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By Terence Tao

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Also, we have no axioms yet on what sets do, or what their elements do. Obtaining these axioms and defining these operations will be the purpose of the remainder of this section. We first clarify one point: we consider sets themselves to be a type of object. 1 (Sets are objects). If A is a set, then A is also an object. In particular, given two sets A and B, it is meaningful to ask whether A is also an element of B. 2. (Informal) The set {3, {3, 4}, 4} is a set of three distinct elements, one of which happens to itself be a set of two elements.

5. Thus, for instance, {1, 2, 3, 4, 5} and {3, 4, 2, 1, 5} are the same set, since they contain exactly the same elements. 1). 4. Thus the "is an element of" relation E obeys the axiom of substitution (see Section 40 3. 7). Because of this, any new operation we define on sets will also obey the axiom of substitution, as long as we can define that operation purely in terms of the relation E. This is for instance the case for the remaining definitions in this section. ) Next, we turn to the issue of exactly which objects are sets and which objects are not.

Thus to begin at the very beginning, we must look at the natural numbers. We will consider the following question: how does one actually define the natural numbers? (This is a very different question from how to use the natural numbers, which is something you of course know how to do very well. ) This question is more difficult to answer than it looks. , that a+b is always equal to b+a) without even aware that you are doing so; it is difficult to let go and try to inspect this number system as if it is the first time you have seen it.

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