Well-ordering theorem

well-ordering principleWell–ordering theorem
It turned out, though, that the well-ordering theorem is equivalent to the axiom of choice, in the sense that either one together with the Zermelo–Fraenkel axioms is sufficient to prove the other, in first order logic (the same applies to Zorn's Lemma). In second order logic, however, the well-ordering theorem is strictly stronger than the axiom of choice: from the well-ordering theorem one may deduce the axiom of choice, but from the axiom of choice one cannot deduce the well-ordering theorem.

Cardinal number

cardinalcardinal numberscardinality
All the remaining propositions in this section assume the axiom of choice: If 2 ≤ κ and 1 ≤ μ and at least one of them is infinite, then: :Max (κ, 2 μ ) ≤ κ μ ≤ Max (2 κ, 2 μ ). Using König's theorem, one can prove κ < κ cf and κ < cf(2 κ ) for any infinite cardinal κ, where cf is the cofinality of κ. Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 0, the cardinal ν satisfying will be κ. Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 1, there may or may not be a cardinal λ satisfying.

Zermelo–Fraenkel set theory

ZFCZFZermelo–Fraenkel axioms
The following axiom is added to turn ZF into ZFC: For any set X, there is a binary relation R which well-orders X. This means R is a linear order on X such that every nonempty subset of X has a member which is minimal under R. Given axioms 1–8, there are many statements equivalent to axiom 9, the best known of which is the axiom of choice (AC), which goes as follows. Let X be a set whose members are all non-empty. Then there exists a function f from X to the union of the members of X, called a "choice function", such that for all Y\in X one has f(Y)\in Y.

Axiom of dependent choice

DCdependent choiceaxiom of dependent choices
The axiom of dependent choice implies the axiom of countable choice and is strictly stronger. *

Axiom of constructibility

V = LV=Lconstructible
The axiom of constructibility implies the axiom of choice (AC), given Zermelo–Fraenkel set theory without the axiom of choice (ZF). It also settles many natural mathematical questions that are independent of Zermelo–Fraenkel set theory with the axiom of choice (ZFC); for example, the axiom of constructibility implies the generalized continuum hypothesis, the negation of Suslin's hypothesis, and the existence of an analytical (in fact, \Delta^1_2) non-measurable set of real numbers, all of which are independent of ZFC.

Continuum hypothesis

generalized continuum hypothesisGCHHilbert's first problem
Like CH, GCH is also independent of ZFC, but Sierpiński proved that ZF + GCH implies the axiom of choice (AC) (and therefore the negation of the axiom of determinacy, AD), so choice and GCH are not independent in ZF; there are no models of ZF in which GCH holds and AC fails. To prove this, Sierpiński showed GCH implies that every cardinality n is smaller than some Aleph number, and thus can be ordered. This is done by showing that n is smaller than which is smaller than its own Hartogs number—this uses the equality ; for the full proof, see Gillman (2002).

Natural number

natural numberspositive integerpositive integers
This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any well-ordered countably infinite set. This assignment can be generalized to general well-orderings with a cardinality beyond countability, to yield the ordinal numbers. An ordinal number may also be used to describe the notion of "size" for a well-ordered set, in a sense different from cardinality: if there is an order isomorphism (more than a bijection!) between two well-ordered sets, they have the same ordinal number. The first ordinal number that is not a natural number is expressed as ω ; this is also the ordinal number of the set of natural numbers itself.

Countable set

countablecountably infinitecountably
Countable sets can be totally ordered in various ways, e.g.: In both examples of well orders here, any subset has a least element; and in both examples of non-well orders, some subsets do not have a least element.

Choice function

selectionselection functionglobal choice function τ
In this case, it was possible to simultaneously well-order every member of X by making just one choice of a well-order of the union, so neither AC nor AC ω was needed. (This example shows that the well-ordering theorem implies AC. The converse is also true, but less trivial.). Axiom of countable choice. Hausdorff paradox. Hemicontinuity.

Axiom of determinacy

certain typeADdeterminacy
With the axiom of choice we can well order the continuum; furthermore, we can do so in such a way that any proper initial portion does not have the cardinality of the continuum. We create a counterexample by transfinite induction on the set of strategies under this well ordering: We start with the set A undefined. Let T be the "time" whose axis has length continuum. We need to consider all strategies {s1(T)} of the first player and all strategies {s2(T)} of the second player to make sure that for every strategy there is a strategy of the other player that wins against it. For every strategy of the player considered we will generate a sequence which gives the other player a win.

Real number

realrealsreal-valued
The well-ordering theorem implies that the real numbers can be well-ordered if the axiom of choice is assumed: there exists a total order on R with the property that every non-empty subset of R has a least element in this ordering. (The standard ordering ≤ of the real numbers is not a well-ordering since e.g. an open interval does not contain a least element in this ordering.) Again, the existence of such a well-ordering is purely theoretical, as it has not been explicitly described. If V=L is assumed in addition to the axioms of ZF, a well ordering of the real numbers can be shown to be explicitly definable by a formula.

Finite set

finitefinitelyfinite sets
S is I-finite or not well-orderable. FinSet. Ordinal number. Peano arithmetic.

Ordinal number

ordinalordinalsordinal numbers
Any well-ordered set having an ordinal as its order-type has the same cardinality as that ordinal. The least ordinal associated with a given cardinal is called the initial ordinal of that cardinal. Every finite ordinal (natural number) is initial, and no other ordinal associates with its cardinal. But most infinite ordinals are not initial, as many infinite ordinals associate with the same cardinal. The axiom of choice is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal.

Von Neumann–Bernays–Gödel set theory

NBGNBG set theoryvon Neumann–Bernays–Gödel axioms
Easton proved that global choice is stronger than the axiom of choice by using forcing to construct a model that satisfies the axiom of choice and all the axioms of NBG except the axiom of global choice. The axiom of global choice is equivalent to every class having a well-ordering, while ZFC's axiom of choice is equivalent to every set having a well-ordering. Axiom of global choice. There exists a function that chooses an element from every nonempty set. : Von Neumann published an introductory article on his axiom system in 1925. In 1928, he provided a detailed treatment of his system. Von Neumann based his axiom system on two domains of primitive objects: functions and arguments.

Transfinite induction

transfinite recursiontransfinitetransfinitely iterating
Proofs or constructions using induction and recursion often use the axiom of choice to produce a well-ordered relation that can be treated by transfinite induction. However, if the relation in question is already well-ordered, one can often use transfinite induction without invoking the axiom of choice. For example, many results about Borel sets are proved by transfinite induction on the ordinal rank of the set; these ranks are already well-ordered, so the axiom of choice is not needed to well-order them.

Group structure and the axiom of choice

There are models of ZF in which the axiom of choice fails. In such a model, there are sets that cannot be well-ordered (call these "non-wellorderable" sets). Let X be any such set. Now consider the set . If Y were to have a group structure, then, by the construction in first section, X can be well-ordered. This contradiction shows that there is no group structure on the set ''Y . If a set is such that it cannot be endowed with a group structure, then it is necessarily non-wellorderable. Otherwise the construction in the second section does yield a group structure. However these properties are not equivalent.

Infinite set

infiniteinfinitelyinfinitely many
The Cartesian product of an infinite number of sets, each containing at least two elements, is either empty or infinite; if the axiom of choice holds, then it is infinite. If an infinite set is a well-ordered set, then it must have a nonempty, nontrivial subset that has no greatest element. In ZF, a set is infinite if and only if the power set of its power set is a Dedekind-infinite set, having a proper subset equinumerous to itself. If the axiom of choice is also true, then infinite sets are precisely the Dedekind-infinite sets.

Tarski's theorem about choice

Tarski's theoremTarski's theorem on choice
Our goal is to prove that the axiom of choice is implied by the statement "For every infinite set A: ". It is known that the well-ordering theorem is equivalent to the axiom of choice, thus it is enough to show that the statement implies that for every set B there exist a well-order. For finite sets it is trivial, thus we will assume that B is infinite. Since the collection of all ordinals such that there exist a surjective function from B to the ordinal is a set, there exist a minimal non-zero ordinal, \beta, such that there is no surjective function from B to \beta. We assume without loss of generality that the sets B and \beta are disjoint.

Axiom of limitation of size

Limitation of Sizemodels of set theory that satisfy von Neumann's axiom
Easton used forcing to build a model of NBG with global choice replaced by the axiom of choice. In Easton's model, V cannot be linearly ordered, so it cannot be well-ordered. Therefore, the axiom of limitation of size fails in this model. Ord is an example of a proper class that cannot be mapped onto V because (as proved above) if there is a function mapping Ord onto V, then V can be well-ordered. The axioms of NBG with the axiom of replacement replaced by the weaker axiom of separation do not imply the axiom of limitation of size.

Morse–Kelley set theory

Kelley–Morse set theoryKMMK
Definition: c is a choice function if c is a function and c(x) \in x for each member x of domain c. '''IX. Choice:' There exists a choice function c'' whose domain is. IX is very similar to the axiom of global choice derivable from Limitation of Size above. Develop: Equivalents of the axiom of choice. As is the case with ZFC, the development of the cardinal numbers requires some form of Choice. If the scope of all quantified variables in the above axioms is restricted to sets, all axioms except III and the schema IV are ZFC axioms. IV is provable in ZFC.

Mathematical logic

formal logicsymbolic logiclogic
Ernst Zermelo (1904) gave a proof that every set could be well-ordered, a result Georg Cantor had been unable to obtain. To achieve the proof, Zermelo introduced the axiom of choice, which drew heated debate and research among mathematicians and the pioneers of set theory. The immediate criticism of the method led Zermelo to publish a second exposition of his result, directly addressing criticisms of his proof (Zermelo 1908a). This paper led to the general acceptance of the axiom of choice in the mathematics community. Skepticism about the axiom of choice was reinforced by recently discovered paradoxes in naive set theory.

Von Neumann cardinal assignment

initial ordinalcardinalinitial
This is a well-ordering of cardinal numbers. Each ordinal has an associated cardinal, its cardinality, obtained by simply forgetting the order. Any well-ordered set having that ordinal as its order type has the same cardinality. The smallest ordinal having a given cardinal as its cardinality is called the initial ordinal of that cardinal. Every finite ordinal (natural number) is initial, but most infinite ordinals are not initial. The axiom of choice is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal.

First uncountable ordinal

ω 1 first uncountable ordinal number&omega; 1
Like any ordinal number (in von Neumann's approach), ω 1 is a well-ordered set, with set membership ("∈") serving as the order relation. ω 1 is a limit ordinal, i.e. there is no ordinal α with α + 1 = ω 1. The cardinality of the set ω 1 is the first uncountable cardinal number, ℵ 1 (aleph-one). The ordinal ω 1 is thus the initial ordinal of ℵ 1. Indeed, in most constructions ω 1 and ℵ 1 are equal as sets. To generalize: if α is an arbitrary ordinal we define ω α as the initial ordinal of the cardinal ℵ α. The existence of ω 1 can be proven without the axiom of choice. (See Hartogs number.) Any ordinal number can be turned into a topological space by using the order topology.

Skeleton (category theory)

skeletonskeletal
The category of all well-ordered sets has the subcategory of all ordinal numbers as a skeleton. A preorder, i.e. a small category such that for every pair of objects A,B, the set Hom(A,B) either has one element or is empty, has a partially ordered set as a skeleton. Glossary of category theory. Thin category. Adámek, Jiří, Herrlich, Horst, & Strecker, George E. (1990). Abstract and Concrete Categories. Originally published by John Wiley & Sons. ISBN: 0-471-60922-6. (now free on-line edition). Robert Goldblatt (1984). Topoi, the Categorial Analysis of Logic (Studies in logic and the foundations of mathematics, 98). North-Holland. Reprinted 2006 by Dover Publications.

Gödel's completeness theorem

completeness theoremcompletenesscomplete
When considered over a countable language, the completeness and compactness theorems are equivalent to each other and equivalent to a weak form of choice known as weak König's lemma, with the equivalence provable in RCA 0 (a second-order variant of Peano arithmetic restricted to induction over Σ 0 1 formulas). Weak König's lemma is provable in ZF, the system of Zermelo–Fraenkel set theory without axiom of choice, and thus the completeness and compactness theorems for countable languages are provable in ZF.