Confused by first order logic and constructing sentences I am still new to stack exchange so please forgive me if something is unclear in my question. I have been working through set theory, logic and their limitations by Machover. I am currently learning about first-order logic and I stumbled upon the following question:
Construct a sentence $\alpha$ containing only logical symbols (that is, no function symbols and no predicate symbols other than '=') such that $\alpha$ holds in a structure $\chi$ iff the domain $U$ of $\chi$ has at least three members.
I have been stuck on this problem for a while now and need some pointers to get started. The reason I feel stuck is that I tried to construct a sentence but how do I prove it holds? For reference, this question is 5.14 pg 161 in the book.
 A: I think a common issue with this sort of problem is the temptation to stare at it and try to guess the right answer rather than playing around with some concrete (if perhaps unrelated) examples first.
Mildly abusing some notational conventions in the interest of readability, here are a few example sentences in the empty language (= no non-logical symbols; note that most modern texts consider "$=$" a logical symbol):


*

*$\forall x,y(x=y)$. ("Every pair of elements are equal.")

*$\exists x(x\not=x)$. ("There is something not equal to itself.")

*$\exists x,y(x\not=y)$. ("There are two things which are not equal to each other.")
A good warm-up exercise is to determine the class of models of each of these sentences. Note that in the empty language, any two structures of the same cardinality are isomorphic (this is a good exercise if you haven't seen it already), so you're really just thinking about the possible cardinalities of models.
These examples should provide a good hint as to how to solve your problem.
