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.