@LeakyNun: Um why did you delete your last message?

@AlessandroCodenotti: Hello!

Hi

How are things?

Very well, I aced the logic exam

Now I have to study for a functional analysis one though...

Ah.

@AlessandroCodenotti How many exams to go?

Functional analysis in January, then commutative algebra and a "foundations of maths" one in February (the last one is a mix of basic set theory and stuff)

I see.

In the oral part of the logic exam (the one I did this morning) the professor asked me to prove that two finite and elementary equivalent structures are actually isomorphic (with some hints), which is a very cool result we didn't have time to see during the course

@AlessandroCodenotti I see. I would simply note that the interpretation of every symbol is captured by some sentence in the theory of the structure, since they are finite mappings.

Hm, what do you mean?

All within suitable existential quantifiers, one for each object.

First the number of elements must be the same.

Let n be that number. Then you look at sentences of the form "∃x[1..n] ( x[1]≠x[2] ∧ x[2]≠x[3] ... ∧ ∀y ( x[1]=y ∨ x[2]=y ∨ ... ) ∧ P(x[1..n])".

@user21820 sure, one can write a sentence saying "there are exactly n elements" and it holds in one iff it holds in the other by elementary equivalence

Okay maybe my intuitive answer has a bigger gap than I thought. If the language is finite, then we can pick some P to capture the interpretation of all the symbols in one structure. That would also give us the isomorphism to the other structure.

The language may be infinite, but there are only finitely many possible distinct interpretations of the symbols since there are only n objects, so we should be able to put them into finitely many equivalence classes.

Then run the earlier argument.

Anyway I got to go for now. If my approach is very different from you and your professor's, I'd be glad to see yours.

It is, mine was longer and more basic, I have to be away for a while but I can write it down later!

@AlessandroCodenotti Sure. No hurry! =)

@user21820 humans do have a tendency to ask why

as if everything has a reason

sure, it may be the case that everything has a reason

but often the left brain just makes up reasons for things that it did