I am working within a first order language with one unary function symbol $f$ and no other nonlogical symbols.
I have written down sentences for 'f is not injective' and 'f is surjective denoted by $\alpha$ and $\beta$ respectively.
$\alpha:\exists a \exists b(f(a)=f(b)\wedge\neg(a=b))$
$\beta:\forall a\exists b(f(b)=a)$
$\sigma_n$ is the statement "There are at least $n$ elements in the domain"
$\sigma_n:\exists x_1 \exists x_2 ... \exists x_n (\neg(x_1=x_2)\wedge\neg(x_1=x_3)\wedge...allPossiblePairs)$
I need to show that $\alpha,\beta\models\sigma_n$ for each $n\in\Bbb{N}$
The semantics I am using is:
$\Sigma\models\sigma$ means for all structures $M$, if $M\models\Sigma$ then $M\models\sigma$
So I need to show that if a structure makes $\alpha$ and $\beta$ true, it also makes $\sigma_n$ true. But how on earth do I introduce $n$ into the argument.