Cross-posted at Brick. A fun exploration, hopefully requiring only a medium level of technical familiarity.

"Not with a bang but a whimper"

It's 1930 and Kurt Gödel has just delivered a talk at Königsberg that carpet bombed Hilbert's Program.

...and no one noticed.

Godel's announcement, delivered during the summarizing session on the third and last day of the conference, was so understated and casual—so thoroughly undramatic—that it hardly qualified as an announcement, and no one present, with one exception, paid it any mind at all.
(source)

Well, almost no one. John von Neumann came up to Gödel after the talk, presumably to double check if he was going crazy or if Gödel had in fact just proved the existence of unprovable mathematical truths.

To understand what Gödel's Incompleteness Theorem says, it helps to see the meta-mathematical opinions of some of his peers. Wittgenstein opens his big famous book (not that I've actually read it) with:

One could put the whole sense of the book perhaps in these words: What can be said at all, can be said clearly; and whereof one cannot speak, thereof one must be silent.

This was a bold move in its own way. Contrast it with the following take, "Everything can be spoken of, you just need to get smarter" which is a deeply held ideal of Strawman STEM-Lord. Interestingly, both Wittgenstein and STEM-Lord agree on something, the exact something that Gödel smashed up in Königsberg. It's a very technical claim, best communicated with the following GIF:

Witt and STEM-Lord disagree on what bleeds, but they are in fervent agreement that what bleeds can be killed.

Unraveling Self-reference

Take the classic liar's paradox:

Oh shit, it's true and false, or neither, paaaaaaaaaaaaaaaradox! The Wittgenstein-esque approach might be to claim the liar's sentence doesn't actually mean anything. There is no reality to it to speak of, so you can't speak of it. You are just getting yourself tied into a linguistic knot, tricking yourself into thinking that just because you can write this sentence it must mean something. Natural language is not a formal system and this sentence need not have a coherent "logical" "truth value". BOOM, checkmate hypothetical opponent!

Hidden in the explanation "Language isn't meant to avoid paradox and self-reference" is the claim "but formal math does avoid them!" The first part is true, the s

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