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axiom of choice

ax·i·om of choice
A a

Transcription

    • US Pronunciation
    • US IPA
    • [ak-see-uh m uhv, ov chois]
    • /ˈæk si əm ʌv, ɒv tʃɔɪs/
    • US Pronunciation
    • US IPA
    • [ak-see-uh m uhv, ov chois]
    • /ˈæk si əm ʌv, ɒv tʃɔɪs/

Definitions of axiom of choice words

  • noun axiom of choice the axiom of set theory that given any collection of disjoint sets, a set can be so constructed that it contains one element from each of the given sets. 1
  • noun Definition of axiom of choice in Technology (logic)   (AC, or "Choice") An axiom of set theory: If X is a set of sets, and S is the union of all the elements of X, then there exists a function f:X -> S such that for all non-empty x in X, f(x) is an element of x. In other words, we can always choose an element from each set in a set of sets, simultaneously. Function f is a "choice function" for X - for each x in X, it chooses an element of x. Most people's reaction to AC is: "But of course that's true! From each set, just take the element that's biggest, stupidest, closest to the North Pole, or whatever". Indeed, for any finite set of sets, we can simply consider each set in turn and pick an arbitrary element in some such way. We can also construct a choice function for most simple infinite sets of sets if they are generated in some regular way. However, there are some infinite sets for which the construction or specification of such a choice function would never end because we would have to consider an infinite number of separate cases. For example, if we express the real number line R as the union of many "copies" of the rational numbers, Q, namely Q, Q+a, Q+b, and infinitely (in fact uncountably) many more, where a, b, etc. are irrational numbers no two of which differ by a rational, and Q+a == {q+a : q in Q} we cannot pick an element of each of these "copies" without AC. An example of the use of AC is the theorem which states that the countable union of countable sets is countable. I.e. if X is countable and every element of X is countable (including the possibility that they're finite), then the sumset of X is countable. AC is required for this to be true in general. Even if one accepts the axiom, it doesn't tell you how to construct a choice function, only that one exists. Most mathematicians are quite happy to use AC if they need it, but those who are careful will, at least, draw attention to the fact that they have used it. There is something a little odd about Choice, and it has some alarming consequences, so results which actually "need" it are somehow a bit suspicious, e.g. the Banach-Tarski paradox. On the other side, consider Russell's Attic. AC is not a theorem of Zermelo Fränkel set theory (ZF). Gödel and Paul Cohen proved that AC is independent of ZF, i.e. if ZF is consistent, then so are ZFC (ZF with AC) and ZF(~C) (ZF with the negation of AC). This means that we cannot use ZF to prove or disprove AC. 1
  • noun axiom of choice (set theory) One of the axioms in axiomatic set theory, equivalent to the statement that an arbitrary direct product of non-empty sets is non-empty. 0
See all 3 definitions of axiom of choice

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Parts of speech for Axiom of choice

noun
adjective
verb
adverb
pronoun
preposition
conjunction
determiner
exclamation
Full grammatical hierarchy of Axiom of choice

axiom of choice popularity

This term is known only to a narrow circle of people with rare knowledge. Only 2% of English native speakers know the meaning of this word.
According to our data most of word are more popular. This word is almost not used. It has a much more popular synonym.

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