can "sets" be interpreted as "predicates"? Instead of writing "$r\in\mathbb{R}$", we can write "$r$ is a real number". In the latter statement we are asserting $P(r)$ where $P(x)$ is the predicate "$x$ is a real number". It seems like sets give rise to predicates. How far can this be taken? Can I replace every mention of a set with a predicate and have an equivalent theory (to set theory)? If not, why not?
The reason for the question is this: the notion of "is in a set", to me, suggests something different than "is a". "is a" seems to just tell me that a certain object has a certain property or type (and we don't have to think about size). "is in a set" tells me that we have to think of all of those objects together in a certain place (and we do have to think about size).
 A: While every set gives rise to a predicate, the reverse is not true. In particular, the predicate "$x$ is a set" does not correspond to a set (which would then be the set of all sets).
If you try to identify predicates with sets, you get naive set theory. But naive set theory is not consistent. For example, the power set of a set is provably larger that the set, but the power set of the set of all sets would consist only of sets, so it cannot be larger than the set of all sets.
A: I'm not sure exactly what you mean by "predicate," but here's one potential pitfall: there are sets which are not "definable" in any nice way. Do these correspond to predicates? If by "predicate" you have in mind some description in a certainl language, I'd argue the answer should be no; in which case most sets do not correspond to predicates.

Re: celtschk's point that some predicates don't define sets, note that some "predicate"-like things can't even be predicates: consider the property "Is a predicate which does not apply to itself" (this is Russell's paradox). Basically, as soon as I have a notion of "set" or "predicate" or "property" or . . . , I'll be able to "diagonalize out of it": there will be some thing, definable in terms of that notion, which is not one of those.
