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Truth and provability, a thread. Let us compare Tarski's theorem on the nondefinability of truth with Gödel's incompleteness theorem. Smullyan advanced the view that much of the fascination with Gödel's theorem should be better directed toward Tarski's theorem.
Both theorems begin with the profound arithmetization idea, Gödel's realization that arithmetic interprets essentially any finite combinatorial process, including arithmetic syntax itself. Every arithmetic assertion φ is thus represented in arithmetic by Gödel code 'φ', a number.
Gödel's theorem (the first incompleteness theorem) is the claim that no computability axiomatizable true arithmetic theory can prove all arithmetic truths. Every such theory will admit true but unprovable assertions.
I am preparing a poster and video about counting beyond infinity—how to to count to ω². Comments welcome—please criticize. Final version will appear later. #PhilMaths 
Updated version.
Final version appears on my blog. Free pdf download available. jdh.hamkins.org/counting-to-in…
