non-standard models of arithmetic in second order arithmetic?
Background: According to Godel's theorem, if we have, in a given consistent system S, a non-provable wff. A, then we can extend the system to either S1 or S2 by including A or ~A as a new axiom, respectively. Both S1 and S2, according to Godel, will be consistent. If S is a first order system, then both S1 and S2 have at least a model, (let us call them M1 and M2) and the two models are essentially different because in M1 A is true, but in M2 ~A is true. If S is PA (peano arithmetic), then that means that there are infinitely many unintended models in addition to the natural numbers, each one obtained by extending PA with a new axiom which is not true for the model of the natural numbers. These unintended interpretations are named non-standard models of arithmetic.
The question(s): The above conclusions refer to first order systems. However, it is unclear to why this argument is restricted to first order systems, as Godel's proof is also valid for second order systems: Any extension of second order PA can be consistently extended into two different systems by including any non-provable wff. A or ~A as a new axiom, respectively. These two systems should in turn describe two essentially different models, and so there should also be non-standard models of arithmetic in second order arithmetic.
The problem is, that it is often stated that second order PA determine the set of natural numbers uniquely, and so there should be not room for non-standard models of arithmetic in second order arithmetic, which contradicts what .
Metaquestion: I am sure there is something wrong in my reasoning leading to this contradiction. But I cannot find any errors, can anybody help me, please!!!!