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

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

by Gödel's second incompleteness theorem, the axioms "finite numbers exist" can't prove their own consistency. • Fin ⊬ Con(Fin) now if we add "ℵ₀ exists", then we can prove the consistency of finite numbers. • Fin + Inf ⊢ Con(Fin) this shows Fin + Inf > Fin.

¹ this indicates "direct implications or relative consistency implications", tho. (e.g., huge < supercompact) in order of size and strength: • inaccessible < measurable < huge < rank-into-rank < 0=1

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

actually, large cardinals refer to large ♾ with specific properties. (the successor of an inaccessible cardinal is not inaccessible by definition.)

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

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).

actually, there's infinitely many infinite numbers.

¹ already known to the Jains of India (400 BC), tho. but they made mistakes such as the number of points on a line |ℝ| and a plane |ℝ²| are not equal. (both are 2^ℵ₀.)

actually, the number of natural, integer, and rational numbers are all the same. that infinite number is called ℵ₀ (aleph zero) or ℶ₀ (beth zero). Cantor (1874) proved that the number of reals is larger than that.¹ that is called 𝔠 = 2^ℵ₀ = ℶ₁ (beth one).

the idea that having the same number is the same as having a 1-1 correspondence is called • Hume's principle.¹ ¹ neither Hume nor Galileo thought it applied to ♾, tho (unlike Cantor).

have a look at this strange pic. there's 2 circles. a big one and a small one. however, both circles consist of the same number of dots. (there's a 1-1 correspondence.) blog.wolframalpha.com/2010/0…

♾ is such a strange thing. the whole and part can be the same size.¹ cf. • Galileo's paradox. ¹ contrary to Euclid's 5th Common Notion (300 BC). although some say that some CNs are by, e.g., Theon of Alexandria (4c).

mathematically, the answer is A✔ they're the same. (even tho evens are part of integers.) this is bc one integer corresponds to one even. 1 ↦ 2 2 ↦ 4 3 ↦ 6 4 ↦ 8 5 ↦ 10 ︙ n ↦ 2n ︙

A) same bc they're infinite B) infinity can't be compared C) integers ofc bc evens are only half of them

let me explain from scratch. Q: which is greater, • the number of all integers • the number of all even numbers both are infinite. there's many ways to think abt it.

mathematicians consider infinity a number. this is bc in the 19c, a man named Cantor found that there are "multiple infinities". i.e., infinity 1, infinity 2, infinity 3, …. just like 1, 2, 3, …. can also do arithmetic with ♾.

im summarizing my tweets here rn min.togetter.com/kb628xT

The largest number is a "contradiction". let me explain this. u may be reminded of a googolplex or graham's number. the problem is: graham's number+1 is larger. they're still finite.