Definition talk:Propositional Expansion/Existential Quantifier

I'm not sure I understand the reason behind saying that the propositional expansion can not exist when the universe is infinite. I have left the comment in (although changed it into less categorically proscriptive language), but I don't immediately see why it "can not exist".

It can surely exist when the universe is countable, it's just that you can't physically write the expansion down. I can see why it can be considered as being unable to exist when the universe is uncountable, because you can not identify the elements as $X_1, X_2, ...$. Is there a reason why infinite expressions, e.g. $X_1 \lor X_2 \lor \cdots$ can not exist? If so we need to indicate it in proofwiki somewhere, and then there will be a fight against the non-intuitionists. And I doubt the intuitionists will win because they don't have admin privs. ;-) --prime mover 05:11, 15 October 2011 (CDT)

Propositional logic does by definition consist of a language, which in turn consists of finite expressions (sentences). There is of course a well-established meaning to this (which is exactly what the existence quantifier expresses) but it cannot be written down in propositional logic. In other words: propositional logic allows only expressions of finite length. For a more detailed treatise, and to refer to what I have learned, check out this text. For this, mainly chapter 2 applies. It probably contains a lot of information worth including on PW as well. --Lord_Farin 05:41, 15 October 2011 (CDT)

I'm going to have to take the time to study this seriously. It seems a good source, but Argh! model theory. Every approach to model theory is different, using a different set of axioms, etc. etc. from every other, and finding a way to unify these approaches is proving challenging. Okay, so I'm prepared to accept that this particular source suggests that propositional statements have to be finite (I'm not up for flogging my way through it today), but if it is true it will need a page specifying it and a solid argument behind it. Then we can reference that page on this one, so that bald statement "The propositional expansion for the existential quantifier can exist in actuality only when the number of objects in the universe is finite" can have something to justify its existence. --prime mover 06:43, 15 October 2011 (CDT)