Axiom of choice - Wikipedia
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family ( 𝑆 𝑖 ) 𝑖 ∈ 𝐼 of nonempty sets, there exists an indexed set ( 𝑥 𝑖 ) 𝑖 ∈ 𝐼 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.[1] In many cases, a set created by choosing elements can be made without invoking the axiom of choice, particularly if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available — some distinguishing property t
Axiom of choice - Wikipedia Jump to content From Wikipedia, the free encyclopedia Axiom of set theory This article is about the mathematical concept. For the band, see Axiom of Choice (band) . "ZF¬C" redirects here; not to be confused with ZFC (disambiguation) . Illustration of the axiom of choice, with each set S i represented as a jar and its elements represented as marbles. Each element x i is represented as a marble on the right. Colors are used to suggest a functional association of marbles after adopting the choice axiom. The existence of such a choice function is in general independent
related reading
- The Axiom of Choiceplato.stanford.edu
- Zermelo–Fraenkel set theoryen.wikipedia.org
- Zermelo–Fraenkel set theoryen.wikipedia.org
- Zorn's lemma - Wikipediaen.wikipedia.org
- Set Theory | Internet Encyclopedia of Philosophyiep.utm.edu
- Why Math’s Final Axiom Proved So Controversial | Quanta Magazinequantamagazine.org
- Set Theory (Stanford Encyclopedia of Philosophy)plato.stanford.edu
- Set theory - Wikipediaen.wikipedia.org
- Schröder–Bernstein theorem - Wikipediaen.wikipedia.org
- Well-ordering theorem - Wikipediaen.wikipedia.org
- How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer. | Quanta Magazinequantamagazine.org
- [2501.02693] Any function I can actually write down is measurable, right?arxiv.org