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Axiom of countable choice

From Wikipedia, the free encyclopedia

Each set in the countable sequence of sets (Si) = S1, S2, S3, ... contains a non-zero, and possibly infinite (or even uncountably infinite), number of elements. The axiom of countable choice allows us to arbitrarily select a single element from each set, forming a corresponding sequence of elements (xi) = x1, x2, x3, ...

The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function with domain (where denotes the set of natural numbers) such that is a non-empty set for every , there exists a function with domain such that for every .

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Transcription

Applications

ACω is particularly useful for the development of mathematical analysis, where many results depend on having a choice function for a countable collection of sets of real numbers. For instance, in order to prove that every accumulation point of a set is the limit of some sequence of elements of , one needs (a weak form of) the axiom of countable choice. When formulated for accumulation points of arbitrary metric spaces, the statement becomes equivalent to ACω.

The ability to perform analysis using countable choice has led to the inclusion of ACω as an axiom in some forms of constructive mathematics, despite its assertion that a choice function exists without constructing it.[1]

Example: finite implies Dedekind-finite

As an example of an application of ACω, here is a proof (from ZF + ACω) that every infinite set is Dedekind-infinite:[2]

Let be infinite. For each natural number , let be the set of all -tuples of distinct elements of . Since is infinite, each is non-empty. Application of ACω yields a sequence where each is an -tuple. One can then concatenate these tuples into a single sequence of elements of , possibly with repeating elements. Suppressing repetitions produces a sequence of distinct elements, where

, with .

This exists, because when selecting it is not possible for all elements of to be among the elements selected previously. So contains a countable set. The function that maps each to (and leaves all other elements of fixed) is a one-to-one map from into which is not onto, proving that is Dedekind-infinite.[2]

Relation to other axioms

Stronger and independent systems

The axiom of countable choice (ACω) is strictly weaker than the axiom of dependent choice (DC),[3] which in turn is weaker than the axiom of choice (AC). DC, and therefore also ACω, hold in the Solovay model, constructed in 1970 by Robert M. Solovay as a model of set theory without the full axiom of choice, in which all sets of real numbers are measurable.[4]

Urysohn's lemma (UL) and the Tietze extension theorem (TET) are independent of ZF+ACω: there exist models of ZF+ACω in which UL and TET are true, and models in which they are false. Both UL and TET are implied by DC.[5]

Weaker systems

Paul Cohen showed that ACω is not provable in Zermelo–Fraenkel set theory (ZF) without the axiom of choice.[6] However, some countably infinite sets of non-empty sets can be proven to have a choice function in ZF without any form of the axiom of choice. For example, has a choice function, where is the set of hereditarily finite sets, i.e. the first set in the Von Neumann universe of non-finite rank. The choice function is (trivially) the least element in the well-ordering. Another example is the set of proper and bounded open intervals of real numbers with rational endpoints.

ZF+ACω suffices to prove that the union of countably many countable sets is countable. These statements are not equivalent: Cohen's First Model supplies an example where countable unions of countable sets are countable, but where ACω does not hold.[7]

Equivalent forms

There are many equivalent forms to the axiom of countable choice, in the sense that any one of them can be proven in ZF assuming any other of them. They include the following:[8][9]

References

  1. ^ Bauer, Andrej (2017). "Five stages of accepting constructive mathematics". Bulletin of the American Mathematical Society. New Series. 54 (3): 481–498. doi:10.1090/bull/1556. MR 3662915.
  2. ^ a b Herrlich 2006, Proposition 4.13, p. 48.
  3. ^ Jech, Thomas J. (1973). The Axiom of Choice. North Holland. pp. 130–131. ISBN 978-0-486-46624-8.
  4. ^ Solovay, Robert M. (1970). "A model of set-theory in which every set of reals is Lebesgue measurable". Annals of Mathematics. Second Series. 92 (1): 1–56. doi:10.2307/1970696. ISSN 0003-486X. JSTOR 1970696. MR 0265151.
  5. ^ Tachtsis, Eleftherios (2019), "The Urysohn lemma is independent of ZF + countable choice", Proceedings of the American Mathematical Society, 147 (9): 4029–4038, doi:10.1090/proc/14590, MR 3993794
  6. ^ Potter, Michael (2004). Set Theory and its Philosophy : A Critical Introduction. Oxford University Press. p. 164. ISBN 9780191556432.
  7. ^ Herrlich, Horst (2006). "Section A.4". Axiom of Choice. Lecture Notes in Mathematics. Vol. 1876. Springer. doi:10.1007/11601562. ISBN 3-540-30989-6. Retrieved 18 July 2023.
  8. ^ a b c d e f g Howard, Paul; Rubin, Jean E. (1998). Consequences of the axiom of choice. Providence, Rhode Island: American Mathematical Society. ISBN 978-0-8218-0977-8. See in particular Form 8, p. 17–18.
  9. ^ a b c d Herrlich, Horst (1997). "Choice principles in elementary topology and analysis" (PDF). Comment. Math. Univ. Carolinae. 38 (3): 545. See, in particular, Theorem 2.4, pp. 547–548.

This article incorporates material from axiom of countable choice on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

This page was last edited on 27 May 2024, at 03:37
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