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
