textsf {I}\varSigma _{1}$-proof of $\neg\, \textit{Prf}(\ulcorner \perp \urcorner ,\overline {n})$. These results are not in conflict with Gödel’s Incompleteness Theorems. Rather, they allow to extend their usual interpretation and show a deep connection to reflections in Hilbert’s last papers of 1931.

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Finding the Limit of Incompleteness I

Cited by 5 publications

References 30 publications

On the Depth of Gödel’s Incompleteness Theorems

On the Depth of Gödel’s Incompleteness Theorems

Weak essentially undecidable theories of concatenation

A new perspective on completeness and finitist consistency

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