Gödel’s Gears

Natalie, once again, publishes something deeply fascinating.

Best sketch of the proof I’ve seen anywhere.

The paper napkin version:

1. Deterministically generate unique keys for every possible statement (and sequence of them) in an axiomatic system. Prime factorization works real good for this.

2. Show that statements about the unique keys themselves (as encoded information about the underlying statement) can still be meaningfully interpreted within the same axiomatic system.

3. Now do a liar’s paradox. Take pretty much any statement as a base and, via a scheme of several substitutions, generate a statement which asserts about itself that it cannot be proven. (In other words, craft a statement that says “there is no sequence of formulas which proves the statement x,” where x is the unique key of that very same statement.)

4. You’ve just blown a hole in the system.

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