Did God make the rules?

[PolyHedral]

Actually, I think I understand you enough to point out how this can't work: If 4, or just {{{{Ø},Ø},Ø},Ø} exists as an inherent part of the universe, then number theory must exist as an inherent part of the universe. This would mean that GIT applies in full force, and I think I pointed out earlier why that leads to contradictions.

 

[Meow Mix]

Actually, I think I understand you enough to point out how this can't work: If 4, or just {{{{Ø},Ø},Ø},Ø} exists as an inherent part of the universe, then number theory must exist as an inherent part of the universe. This would mean that GIT applies in full force, and I think I pointed out earlier why that leads to contradictions.

Not necessarily.

These consistent and complete axioms can get us to four:

1) ∀x : ¬(0 = x + 1)
2) ∀x∀y : ¬(x = y) --> ¬(x + 1 = y + 1)
3) ∀x : x + 0 = x
4) ∀x∀y : (x + y) + 1 = x + (y + 1)

GIT doesn't apply to that. These axioms can't produce multiplication, but addition through infinity is no problem.

 

[PolyHedral]

But producing multiplication from addition is merely a case of defining the function correctly.

Let:
mul(x,1) = x
mul(x,y) = x + mul(x,y-1)

 

[Meow Mix]

But producing multiplication from addition is merely a case of defining the function correctly.

Let:
mul(x,1) = x
mul(x,y) = x + mul(x,y-1)

I haven't taken a whole lot of proofing at this point but you'd have to prove that your definition can't lead to contradictions (consistency) and that it's complete (all your statements can be proven true or false).

For GIT to kick in you must have enough in the axiomatic system to make it effective at checking itself for axioms.

I could crack open my old books or take this to school with me and see if indeed it's consistent and complete but something tells me that the system I was describing earlier doesn't include multiplication for a reason (because it's a famous system for NOT being subject to GIT).