A set is a collection of elements, the members of the set. The elements can be anything, including other sets, except in a pure set theory, in which case they can only be sets. The nature of sets is determined by the principle of extensionality, which states that two sets are not distinct unless one has an element which is not in the other. (i.e. two sets are in fact the same set if and only if they have the same elements). 
The problem in set theory is not in knowing what a set is, but in knowing what sets there are (in the domain of discourse).
This is a problem because the most natural assumption, that to every property of sets expressible in first order logic there corresponds a set whose elements are just those sets having that property, turns out to be inconsistent.
(Russell's paradox demonstrates this).
There is no single answer to this question, there are many different interpretations of set theory. However, mathematics is normally conducted in the context of a set theory in which the sets are the pure wellfounded collections and this domain is what is described by "the iterative conception of set". 
A simple way one might hope to describe the iterative heirarchy is as follows:


There are two main problems.
The answer to problem 2 is: at least enough to make a model for the theory you want to work with (e.g. ZFC) but as many as you like after that. 
Instead of using the standard definition of truth, which I take to be the concept of validity, the following slight modification can be used.
A statement converges if after some point in the sequence of standard interpretations its truth value is always the same, and it converges to that value. A statement is true if it converges to true and false if it converges to false. Otherwise it has no truth value. This is only relevant at the judgement level, and doesn't affect the standard way in which the semantics of the propositional connectives is rendered. i.e. the logic is still two valued. 
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Are Foundations Necessary? 
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