I'm trying to axiomatize the following statement in first order language: "every polynomial of an odd degree has a root" using infix notation.
For example, I can write "every polynomial of degree n has a root" as follows:
$\forall a_1\forall a_2...\forall a_n(a_n\neq 0 \rightarrow \exists x(a_1x^1+a_2x^2+...+a_nx^n=0))$
But how can I make $n$ to be odd? i.e. $\exists m\in\mathbb{N} (n=2m+1)$. Obviously we can't specify the variable $m$ to be a natural number.
This is not a single statement. It is a schema. For every odd integers $n$, we write the following sentence: $$\forall a_0\ldots\forall a_n\exists x(a_n\neq 0\rightarrow a_n x^n+\ldots+a_0=0),$$ Presumably, we are in the language of rings, so the exponentiation is just a shorthand for repeated multiplication.
But the oddity of $n$ is in the meta-theory. Not in the theory.