finiteness and first order sentences Lets consider a set of sentences $T$ and a signature $\sigma$. I proved (using compactness theorem) that when $T$ has arbitrary large models than also an infinite model. Now there are several consequences from this statement but I have problems formulating the connection to the statemant above.


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*Finiteness is not first order characterizable, i.e there is no set of sentences $T$ such that $M$ is a $T$-model if and only if $M$ is finite. 


In my opinion this is a direct consequence because $T$ also has infinte models, therefore the finiteness of $M$ is no sufficient condition being a $T$-model.


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*Infinity is not characterizable through one single sentence, i.e there is no sentence $\phi$ such that $M$ is a $\{\phi\}$-model if and only if $M$ is infinity.


I do not know how to argue here. 
 A: As for your first point, you are correct.
For, suppose there were such a set of sentences $T$ (hereafter, I speak of a theory). Then $T$ has arbitrarily large finite models, but no infinite model. This contradicts the result you have obtained.
For the second, suppose $M \models \phi$ iff $M$ is infinite. Then what can we say about the theory $T = \{\neg \phi\}$ consisting only of the negation of $\phi$?

When dealing with a certain fixed theory $T$ (e.g. the theory of groups), we can follow a similar strategy to prove that (essentially) the same results hold for models of $T$:
Using Compactness, we prove that $T \cup \{\neg\phi_n:n \in \Bbb N\}$ (where $\phi_n$ is the standard "there are at most $n$ distinct elements" sentence) is consistent, i.e. that $T$ admits infinite models, from the fact that arbitrarily large finite models of $T$ exist.
Analogous to the first point, we prove that "being a finite model of $T$" is not first-order axiomatisable.
Finally, we prove that there is no single sentence axiomatising infinite $T$-models. Suppose $\phi$ were a single sentence expressing "being an infinite model of $T$". Then $T \cup \{\neg \phi\}$ is a first-order axiomatisation of "being a finite model of $T$", which we had shown to be impossible.
