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Pure sets in set theory

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With the concept of the pure set, we are able to construct higher-ranked sets from lower-ranked sets.

One way to build up a universe of sets goes like this: We start with some individuals on rank 0. Given these individuals, we build sets on rank 1. Then we build sets on rank 2. And we continue like this, building up our universe of sets rank by rank. This way of building a universe of sets is clearly useful. But the price of utility is uncertainty. What are the individauls on rank 0? Should we also include space-time points? Should we consider possible individuals, or just actual individuals? To avoid any uncertainties, we can ignore individuals altogether(Steinhart, p. 16)

The absence of individuals is what makes the concept of the pure set necessary. A pure set is a set without individuals. The simplest pure set is an empty set, and we can build other pure sets out of this simple, pure set ad infinitum. The original pure set, the empty set, is symbolized {}. Suppose we have an empty set {} and a pure set ranked higher than an empty set, {{}}. We can use this combination to build a set of these two very sets, by symbolizing them {{}, {{}}}. A universe of sets of infinite complexity can be constructed of a set or set of sets of relatively unimpressive simplicity. To speak in definite rather than purely hypothetical terms, "We can say that for any two pure sets x and y, there exists the pure set {x, y}. Or, for every pure set x, there exists the power set of x. This power set is also pure"(Steinhart, p. 16).

In set theory, a universe of pure sets is symbolized with "V." We construct V, the entire universe of pure sets, by working from the most basic simplicity and defining a series of "partial universes"(Steinhart, p. 17). The most basic partial universe is symbolized "V" with a subscript 0. This is merely the empty set; that is, a basic pure set absent of any individuals. We go from V0 (keeping in mind that the subscript is used in such a context) to V1, which is a larger set than V0 (Steinhart, p. 17). Since V1 is constructed from V0, it is necessary for the former to be the power set of V0. "Since the only subset of {} is {} itself, the set of all subsets of {} is the set that contains {}. It is {{}}"(Steinhart, p. 17). It is for this reason that V1 = {{}}.

Next comes the partial universe V2. Just as V1 was the power set of V0, V2 will be the power set of V1. This means that V2 will be {{}, {{}}}. That is, it includes the empty set as well as the power set of that empty set. This is just another way of saying that it includes both V0 and V1, keeping in mind that each partial universe must be constructed from the power set of the antecedent partial universe. Each construction of a subsequent larger partial universe is called an "iteration"(Steinhart, p. 17). We represent this subsequent partial universe by writing Vn+1(keeping in mind that the "n+1" is a subscript). Each partial universe, that is, each Vn, has a Vn+1 that constitutes its subsequent iteration. This subsequent iteration is the partial universe that is constructed from the power set of the antecedent partial universe.

Steinhart, Eric. "More Precisely: The Mathematics You Need To Do Philosophy." Broadview Press, 2009.

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