Gödel's Incompleteness Theorems and the Nature of Truth

Prof. Joel David Hamkins expounds on the philosophical implications of Gödel's incompleteness theorems and what they tell us about mathematical logic, formal systems, and possibly, the nature of truth as such.

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[Robert_McGarry_Poems]

Taking that to the extreme... We can never describe the universe as we are imbedded in it. We can never describe the plank realm either... But we can talk about everything in-between, we just can't prove the usefulness of anything before it happens.

[ShaunLovesMaths]

Some systems, necessarily that do not rely on (i.e., "contain" a copy of) Peano Arithmetic, can get around Gödel's Incompleteness Theorem (the first one). For example, there's the trivial system in which everything is reguarded as true. Its axioms are simply described by the family A for all statements A. Is A true in that system? Yes. Proof: A. Q.E.D. I think some nontrivial systems exist, too, of course, like some of those concerning restricted versions of graph theory.