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11月18日のツイート

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the nature of • the size of a number is • how many other numbers can it prove to be consistent. the largest number proves the consistency of all numbers, including itself. by Gödel's theorem, it is a "contradiction".

posted at 23:34:50

let us write Fin + Inf as ZFC. similarly, if we add "a large infinite number¹ exists", then we can prove the consistency of ZFC. • ZFC + LC ⊢ Con(ZFC) ¹ an inaccessible cardinal etc.

posted at 23:03:01

this means that Fin + Inf is a stronger theory than Fin. and this corresponds to • infinite numbers > finite numbers.

posted at 22:46:10

however, if we add "an infinite number exists", then we can prove the consistency of finite numbers. • Fin + Inf ⊢ Con(Fin)

posted at 22:39:17

the axioms "finite numbers exist" can't prove their own consistency.¹ • Fin ⊬ Con(Fin) ¹ Gödel's second incompleteness theorem

posted at 22:34:39

so the Q is, what tf is the largest infinite number. n it's a "contradiction" (aka 0=1). as drawn in this pic. let me explain from scratch. pic.twitter.com/ehKrmCRpgd

posted at 22:22:31

¹ it is shown • ℶ₁ ≠ ℵ₀ (Cantor) • ℶ₁ ≠ ℵ_ω etc. (König's theorem) tho.

posted at 21:58:42

infinities are written • ℵ₀, ℵ₁, ℵ₂, …, ℵ_ω, … in order from smallest to largest. for the number of reals ℶ₁, • ℶ₁ = ℵ_? is not provable from the standard axioms of mathematics.¹ the ℶ₁ = ℵ₁ conjecture is called the • continuum hypothesis (CH).

posted at 21:50:24


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