The foundations crisis

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The foundations crisis of formal reasoning: Cantor's different sizes of infinity, Russell's paradox, Hilbert's formalist program, and Gödel's 1931 proof that the program is impossible. Leibniz dreamed of a formal language that could decide any question by calculation — Gödel killed that dream, and this unit is about that story.

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done when you can

State Gödel's first incompleteness theorem in your own words, including what "sufficiently powerful" means.Explain Gödel numbering — how arithmetic encodes statements about itself.Explain what the second incompleteness theorem adds.Say what Gödel did not prove (Penrose's argument in Unit 5 of the Philosophy of Mind page depends on overclaiming here; be able to spot the overclaim).Place the crisis historically: what Hilbert wanted, what Russell's paradox forced, why the 1920s felt urgent.

resources

●Gödel's Proof— Ernest Nagel & James R. Newmananchor · book
●Gödel, Escher, Bach— Douglas Hofstadtercompanion · book
●Infinite Powers— Steven Strogatzcompanion · book
○LessWrong review of Gödel, Escher, Bachcompanion · blog post
○Before You Read GEBcompanion · blog post

sessions

unlocks Unit 2 (Turing is the computational version of the same story). Unit 5 of the Philosophy of Mind page (Penrose's argument about consciousness).

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