0

First of all, the statement that I'd like to prove is basically similar to the statement in these three links below:

  1. Prove there is no theory whose models are exactly the interpretations with finite domains
  2. Compactness and axiomatisability
  3. compactness model theory question

Let me write down what the statement that I want to prove: There is no theory $K$ whose models are exactly the interpretations with finite domains. In all of these three links, they use compactness theorem and they consider a wf $$\varphi_n=\exists x_1\cdots\exists x_n\left(\bigwedge_{0<j<i\le n}x_i\ne x_j\right).$$

I kind of understand their proofs, but I do have a doubt about these proofs. Can I really consider the above wf in a general theory $K$? It seems like I can consider that wf when $K$ is a first-order theory with equality.

If I can't consider the above wf in a general theory $K$, how can I prove the statement? I'd like to have an idea how to show it, so a hint would be appreciated.

Vicky
  • 971
  • 7
  • 11
  • 2
    Are you sure you're not missing an implicit or explicit assumption that $K$ is a first-order theory with equality? You can, for instance, find a set of sentences in 2nd order arithmetic modeled exactly by the finite sets, so this doesn't hold in full generality (not to mention it's pretty rare to prove anything substantial about "theories" in total generality). – Alexander Tenenbaum Oct 04 '20 at 19:05
  • Well it is possible that I missed that assumption. But I found this statement in a section where the first-order theory with equality has not been introduced yet, that's why I had my doubts. Also, what do you mean by 2nd order arithmetic? – Vicky Oct 04 '20 at 19:13
  • 4
    In the context of model theory, "theory" typically refers to a first-order theory unless specified otherwise (and first-order logic is typically taken to always include equality). – Eric Wofsey Oct 04 '20 at 19:21
  • @VicRyan Second order arithmetic is like Peano Arithmetic, but you are allowed to quantify over predicates (or equivalently sets) as well as just numbers. – Alexander Tenenbaum Oct 04 '20 at 19:44

0 Answers0