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Quantifiers and Infinite Sets


#1

I noticed that quantifiers are defined in a way resembling induction: we define the quantifier for the empty set, for singletons and then for the union of sets.

The problem is, such a method doesn’t work for infinite sets (for an example of why, consider the fact that finite intersections of open sets are open, but arbitrary intersections are not necessarily so. In topology, a similar statement about the intersection of two open sets is taken as an axiom, but many infinite intersections can easily be shown to be singletons and hence not open). Yet, in the lectures, we saw examples of both infinite sums (over the naturals), universal quantifiers (over the rationals), existential quantifiers (over the naturals) and even products (over the naturals), none of which can be handled by the definitions given, at least according to my understanding.

Are we going to be given different definitions, or am I making a mistake here?


#2

You are exactly right. Empty, singleton, and union gives us quantification over finite domains. The textbook says, on page 28,

There are further axioms to say how each quantifier behaves when the domain is a result of the § quantifier; they are listed at the back of the book, together with other laws concerning quantification.

If you look at the definition of quantifiers on page 237, you will find the missing axioms.


#3

Thanks! I just got confused because they weren’t listed, as far as I remember, in the lecture.


#4

I think you are right - the lecture did not mention them.