# Difine a first order formula for positive numbers

• Jul 5th 2011, 02:11 AM
hmmmm
Difine a first order formula for positive numbers
Let $\displaystyle \mathcal{V}_{ar}={+,.,0,1}$ be the vocabulary of arithmetic. Let R be the structure that has universe $\displaystyle \mathbb{R}$ and interprets the vocabulary in the usual manner.

Define a $\displaystyle \mathcal{V}_{ar}$ formula $\displaystyle \alpha(x)$ such that for any
$\displaystyle a\in\mathbb{R}, R\models\alpha(a)$ if and only if a is positive.

Im sure I really should be able to do this but I cannot think how to.

Thanks for any help

For clarity $\displaystyle R$ is the structure and $\displaystyle \mathbb{R}$ is the real numbers.

(what is the LaTex for the reals?)

Thanks again
• Jul 5th 2011, 04:42 AM
emakarov
Re: Difine a first order formula for positive numbers
How about $\displaystyle a\ne0\land\exists x\,x\cdot x=a$?
• Jul 5th 2011, 07:57 AM
hmmmm
Re: Difine a first order formula for positive numbers
haha yeah thanks I was thinking about that for ages!