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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 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 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 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 ...
The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows (also called morphisms, although this term also has a specific, non category-theoretical sense), where these collections satisfy certain ...
A Carroll diagram, Lewis Carroll's square, biliteral diagram or a two-way table is a diagram used for grouping things in a yes/no fashion. Numbers or objects are either categorised as 'x' (having an attribute x) or 'not x' (not having an attribute 'x'). They are named after Lewis Carroll, the pseudonym of polymath Charles Lutwidge Dodgson. [1] [2]
Tegmark's MUH is the hypothesis that our external physical reality is a mathematical structure. [3] That is, the physical universe is not merely described by mathematics, but is mathematics — specifically, a mathematical structure. Mathematical existence equals physical existence, and all structures that exist mathematically exist physically ...