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.
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.