Current Research on Gödel’s Incompleteness Theorems

Abstract

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.

Finding the limit of incompleteness II

“…However, G2 is intensional and "whether the consistency of T is provable in T " depends on many factors such as the way of formalization, the base theory we use, the way of coding, the way to express consistency, the provability predicate we use, the way we enumerate axioms of the base theory, etc. For the discussion of the intensionality of G2, we refer to [2].…”

“…The notion of interpretation provides us a method to compare different theories in different languages. If T is interpretable in S, then all sentences provable (refutable) in T are mapped, by the interpretation function, to sentences provable (refutable) in S. 2 The interpretation relation among first order theories (✂) is reflexive and transitive. The equivalence classes of theories, under the equivalence relation ≡ I , are called the interpretation degrees.…”

Exploring the Foundational Significance of Gödelʼs Incompleteness Theorems

“…Gödel's sentence does not lead to a contradiction as the liar sentence does. In addition to the liar paradox, many other paradoxes have been used to give new proofs of incompleteness theorems: for example, Berry's paradox, the Grelling-Nelson paradox, the unexpected examination paradox, and Yablo's paradox [see Cheng (2021)]. The key point of proofs of Gödel's incompleteness theorems based on logical paradox is to properly formalize the logical paradox used.…”

“…These sentences have a clear mathematical flavor and do not refer to the arithmetization of syntax and provability. Paris and Harrington (1977) proposed the Paris-Harrington principle from ordinary mathematics with combinatorial content and proved that it is independent of PA. For more examples of independent sentences of PA from ordinary mathematics, refer to Bovykin (2006), Cheng (2019Cheng ( , 2021, and Friedman (2022).…”

Goedel's Incompleteness Theorem

“…3 S formally is a unitary operation whose domain are the terms of the language. 4 True, as it says 1 + 1 = 2.…”