As I've recently been genning up on some of my undergraduate courses, I took a passing interest in the Part II Cambridge Maths Tripos questions from the exams sat earlier this year.
One question, from the Logic and Set Theory course (p63 at the above link), that caught my eye was the following:
If a theory $T$ in a language $L$ has arbitrarily large finite non-rigid models, show that $T$ also has an infinite non-rigid model.
I'm somewhat stumped by this!
Without the "non-rigid" bit, I'm familiar with the standard compactness argument for showing that arbitrarily large models imply infinite models -- essentially we add to the theory $T$ axioms of the form "$\phi_n$ :the model is at least size $n$" to form $T'$; as any finite subset of $T'$ contains only finitely many $\phi_n$s, it is satisfiable by a sufficiently large (finite) model for $T$, which we are told exists; hence by compactness $T'$ has a model, which will necessarily be infinite to satisfy all the $\phi_n$s.
The above question appears to be wanting a slight adaption to this argument, but that's easier said than done. For example, it would be nice if we could attach axioms to the theory that would encapsulate "the model is non-rigid", whereupon we could use the same argument outlined above. The problem however is that this would involve axiomatising "the model is non-rigid" and I'm not sure how to show that's possible in first-order logic -- indeed, I'm not even sure if it is possible.
Given that the compactness argument would only require axioms to be satsifiable in sufficiently large finite models, I wondered if I could somehow axiomatise the idea of a non-trivial automorphism on a finite subset of the model, but I could only do this by specifying a particular size for the subset, which then became too strong a condition to be guaranteed satisfiable by the larger finite models that the argument relies on.
Is anyone able to point me in the right direction?