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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 ...
Properly describing or enumerating the contents of a free object can be easy or difficult, depending on the particular algebraic object in question. For example, the free group in two generators is easily described. By contrast, little or nothing is known about the structure of free Heyting algebras in more than one generator. [1]
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 free group F S with free generating set S can be constructed as follows. S is a set of symbols, and we suppose for every s in S there is a corresponding "inverse" symbol, s −1, in a set S −1. Let T = S ∪ S −1, and define a word in S to be any written product of elements of T. That is, a word in S is an element of the monoid ...
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.
f 2 = 1. Informally, we can consider these products on the left hand side as being elements of the free group F = r, f , and let R = rfrf, r 8, f 2 . That is, we let R be the subgroup generated by the strings rfrf, r 8, f 2, each of which is also equivalent to 1 when considered as products in D 8.
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 ...
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 ...