2021 Bulletin of Symbolic Logic  
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. Introduction Gödel's first and second incompleteness theorem are some of the most important and profound results in the foundations of mathematics and have had wide
more » ... e 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 [46] 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 [153], p.431). Gödel comments in a footnote of [46] 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 2000 Mathematics Subject Classification. 03F40, 03F30, 03F25.
doi:10.1017/bsl.2020.44 fatcat:uqklozcs3rapxc6t25fylacf7i