Proof of the Diagonal Lemma in Logic

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fairly easy
Why is the proof so short yet so difficult? Saeed Salehi is a logician at the University of Tabriz in Iran. Three years ago he gave a presentation at a Moscow workshop on proofs of the diagonal lem…
Proof of the Diagonal Lemma in Logic

Why is the proof so short yet so difficult?

Saeed Salehi is a logician at the University of Tabriz in Iran. Three years ago he gave a presentation at a Moscow workshop on proofs of the diagonal lemma.

Today I thought I would discuss the famous diagonal lemma.

The lemma is related to Georg Cantor's famous diagonal argument yet is different. The logical version imposes requirements on when the argument applies, and requires that it be expressible within a formal system.

The lemma underpins Kurt Gödel's famous 1931 proof that arithmetic is incomplete. However, Gödel did not state it as a lemma or proposition or theorem or anything else. Instead, he focused his attention on what we now call Gödel numbering. We consider this today as "obvious" but his paper's title ended with "Part I". And he had readied a "Part II" with over 100 pages of calculations should people question that his numbering scheme was expressible within the logic.

Only after his proof was understood did people realize that one part, perhaps the trickiest part, could be abstracted into a powerful lemma. The tricky part is not the Gödel numbering. People granted that it can be brought within the logic once they saw enough of Gödel's evidence, and so we may write for the function giving the Gödel number of any formula and use that in other formulas. The hard part is what one does with such expressions.

This is what we will try to motivate.

Tracing the Lemma

Rudolf Carnap is often credited with the first formal statement, in 1934, for instance by Eliott Mendelson in his famous textbook on logic. Carnap was a member of the Vienna Circle, which Gödel frequented, and Carnap is considered a giant among twentieth-century philosophers. He worked on sweeping grand problems of philosophy, including logical positivism and analysis of human language via syntax before semantics. Yet it strikes us with irony that his work on the lemma may be the best remembered.

Who did…
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