On the depth of G\"{o}del's incompleteness theorem

In this paper, we use G\"{o}del's incompleteness theorem as a case study for investigating mathematical depth. We take for granted the widespread judgment by mathematical logicians that G\"{o}del's incompleteness theorem is deep, and focus on the philosophical question of what its depth consists in. We focus on the methodological study of the depth of G\"{o}del's incompleteness theorem, and propose three criteria to account for its depth: influence, fruitfulness, and unity. Finally, we give some explanations for our account of the depth of G\"{o}del's incompleteness theorem.Comment: 23 pages, revised version. arXiv admin note: text overlap with arXiv:2009.0488

Incompleteness via paradox and completeness

This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) andWang (1955) in order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell’s paradox, a variant of Mirimano’s paradox, the Liar, and the Grelling-Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth

Current research on G\"odel's incompleteness theorems

We give a survey of current research on G\"{o}del's incompleteness theorems from the following three aspects: classifications of different proofs of G\"{o}del's incompleteness theorems, the limit of the applicability of G\"{o}del's first incompleteness theorem, and the limit of the applicability of G\"{o}del's second incompleteness theorem.Comment: 54 pages, final accepted version, to appear in The Bulletin of Symbolic Logi

Finding the limit of incompleteness I

In this paper, we examine the limit of applicability of G\"{o}del's first incompleteness theorem ($\sf G1$ for short). We first define the notion "$\sf G1$ holds for the theory $T$". This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which $\sf G1$ holds. To approach this question, we first examine the following question: is there a theory $T$ such that Robinson's $\mathbf{R}$ interprets $T$ but $T$ does not interpret $\mathbf{R}$ (i.e. $T$ is weaker than $\mathbf{R}$ w.r.t. interpretation) and $\sf G1$ holds for $T$? In this paper, we show that there are many such theories based on Je\v{r}\'{a}bek's work using some model theory. We prove that for each recursively inseparable pair $\langle A,B\rangle$, we can construct a r.e. theory $U_{\langle A,B\rangle}$ such that $U_{\langle A,B\rangle}$ is weaker than $\mathbf{R}$ w.r.t. interpretation and $\sf G1$ holds for $U_{\langle A,B\rangle}$. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree $\mathbf{0}< \mathbf{d}<\mathbf{0}^{\prime}$, there is a theory $T$ with Turing degree $\mathbf{d}$ such that $\sf G1$ holds for $T$ and $T$ is weaker than $\mathbf{R}$ w.r.t. Turing reducibility. As a corollary, based on Shoenfield's work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which $\sf G1$ holds.Comment: 18 pages. Accepted and to appear in Bulletin of Symbolic Logi

The fanciful optimism of Miguel Sánchez-Mazas. Let us calculate... = Freedom and Justice

May 2020 marked the 25th anniversary of the death of Miguel Sánchez-Mazas, founder of Theoria. An International Journal of Theory, History and Foundations of Science, and regarded as the person who brought mathematical logic to Spain. Here we present some of his biographical features and a summary of his contributions, from his early work in the 1950s - introducing contemporary advances in logic and philosophy of science in a philosophically backward milieu dominated by the scholasticism of that era in Spain - to the development of a project of Lebnizian lineage aimed at producing an arithmetic calculation that would elude some of the difficulties confronting Leibniz’s calculus.; En mayo de 2020 se cumplen 25 años del fallecimiento de Miguel Sánchez-Mazas, fundador de Theoria. An International Journal of Theory, History and Foundations of Science y considerado como el introductor de la lógica matemática en España. En esta contribución presentamos algunos sus rasgos biográficos, así como un resumen de sus aportaciones, desde las iniciales en la década de los cincuenta del siglo XX introduciendo los avances en lógica y filosofía de la ciencia contemporáneos en un medio filosóficamente retrasado como el dominado por la escolástica de aquel tiempo en España hasta el desarrollo de un proyecto de estirpe lebniziana orientado a elaborar un cálculo aritmético que eludiera algunos de los problemas con los que se vio confrontado el calculo de Leibniz

Decidability vs. undecidability. Logico-philosophico-historical remarks

The aim of the paper is to present the decidability problems from a philosophical and historical perspective as well as to indicate basic mathematical and logical results concerning (un)decidability of particular theories and problems