Axiom of choice | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-05-06

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Axiom_of_choice


00:03:38 1 Statement
00:05:20 1.1 Nomenclature ZF, AC, and ZFC
00:05:53 1.2 Variants
00:07:46 1.3 Restriction to finite sets
00:08:41 2 Usage
00:10:46 3 Examples
00:13:58 4 Criticism and acceptance
00:17:22 5 In constructive mathematics
00:19:16 6 Independence
00:22:08 7 Stronger axioms
00:22:46 8 Equivalents
00:26:24 8.1 Category theory
00:27:45 9 Weaker forms
00:28:47 9.1 Results requiring AC (or weaker forms) but weaker than it
00:31:58 9.2 Possibly equivalent implications of AC
00:33:53 10 Stronger forms of the negation of AC
00:35:35 11 Statements consistent with the negation of AC
00:38:01 12 Axiom of choice in type theory
00:39:30 13 Quotes



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SUMMARY
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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 exactly one object from each bin, even if the collection is infinite. Formally, it states that for every indexed family 
  
    
      
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   of nonempty sets there exists an indexed family 
  
    
      
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   of elements such that 
  
    
      
        
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  .  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; this is in particular the case if the number of sets is finite, or if a selection rule is available – some distinguishing property that happens to hold for exactly one element in each set. An illustrative example is sets picked from the natural numbers. From such sets, one may always select the smallest number, e.g. in {{4, 5, 6}, {10, 12}, {1, 400, 617, 8000}} the smallest elements are {4, 10, 1}. In this case, "select the smallest number" is a choice function. Even 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 (if there are non-constructible reals). In that case, the axiom of choice must be invoked. 
Bertrand Russell coined an analogy: for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection; this makes it possible to directly define a choice function. For an infinite collection of pairs of socks (assumed to have no distinguishing features), there is no obvious way to make a function that selects one sock from each pair, without invoking the axiom of choice.Although originally controversial, the axiom of choice is now used without reservation by most mathematicians, and it is included in the standard form of axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice (ZFC). One mot ...