Abstract
It is rightly said that the principle of set theory known as the Axiom of Choice is “probably the most interesting and in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid’s Axiom of Parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel, & Levy, 1973).
This section is based on (Clerbout & Rahman, 2015) and (Rahman, Clerbout, & Jovanovic, 2015), where we developed a complete demonstation of AC but with a slighlty different dialogical setting.
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Notes
- 1.
See also (Bell, 2009).
- 2.
- 3.
- 4.
See for instance this observation of (Martin-Löf, 2006, p. 349): “[…] this is not visible within an extensional framework, like Zermelo-Fraenkel set theory, where all functions are by definition extensional.”
- 5.
- 6.
Making use of C(x, y) : set (x : A, y : B(x)).
- 7.
That is, C(x, Ap((λx) p (Ap(z, x)), x)) = C(x, p(Ap(z, x)) : set.
- 8.
(Martin-Löf, 1984, pp. 50–51).
- 9.
See (Jovanovic, 2013).
- 10.
See Sect. 9.1 for further details on game definiteness and its crucial role in the dialogical framework.
- 11.
(Poincaré, 1905, p. 27). “If you are present at a game af chess, it will not suffice, for the understanding of the game, to know the rules for moving the pieces. That will only enable you to recognize that each move has been made conformably to these rules, and this knowledge will truly have very little value. Yet this is what the reader of a book on mathematics would do if he were a logician only. To understand the game is wholly another matter; it is to know why the player moves this piece rather than that otber which he could have moved without breaking the rules of the game. It is to perceive the inward reason which makes of this series of successive moves a sort of organized whole. This faculty is still more necessary for the player himself, that is, for the inventor” (Poincaré, 1907, p. 22).
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Rahman, S., McConaughey, Z., Klev, A., Clerbout, N. (2018). The Remarkable Case of the Axiom of Choice. In: Immanent Reasoning or Equality in Action. Logic, Argumentation & Reasoning, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-319-91149-6_8
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