Prove $\{\neg\tau\}\cup\{\sigma_n:n\in\Bbb{N}\}$ has a model by compactness

Question

$\sigma_n$ is the statement "There are at least $n$ elements in the domain"

$\sigma_n:\exists x_1 \exists x_2 ... \exists x_n (\neg(x_1=x_2)\wedge\neg(x_1=x_3)\wedge...allPossiblePairs)$

$\tau$ is a sentence which satisfies:

$M\models\tau\Leftrightarrow\textrm{M is infinite}$

I need to use the compactness theorem so I need to show that every finite subset of $ \{ \neg\tau\}\cup\{\sigma_n:n\in\Bbb{N}\}$ has a model. Any ideas?

Answer 1

Let $F$ be a finite subset of the $\sigma_n$. Then there is a $b$ such that all the $\sigma_n$ in $F$ have $n\le b$. Let $M_b$ be any $L$-structure for our first-order language $L$ whose underlying set has $b$ elements. Then $\lnot\tau$ and all the $\sigma_n$ in $F$ are true in $M_b$.