User talk:Hakerh400/Conjectures
Conjecture 4 is underspecified. There are multiple models of first-order presentations of Peano arithmetic, by Gödel's incompleteness results; for any Gödel sentence G which is true but unprovable in Peano arithmetic, there is a model where G is an axiom and another where ~G is an axiom.
The conjecture is true for the standard model, where the only natural numbers are zero and its successors. This follows from its provability in a second-order presentation; all second-order presentations of the natural numbers are categorical.
By definition, the conjectured expression gives addition of natural numbers. Addition of natural numbers is commutative; this can be shown with first-order Peano axioms alone, e.g. [1]. Indeed, the initial segment of natural numbers always forms a commutative semiring, and any non-standard number has a commutative ring around it, generated by polynomials on that number. I would guess that the statement is provable in first-order PA, and thus true in all models. This would match my understanding of non-standard models. However, I can easily be wrong, so we should desire a proof.