Von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets. This collection, formalized by Zermelo–Fraenkel set theory, is used to provide an interpretation or motivation of the axioms of ZFC; the rank of a well-founded set is defined inductively as the smallest ordinal number greater than the ranks of all members of the set. In particular, the rank of the empty set is zero, every ordinal has a rank equal to itself; the sets in V are divided into the transfinite hierarchy Vα , called the cumulative hierarchy, based on their rank. The cumulative hierarchy is a collection of sets Vα indexed by the class of ordinal numbers, thus there is one set Vα for each ordinal number α. Vα may be defined by transfinite recursion as follows: Let V0 be the empty set: V 0:= ∅. For any ordinal number β, let Vβ+1 be the power set of Vβ: V β + 1:= P. For any limit ordinal λ, let Vλ be the union of all the V-stages so far: V λ:= ⋃ β < λ V β.
A crucial fact about this definition is that there is a single formula φ in the language of ZFC that defines "the set x is in Vα". The sets Vα are called ranks; the class V is defined to be the union of all the V-stages: V:= ⋃ α V α. An equivalent definition sets V α:= ⋃ β < α P for each ordinal α, where P is the powerset of X. The rank of a set S is the smallest α such that S ⊆ V α. Another way to calculate rank is: rank = ⋃; the first five von Neumann stages V0 to V4 may be visualized. The set V5 contains 216 = 65536 elements; the set V6 contains 265536 elements, which substantially exceeds the number of atoms in the known universe. So the finite stages of the cumulative hierarchy cannot be written down explicitly after stage 5; the set Vω has the same cardinality as ω. The set Vω+1 has the same cardinality as the set of real numbers. If ω is the set of natural numbers Vω is the set of hereditarily finite sets, a model of set theory without the axiom of infinity. Vω+ω is the universe of "ordinary mathematics", is a model of Zermelo set theory.
A simple argument in favour of the adequacy of Vω+ω is the observation that Vω+1 is adequate for the integers, while Vω+2 is adequate for the real numbers, most other normal mathematics can be built as relations of various kinds from these sets without needing the axiom of replacement to go outside Vω+ω. If κ is an inaccessible cardinal Vκ is a model of Zermelo-Fraenkel set theory itself, Vκ+1 is a model of Morse–Kelley set theory. V is not "the set of all sets" for two reasons. First, it is not a set. Second, the sets in V are only the well-founded sets; the axiom of foundation demands that every set be well founded and hence in V, thus in ZFC every set is in V. But other axiom systems may replace it by a strong negation; these non-well-founded set theories are not employed, but are still possible to study. A third objection to the "set of all sets" interpretation is that not all sets are "pure sets", which are constructed from the empty set using power sets and unions. Zermelo proposed in 1908 the inclusion of urelements, from which he constructed a transfinite recursive hierarchy in 1930.
Such urelements are used extensively in model theory in Fraenkel-Mostowski models. The formula V = ⋃αVα is considered to be a theorem, not a definition. Roitman states that the realization that the axiom of regularity is equivalent to the equality of the universe of ZF sets to the cumulative hierarchy is due to von Neumann. Since the class V may be considered to be the arena for most of mathematics, it is important to establish that it "exists" in some sense. Since existence is a difficult concept, one replaces the existence question with the consistency question, that is, whether the concept is free of contradictions. A major obstacle is posed by Gödel's incompleteness theorems, which imply the impossibility of proving the consistency of ZF set theory in ZF set theory itself, provided that it is in fact consiste
Axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of one object from each bin if the collection is infinite. Formally, it states that for every indexed family i ∈ I of nonempty sets there exists an indexed family i ∈ I of elements such that x i ∈ S i for every i ∈ I; the axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, such a selection can be made without invoking the axiom of choice. An illustrative example is sets picked from the natural numbers. From such sets, one may always select the smallest number, e.g. in the smallest elements are. In this case, "select the smallest number" is a choice function. If infinitely many sets were collected from the natural numbers, it will always be possible to choose the smallest element from each set to produce a set.
That is, the choice function provides the set of chosen elements. However, no choice function is known for the collection of all non-empty subsets of the real numbers. In that case, the axiom of choice must be invoked. Bertrand Russell coined an analogy: for any collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection. For an infinite collection of pairs of socks, there is no obvious way to make a function that selects one sock from each pair, without invoking the axiom of choice. Although controversial, the axiom of choice is now used without reservation by most mathematicians, it is included in the standard form of axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice. One motivation for this use is that a number of accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs. Contemporary set theorists study axioms that are not compatible with the axiom of choice, such as the axiom of determinacy.
The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced. A choice function is a function f, defined on a collection X of nonempty sets, such that for every set A in X, f is an element of A. With this concept, the axiom can be stated: Formally, this may be expressed as follows: ∀ X. Thus, the negation of the axiom of choice states that there exists a collection of nonempty sets that has no choice function; each choice function on a collection X of nonempty sets is an element of the Cartesian product of the sets in X. This is not the most general situation of a Cartesian product of a family of sets, where a given set can occur more than once as a factor; the axiom of choice asserts the existence of such elements. In this article and other discussions of the Axiom of Choice the following abbreviations are common: AC – the Axiom of Choice. ZF – Zermelo–Fraenkel set theory omitting the Axiom of Choice.
ZFC – Zermelo–Fraenkel set theory, extended to include the Axiom of Choice. There are many other equivalent statements of the axiom of choice; these are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it. One variation avoids the use of choice functions by, in effect, replacing each choice function with its range. Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains one element in common with each of the sets in X; this guarantees for any partition of a set X the existence of a subset C of X containing one element from each part of the partition. Another equivalent axiom only considers collections X that are powersets of other sets: For any set A, the power set of A has a choice function. Authors who use this formulation speak of the choice function on A, but be advised that this is a different notion of choice function, its domain is the powerset of A, and