I need to prove that there is no set of wff Σ of the language with equality (i.e. the language the only symbol of which is that of equality and the interpretations of this language are sets) such that:
A is a model of Σ iff A is a finite set
I have been hinted that I need to use the theorem of compactness, but I can not figure it out... Any ideas?
Consider the theory $T$ that contains a formula $\phi_n$ for every $n\in \mathbb N$ such that $\phi_n$ states there exist $n$ elements that are all unequal to each other.
It should be easy to see that any finite subset of $T$ is satisfied in a finite model. But then by compactness there exists a model of $T$. Can such a model be finite?
Now let $\Sigma$ be a theory such that $\Sigma$ is true in a model if and only if the model is finite. Then show that $\Sigma\cup T$ is consistent.
The title of this question is not provable, since the theory $\{\forall xy(x=y)\}$ clearly has only finite models of size 1.