On constructivity and the Rosser property: a closer look at some Gödelean proofs

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.

Section: Recursion-theoretic Proofs Gödel's First Incompleteness Thementioning

confidence: 99%

Section: Recursion-theoretic Proofs Gödel's First Incompleteness Thementioning

confidence: 99%

Current Research on Gödel’s Incompleteness Theorems

Cheng¹

2021

“…Then <v (u) says that u is the least number not definable by any formula with length less than v. It is rather easy to see that the length of the formula B(x) is less than 5ℓ (cf. [26]). So, the relation of the formulas <v (u) and B(u) with the Berry's paradox is apparent now.…”

Section: The Semantic Form Of the Diagonal Lemmamentioning

confidence: 99%

“…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]. Before Chaitin and Boolos, Rosser [24] (in 1936) and Kleene [18,19] (in 1936 and 1950) had given alternative proofs for the first incompleteness theorem.…”

mentioning

confidence: 99%

“…Before Chaitin and Boolos, Rosser [24] (in 1936) and Kleene [18,19] (in 1936 and 1950) had given alternative proofs for the first incompleteness theorem. Their proofs did not use Berry's paradox (see [26] and the references therein), and, instead, used ideas of computability theory. The computability theory approach to proving the incompleteness has the conceptual advantage of linking the study of incompleteness with the study of computability in a smooth way.…”

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confidence: 99%

See 1 more Smart Citation

On the Diagonal Lemma of Gödel and Carnap

Salehi¹

2020

Self Cite

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.

Gödel’s Second Incompleteness Theorem: How It Is Derived and What It Delivers

Salehi

2020

Self Cite

The proofs of Gödel (1931), Rosser (1936), Kleene (first 1936 and second 1950), Chaitin (1970), and Boolos (1989) for the first incompleteness theorem are compared with each other, especially from the viewpoint of the second incompleteness theorem. It is shown that Gödel’s (first incompleteness theorem) and Kleene’s first theorems are equivalent with the second incompleteness theorem, Rosser’s and Kleene’s second theorems do deliver the second incompleteness theorem, and Boolos’ theorem is derived from the second incompleteness theorem in the standard way. It is also shown that none of Rosser’s, Kleene’s second, or Boolos’ theorems is equivalent with the second incompleteness theorem, and Chaitin’s incompleteness theorem neither delivers nor is derived from the second incompleteness theorem. We compare (the strength of) these six proofs with one another.