Do I understand it correctly that in an axiomatic system that includes the natual numbers defined by using Peano's axiom, where the 9th axiom (induction) is formulated using 2nd-order logic, then Gödels incompleteness theorem is false ?
Can I say that if an axiomatic system defines the natural numbers properly (i.e. not allowing non-standard models), then Gödels incompleteness theorem will be false ?
Why do mathematicians accept axiomatic systems for the natural numbers, such as Peano arithmetic, which allow non-standard models ? If the axioms allow for multiple non-isomorphic models, then we are missing some axiom. Or am I throwing babies out with bathwater having this attitude ?