Let $L=\emptyset$ find a theory $T$ in $L$ so that for all structures $M$ in any language $M\models T$ if and only if $M$ is infinite

Question

Let $L=\emptyset$ find a theory $T$ in $L$ so that for all structures $M$ in any language, $M\models T$ if and only if $M$ is infinite

I'm not sure how to do this. I'm not even sure how its possible since I thought you can't create a finite number of sentences which if true imply a set is infinite, without relations.

Answer 1

Following the commentaries on your question. Let $T$ be the set of sentences $\bigcup \limits_{i\in \mathbb{N}} S_{i}$ where each $S_i$ is:

$S_1 = \exists x_1$

$S_2= \exists x_1 \exists x_2 (x_1\not = x_2)$

$S_3= \exists x_1 \exists x_2 \exists x_3 (x_1\not = x_2 \wedge x_1\not = x_3)$

And so on. Any model $M$ is clearly infinite, as assume it is finite and has only $n$ elements. Then, it doesn't satisfy $S_{n+1}$.