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Thus, even if the primary interest is X, the universe may need to be considerably larger than X. Following the above ideas, one may want the superstructure over X as the universe. This can be defined by structural recursion as follows: Let S 0 X be X itself. Let S 1 X be the union of X and PX. Let S 2 X be the union of S 1 X and P(S 1 X).
The set V 5 contains 2 16 = 65536 elements; the set V 6 contains 2 65536 elements, which very substantially exceeds the number of atoms in the known universe; and for any natural n, the set V n+1 contains 2 ⇈ n elements using Knuth's up-arrow notation. So the finite stages of the cumulative hierarchy cannot be written down explicitly after ...
The domain of a structure is an arbitrary set; it is also called the underlying set of the structure, its carrier (especially in universal algebra), its universe (especially in model theory, cf. universe), or its domain of discourse. In classical first-order logic, the definition of a structure prohibits the empty domain. [citation needed] [5]
The sets of the constructible universe form a cumulative hierarchy. The Boolean-valued models constructed by forcing are built using a cumulative hierarchy. The well founded sets in a model of set theory (possibly not satisfying the axiom of foundation ) form a cumulative hierarchy whose union satisfies the axiom of foundation.
The picture of the universe of sets stratified into the cumulative hierarchy is characteristic of ZFC and related axiomatic set theories such as Von Neumann–Bernays–Gödel set theory (often called NBG) and Morse–Kelley set theory. The cumulative hierarchy is not compatible with other set theories such as New Foundations.
The term hierarchy is used to stress a hierarchical relation among the elements. Sometimes, a set comes equipped with a natural hierarchical structure. For example, the set of natural numbers N is equipped with a natural pre-order structure, where n ≤ n ′ {\displaystyle n\leq n'} whenever we can find some other number m {\displaystyle m} so ...
Equivalent structures may be treated as a single structure, as shown by a large box on Fig. 4. The transitions denoted by the arrows obey isomorphisms. That is, two isomorphic A-spaces lead to two isomorphic B-spaces. The diagram on Fig. 4 is commutative. That is, all directed paths in the diagram with the same start and endpoints lead to the ...
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. L {\displaystyle L} is the union of the constructible hierarchy L α {\displaystyle L_{\alpha }} .