Observations concerning Gödel's 1931

http://arxiv.org/abs/2403.19665, 2024

In this article we discuss the proof in the short unpublished paper appeared in the 3rd volume of Gödel's Collected Works entitled "On undecidable sentences" (*1931?), which provides an introduction to Gödel's 1931 ideas regarding the incompleteness of arithmetic. We analyze the meaning of the negation of the provability predicate, and how it is meant not to lead to vicious circle. We show how in fact in Gödel's entire argument there is an omission regarding the cases of non-provability, which, once taken into consideration again, allow a completely different view of Gödel's entire argument of incompleteness. Previous results of the author are applied to show that the definition of a contradiction is included in the argument of *1931?. Furthermore, an examination is also briefly presented in order to the application of the substitution in the well-known Gödel formula as a violation of the uniqueness, calling into question its very derivation.

I am delighted to be able to join the other authors of this Festschrift in honouring John Bell, and to be able to express my gratitude to John for his exuberant and generous friendship ever since we first met, in 1967. Even before then, John’s fame had preceded him by way of an article about his plans to study mathematics in Oxford or Cambridge that had appeared in the San Francisco Chronicle 6 years earlier and impressed me enough to keep. It is wonderful to think back to those days of extraordinary promise from the vantage point of John’s tremendous accomplishments in the nearly 50 years since, and his promise of yet more.

This notes provides a brief survey that reflects the early development of provability logic. It includes a detailed exposition of Gödel's incompleteness results, the Löb theorem, and the arithmetical completeness theorems of Solovay.

Modern Logic, 1997

2023

The aim of this text is to offer an explanation of Gödel's Theorem according to the schemes and notations of the original article. There are many good didactic explanations of the theorem that reveal its central points and implications, but these are difficult to recognize when reading the original work, due to the complexity of its formulation and the author's economical style in explaining the steps of his argument. An exposition of the central concepts will be made, as well as a detailed explanation of the main points of the algebraic development of the proof, which will allow the non-specialist reader to find the well-known paradox from them.

2007

This paper contains results that are not easy to find in the literature or unpublished, with linking material that is well known and published but which I include to tell a coherent story. Almost none of these results is original with me and those that I figured out for myself I am sure are known to others. My contribution is in bringing this material together. A number of these results I know through Georg Kreisel’s generous answers to my questions, for which I am very grateful. I presented some of this material at a conference to celebrate Godel’s centenary in 2006 organized by Russell Howell at Westmont College in Santa Barbara, California.

Social Science Research Network, 2003

I review the classical conclusions drawn from Gödel's meta-reasoning establishing an undecidable proposition GUS in standard PA. I argue that, for any given set of numerical values of its free variables, every recursive arithmetical relation can be expressed formally in PA by different, but formally equivalent, propositions. I argue that this asymmetry yields alternative Representation and Self-reference meta-Lemmas. I argue that Gödel's meta-reasoning can thus be expressed avoiding any appeal to the truth of propositions in the standard interpretation IA of PA. I argue that this now establishes GUS as decidable, and PA as omega-inconsistent. I argue further that Rosser's extension of Gödel's meta-reasoning involves an invalid deduction. The first is a direct expression of the relation as the PA-formula [q(k, m)] 3-since a recursive arithmetical relation q(x, y) can be expressed as a proposition in only the primitive symbols of PA once the variables x and y are replaced by the numerals k and m 4 , that represent the specific natural numbers k and m. The second is as a formally equivalent formula 5 [Q(k, m)] of PA that is defined in terms of the Gödel-Beta function by the usual Representation meta-Lemma ([Me64], p131). This meta-Lemma establishes that every recursive arithmetical relation, such as say q(x, y), is instantiationally equivalent to an arithmetical relation Q(x, y) that can be expressed in only the primitive symbols of PA. Thus, although q(x, y) may not necessarily be reducible to an expression that consists of only the primitive symbols of PA, Q(x, y) is always a well-formed formula of PA. Further, q(k, m) and Q(k, m) are equivalent arithmetical propositions that can both be expressed in PA by the formally equivalent PA-propositions [q(k, m)] and [Q(k, m)], respectively, for any given natural numbers k and m. It follows that though the relations q(x, y) and Q(x, y) are "arithmetically" equivalent, they are not "formally" equivalent. I argue that this asymmetry yields alternative Representation and Self-reference meta-Lemmas that are critical to any exposition of Gödel's meta-reasoning. This reasoning can 2 We use the terms "proposition" and "sentence" interchangeably. When referring to a well-formed symbolic expression, both terms imply that the expression has no free variables. 3 We use square brackets to indicate that the expression (including square brackets) only denotes the PAstring that is named by the expression within the brackets. Thus, "[q(k, m)]" is not part of the formal system PA. In this case, the PA-string named by "q(k, m)" is obtained by replacing the symbols constituting the arithmetical proposition "q(k, m)" by the PA-symbols of which they are the interpretations. The result is a PA-string of which "q(k, m)" is the arithmetic interpretation. 4 We denote by "n" the formal numeral in PA that represents the natural number "n" of IA. 5 We use the formal term "formula" as corresponding to the intuitive term "expression". By "well-formed formula" we mean a symbolic expression that is constructed according to some grammatical rules of a system for the formation of symbol strings, by concatenation of the primitive symbols of the system, that are to be considered as "well-formed".

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1978

Clin Chim Acta, 2003

Automated Deduction – CADE 27, 2019

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