# Are the natural numbers definable in the (2nd-order) theory of complete ordered fields?

I read that the natural numbers are not definable in the theory of real closed fields (RCF), which captures the 1st-order properties of the real numbers. That's why decidability of RCF doesn't contradict Gödel's theorem.

But are the natural numbers definable in the 2nd-order theory of complete ordered fields (COF), which categorically captures the real numbers?

Edit : Looks like the answer was simple, after some thinking. Since COF is a 2nd-order theory, we can quantify over subsets of $$\mathbb R$$. Then expressing that $$\mathbb N$$ is the smallest inductive subset of $$\mathbb R$$ should do.

Essentially, for a subset $$S\subseteq\mathbb R$$, define the (meta-)property $$I(S)$$ of being inductive : $$I(S) := (0\in S \wedge (\forall x\in\mathbb R : x\in S\rightarrow x+1\in S))$$ Define the (meta-)property $$N(S)$$ of being the smallest inductive set : $$N(S) := (I(S) \wedge (\forall T\subseteq\mathbb R : I(T)\rightarrow S\subseteq T))$$ Then $$\mathbb N$$ is the unique subset $$S$$ of $$\mathbb R$$ such that $$N(S)$$.

I hope there is no mistake.

• That looks like the way I've seen the natural numbers defined in textbooks that start from the axioms for a complete ordered field. Now sure what you mean by (meta-). By "meta-property" do you just mean a second order property? What's "meta" about that?
– bof
Commented Nov 30, 2018 at 10:33
• It's just that the "predicates" $I$ and $N$ are not symbols of the language of COF, they are merely abbreviations. They are not predicates inside of COF. That's what I meant. Commented Nov 30, 2018 at 10:36
• So every defined notion, like $2$ or $-$ or $!$ or $\exp$, is "meta"? All right.
– bof
Commented Nov 30, 2018 at 10:40
• If you define 1st-order objects (terms, functions, relations), you can add them to the language. For example, add the term symbol "2" and the defining axiom "2=1+1", and "2" is now internal to the theory. But if you conceive a property about a subset (like $I(S)$ above, which is a property of the subset $S$), you can't add the symbol $I$ to the language since in a 2nd-order language, there are no such kind of symbols... at least, that I know of. Commented Nov 30, 2018 at 12:45
• Aaaaaand another question in the huge pile of unanswered questions which get bumped but which are actually already answered in the comments and even in the post itself. People, please write answers. Commented Jan 4 at 1:15

In second-order logic (with the standard semantics) we can define $$\mathbb{N}$$ in the ordered field $$(\mathbb{R};+,\times,0,1,<)$$ as the smallest inductive set: $$n$$ is a natural number iff $$n$$ is in every inductive set.
Is there a formula of the form $$\exists X\theta(x,X)$$ with $$\theta$$ involving only first-order quantifiers such that $$\{r: \mathbb{R}\models\exists X\theta(r,X)\}=\mathbb{N}$$?
$$n$$ is a natural number iff there is some set $$X$$ containing only nonnegative reals such that $$0\in X$$, $$m\in X\implies m+1\in X$$, and $$m\in X$$ and $$0<\epsilon<1$$ implies $$m+\epsilon\not\in X$$.