On the Diagonal Lemma of Gödel and Carnap

Abstract

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.

Tarski's Undefinability Theorem and Diagonal Lemma

“…By incorporating the proof of Theorem 2.4 into this proof (e.g. by taking Υ ≡ ¬Ψ), one can get a proof for the Semantic Diagonal Lemma, which is different from the standard (textbook) proofs (see [10]).…”

“…ONE OF THE CORNERSTONES of modern logic (and theory of incompleteness after Gödel) is the Diagonal Lemma (aka Self-Reference, or Fixed-Point Lemma) due to Gödel and Carnap (see [10] and the references therein). The lemma states that (when α → α is a suitable Gödel coding which assigns the closed term α to a syntactic expression or object α) for a given formula Ψ(x) with the only free variable x, there exists some sentence θ such that the equivalence Ψ( θ ) ↔ θ holds; "holding" could mean either being true in the standard model of natural numbers N or being provable in a suitable theory T (which is usually taken to be a consistent extension of Robinson's arithmetic).…”

An incompleteness theorem via ordinal analysis

“…Moreover, Gentzen's proof refines Gödel's result by exhibiting the first example of a non-meta-mathematical arithmetic statement-namely, the statement that ε 0 lacks primitive recursive descending sequences-that is not provable from the Peano axioms. Though Gödel's result is highly general, his proof relies on selfreference, rendering it opaque and mysterious [5,14,15]. By contrast, Gentzen's proof is concrete but his results are specific to the case of Peano arithmetic.…”

An Incompleteness Theorem via Ordinal Analysis

“…Moreover, Gentzen’s proof refines Gödel’s result by exhibiting the first example of a non-meta-mathematical arithmetic statement—namely, the statement that lacks primitive recursive descending sequences—that is not provable from the Peano axioms. Though Gödel’s result is highly general, his proof relies on self-reference, rendering it opaque and mysterious [6, 15, 16]. By contrast, Gentzen’s proof is concrete but his results are specific to the case of Peano arithmetic.…”