Exploring the Foundational Significance of Gödelʼs Incompleteness Theorems

Arithmetic and the Incompleteness Theorems

Gödel’s Incompleteness Theorems

In 1931, when he was only 25 years of age, the great Austrian logician Kurt Gödel (1906– 1978) published an epoch-making paper [16] (for an English translation see [8, pp. 5–38]), in which he proved that an effectively definable consistent mathematical theory which is strong enough to prove Peano’s postulates of elementary arithmetic cannot prove its own consistency. In fact, Gödel first establ...

Gödel's Incompleteness Theorems

Gödel’s two incompleteness theorems [2] are formalised, following a careful presentation by Świerczkowski [3], in the theory of hereditarily finite sets. This represents the first ever machine-assisted proof of the second incompleteness theorem. Compared with traditional formalisations using Peano arithmetic [1], coding is simpler, with no need to formalise the notion of multiplication (let alo...

Teaching Gödel’s incompleteness theorems

The basic notions of logic—predicate logic, Peano arithmetic, incompleteness theorems, etc.—have for long been an advanced topic. In the last decades, they became more widely taught, in philosophy, mathematics, and computer science departments, to graduate and to undergraduate students. Many textbooks now present these notions, in particular the incompleteness theorems. Having taught these noti...

Model theoretic proofs of the incompleteness theorems

Godel's proofs of the incompleteness theorems are frequently discussed in the context of constructive or nitary viewpoint. Godel showed a concrete independent statement of arithmetic and the proof does not contain any transcendental argument. On the other hand, the completeness theorem asserts that a sentence is not derivable from a theory if and only if there exists a model of the theory whi...