well i'm off to bed, but i'll just go ahead and say

if you've defined the numbers like i did here >>7995958

and assigned the symbol "1" to the element [math] \{ \emptyset \} [/math]

then 1 = 1 follows directly from the axiom of extensionality

[math] \forall{A}. \forall{B}. (\forall{C}.(C \in A \iff C \in B) \implies A=B) [/math]

which says "if you take any sets A and B, and if for every set C, C is in A only if C is in B, then A is equal to B"

which, put less autistically, says "if A and B have the same elements, then they are equal"

so there's my first proof that 1 = 1

i get bored in class and do a lot of messing around with foundations and low-level stuff so i've got a lot more of this if anybody's interested

i kinda want to post it all TЬH but i'd rather post it in a thread where there's interest

i have category theoretic and type theoretic proofs that 1=1 as well, but they're pretty much the same as that proof up there^

also can somebody tell me where pic related is from?

i've image searched it and stuff but i can't find anything

i just want to know why the dodecahedron group is off by itself as if it's as important or even as interesting as