I just learned today about Tarski Seidenberg theorem which implies the decidability of the reals (only with the field operations). I also know about Gödel incompleteness theorem, which implies that Peano arithmetic is not decidable.
I was thinking then what is wrong with statements of the form $$ x \text{ is a natural number} \wedge \phi(x) $$ Because then you would be able to encode statements about the naturals in the reals. Of course the problem is how to define something to be a natural number, and I was thinking by doing it like $$x \text{ is a natural number} \iff x \ge 0 \wedge \sin(\pi x) = 0$$ And $\sin(x)$ can be define as an infinite series. I guess the problem in this case would be on how to define limits (I am not sure where is the problem here, but there has to be one).
My question is then, what extra thing would you need in order to define the statement $ x \text{ is a natural number}$? and why can't you define $\sin(x)$ in the reals using only the field operations?
Thanks,
If nothing else, in non-standard models of the reals, the statement "natural number" will not mean what you want. There is a non-standard model of the naturals embedded in every non-standard model of the reals.
Looking at an ultra power of $\mathbb{R}$, (the equivalence of) the sequence $n\mapsto n$ would satisfy any reasonable first order attempt to define "is a natural number" by Los's theorem. But this member of the ultra product fails to be a standard real.
First order simply isn't expressive enough to express "obvious" notions sometimes. Alternatively, our intuition about obvious is wrong and statements like "is a natural number" are more complicated than we naively realize.