# Why is the "finitely several" quantifier not definite in First Order Logic?

In First Order Logic with Identity (FOL+I), one can share the suggestion that there are specifically 3 things that have the building P.

Why is it not feasible to share the suggestion that there is a limited variety of things that have the building P (in FOL+I)?

We can specify formula $P_i$ that claims "there go to the majority of $i$ components pleasing $P$". Currently, if the boundless disjunction of the $P_i$ was definite in FO, it would certainly (by density) indicate a combination of some limited part of the $P_i$, therefore it would indicate $P_i$ for some $i$. That is not real, if $P$ can have (claim) $i+1$ components pleasing it.

Any building expressible under first-order logic is shut under ultraproducts. The building of limited collections is not, nonetheless, shut under ultraproducts.

Well, it would certainly suggest that you have the declaration $P_1\vee P_2\vee\ldots$ where $P_i$ represents "there are $i$ things with building $P$", and also boundless disjunctions aren't permitted. When it comes to confirming that this isn't equal to anything else you can write that IS permitted, I do not recognize just how to do that.

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