Why is the "finitely many" quantifier not definable in First Order Logic?

Question

In First Order Logic with Identity (FOL+I), one can express the proposition that there are exactly 3 items that have the property P.

Why is it not possible to express the proposition that there is a finite number of items that have the property P (in FOL+I)?

Answer 1

We can define formula $P_i$ that says "there are at most $i$ elements satisfying $P$". Now, if the infinite disjunction of the $P_i$ was definable in FO, it would (by compactness) imply a conjunction of some finite subset of the $P_i$, hence it would imply $P_i$ for some $i$. That is not true, if $P$ can have (say) $i+1$ elements satisfying it.

Answer 2

Any property expressible under first-order logic is closed under ultraproducts. The property of finite sets is not, however, closed under ultraproducts.

Answer 3

Well, it would mean that you have the statement $P_1\vee P_2\vee\ldots$ where $P_i$ stands for "there are $i$ objects with property $P$", and infinite disjunctions aren't allowed. As for proving that this isn't equivalent to anything else you can write that IS allowed, I don't know how to do that.

Answer 4

I think you can express "forall but finitely many" in FOL, which means "for all sufficiently large". But I am not sure whether this is equivalent to "finitely many", it seems to be a little bit different. See http://arxiv.org/pdf/math/0602415.pdf.