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The Von Neumann Heirarchy

ecall that sets can contain other sets as elements, and that the empty set is the set with nothing in it. We construct the Von Neumann heirarchy by beginning with the empty set, and then at each step of the construction we form the set that contains all the previous sets as elements:

 step 0: { } (empty set) step 1: { { } } (set containing empty set) step 2: { { }, { { } } } (set containing previous two sets) step 3: { { }, { { } } , { { }, { { } } } } (set containing previous three) step 4: { { }, { { } } , { { }, { { } } }, { { }, { { } } , { { }, { { } } } } } (etc.)

This goes on, of course, forever. If you take the entire (infinite) collection of all such sets and make a collection of them (a set!), then you have the Von Neumman heirarchy of finite ordinals. Since keeping track of all those braces quickly becomes impractical, we typically write “0” for the empty set, “1” for the set containing the empty set, and so on. Thus we represent the ordinal “2” as the set {0, 1} – that is, as the set containing 0 and 1 – the ordinal “3” as the set {0, 1, 2}, and any ordinal n as {0, 1, 2, ..., n - 1}. In short, each ordinal is defined to be the set containing all lesser ordinals.
The fascinating things is, if you define suitable operations on the ordinals (which are sets of sets), based upon simple union and intersection, then they behave just like the natural numbers. Indeed, this set may be considered to be the set of natural numbers, in the sense that structurally there is no difference between the two. Both sets are well founded, and both are closed under suitably defined operations of “addition.”
This construction may also be performed by starting with the empty set, and forming each successor set by taking the union of the current set with the set containing the current set. After forming the complete infinite heirarchy of finite sets in this way, we may pass into the transfinite realm by taking the union of all of these sets, and then continuing as before. Thus, the Von Neumann heirarchy passes beyond all finite stages, and beyond all infinite stages as well. In this way the entire class of ordinals is created.

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