# Thread: First-order formulae for b is greater than or equal to a.

1. ## First-order formula for b is greater than or equal to a.

Let $\mathcal{V}_{ar}=\{+,.,0,1\}$ be the vocabulary of arithmetic. Let R be the structure that has universe $\mathbb{R}$ and interprets the vocabulary in the usual manner.

Define a $\mathcal{V}_{ar}$ formula $\beta(x,y)$ such that for any
$a,b\in\mathbb{R}, R\models\beta(a,b)$ if and only if $b\geq{a}$.

Is this correct (i think that it is but im just a little unsure)

$a=b\vee (\exists{y}(y\neq0\wedge\exists{x}(x.x=y))\wedge a+y=b)$

thanks for any help

2. ## Re: First-order formula for b is greater than or equal to a.

Yes, this would work.