On Godel's treatment of the undecidable in 1931

In this article we discuss the proof in the short unpublished paper appeared in the 3rd volume of Godel’s Collected Works entitled “On undecidable sentences” (*1931?), which provides an introduction to Godel’s 1931 ideas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 Godel’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 Godel’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 of the application of substitution in the well-known Godel formula as a violation of uniqueness is also briefly presented, questioning its very derivation.

💡 Research Summary

The paper undertakes a close reading of the little‑known manuscript “On undecidable sentences” that appears in the third volume of Gödel’s Collected Works. The author’s aim is to show that Gödel’s original 1931 argument contains hidden assumptions about the negation of the provability predicate and that the treatment of the non‑provable case is incomplete. By re‑examining the definition of the class K (the set of Gödel numbers n such that ¬Bew(ϕₙ(n))) the author argues that Gödel’s informal reading of ¬Bew as “not provable” leads to a vicious‑circle interpretation: from ⊢ ¬Bew(ϕₖ(k)) one would immediately infer ⊢ ¬ϕₖ(k), and together with ⊢ ϕₖ(k) this yields a direct contradiction. The paper proposes instead that Gödel meant ¬Bew(x) to express the arithmetic fact “the Gödel number x is not the number of a provable formula”, i.e., a meta‑mathematical statement about coding rather than a claim about truth.

To fill the gap concerning the non‑provable side, the author introduces two new primitive‑recursive relations: xW y (x is a refutation of y) and Wid(x) (∃z zW x, i.e., x is refutable). These are shown to be decidable in the same way as the original provability predicate Bew. Four lemmas establish that a number cannot simultaneously be a proof and a refutation, and that the characteristic functions of proof and refutation are complementary.

Using these tools the author rewrites the definition of K as a chain of logical equivalences:

n∈K ⇔ ¬Bew(ϕₙ(n)) ⇔ ¬∃y yB ϕₙ(n) ⇔ ∀y ¬yB ϕₙ(n) ⇔ mW ϕₙ(n) ⇔ Wid ϕₙ(n) ⇔ ⊢ ¬ϕₙ(n).

Thus membership in K is equivalent to the formal derivability of the negation of the corresponding self‑referential sentence, not merely to the failure of provability. Consequently the step in Gödel’s original proof where one substitutes k=n to obtain ϕₖ(k) ⇒ ¬ϕₙ(n) is shown to violate the uniqueness principle (x=y ⇒ (φ(x)↔φ(y))) that holds in arithmetic. Because ϕₖ(n)↔¬ϕₙ(n) cannot be true when k=n, the substitution is illegitimate, and the whole “Gödel sentence” construction collapses under this scrutiny.

The paper therefore reaches three main conclusions:

1. The negation of the provability predicate must be interpreted as a meta‑mathematical coding condition, avoiding circularity.
2. By extending Gödel’s arithmetisation to include a primitive‑recursive refutation predicate, the previously ignored “non‑provable” case can be handled rigorously.
3. The standard derivation of the Gödel sentence contains an implicit uniqueness violation, rendering the original argument incomplete.

Overall, the author argues that Gödel’s 1931 incompleteness result does not rest on a simple self‑referential paradox but on a more intricate interplay of provability, refutability, and coding within arithmetic. The paper invites a reassessment of the foundational assumptions behind Gödel’s theorem and suggests that a corrected formalisation, incorporating both provability and refutability predicates, yields a clearer picture of why any sufficiently strong formal system must contain undecidable sentences.