Again: every mathematical error is a real physical even in someone’s brain, so , again, physics guarantees nothing.
I don’t get what you’re trying to show with this. If I mistakenly derive in Peano Arithmetic that 2 + 2 = 3, I will find myself shocked when I put 2 apples inside a bag that already contains 2 apples and find that there are now 4 apples in that bag. Incorrect mathematical reasoning is physically distinguishible from correct mathematical reasoning.
There are of course, lots of infinities in maths.
Everything we know about all other infinities can be built on top of just FORALL in first-order logic.
In ZFC, the Axiom of Infinity can be written entirely in terms of ∈, ∧, ¬, and ∀. Since all of math can be encoded in ZFC (plus large cardinal axioms as necessary), all our knowledge about infinity can be described with ∀ as our only source of infinity.
Only for the subset of maths that’s also physical. You can’t resolve the Axiom of Choice problem that way.
You can’t resolve the Axiom of Choice problem in any way. Both it and its negation are consistent.
I don’t get what you’re trying to show with this. If I mistakenly derive in Peano Arithmetic that 2 + 2 = 3, I will find myself shocked when I put 2 apples inside a bag that already contains 2 apples and find that there are now 4 apples in that bag. Incorrect mathematical reasoning is physically distinguishible from correct mathematical reasoning.
Everything we know about all other infinities can be built on top of just FORALL in first-order logic.
Only for the subset of maths that’s also physical. You can’t resolve the Axiom of Choice problem that way.
Whatever “built on top of” means. Clearly, we can intend transfinite models.
In ZFC, the Axiom of Infinity can be written entirely in terms of ∈, ∧, ¬, and ∀. Since all of math can be encoded in ZFC (plus large cardinal axioms as necessary), all our knowledge about infinity can be described with ∀ as our only source of infinity.
You can’t resolve the Axiom of Choice problem in any way. Both it and its negation are consistent.