Comment by bo1024

Oof @ many of these answers. The reason is that there are multiple "models" of the natural numbers that satisfy the axioms, and the statement is true in the usual model that we all know, but in some other models the statement is false.

See https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_... and https://en.wikipedia.org/wiki/Peano_axioms#Nonstandard_model...

Godel's "completeness" theorem gives a converse. If the statement were true about all possible models, then it would be provable.