1. Preburger arithmetic is a very limited subset of Peano arithmetic which does not even contain multiplication. When we speak of natural numbers we mean Peano arithmetic. So, you have to consider all functions `f : NN -> 2` definable in Peano arithmetic, and it is impossible to decide whether such a function in general will always output `1` (actually, this already cannot be done for primitive recursive functions, which are a small subfamily of all number-theoretic functions). To put it in another way, of course by changing the assumptions you can draw a different conclusion.

2. Tarski’s axioms are for real closed fields, not natural numbers. The set of natural numbers is not definable in Tarski’s theory. So your second remarks is not relevant because I was talking about function `NN -> 2`, not reals.

2. Another option of existing more than two truth values on `Omega` is when there are independent sentences on the theory, like the ‘Goodstein’s theorem’ on ‘Peano axioms’

There is a lot of confusion about what it means to have two-valued logic (which is why I wrote this post). Lukasiewicz’s three-valued logic is not a model of intuitionistic logic, because some rules of inference are not valid in it. I was not talking about just any arbitrary thing that anyone might call logic, but rather specifically about intuitionistic logic. But that does not matter since we can find models of intuitionistic logic which seemingly have “three values”, for example the topos of presheaves on the category consistig of two objects and an arrow between them. In fact, given any Heyting algebra `H` with as many points as you like, you can find a model of intuitionistic logic, namely the `H`-valued sets, such that `Omega` is the set `H` (with a suitably defined `H`-valued equality predicate). But now it is important not to confuse the question “How many points does `H` have?” with the question “Inside the model, how many points does `Omega` have?”. Inside the model you can prove the validity of the statement `not exists p in Omega . (p != TT and p != _|_)`, but in the metatheory you can easily find a Heyting algebra `H` with more than two points. Please do not confuse a theory with meta-theory.