Abstract
The principle of set theory known as the Axiom of Choice ( AC) has been hailed as “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”. It has been employed in countless mathematical papers, a number of monographs have been exclusively devoted to it, and it has long played a prominently role in discussions on the foundations of mathematics.
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Notes
- 1.
Fraenkel, Bar-Hillel, Levy (1973), Section II.4.
- 2.
Zermelo does not actually give the principle an explicit name at this point, however. He does so only in his (1908), where he uses the term “postulate of choice”.
- 3.
Zermelo’s formulation reads literally: A set S that can be decomposed into a set of disjoint parts A,B,C…, each containing at least one element, possesses at least one subset S 1 having exactly one element with each of the parts A,B,C,…, considered.
- 4.
This argument, suitably refined, yields a rigorous derivation of AC in this formulation from Zorn’s lemma.
- 5.
It is this formulation of AC that Russell and others refer to as the multiplicative axiom, since it is easily seen to be equivalent to the assertion that the product of arbitrary nonzero cardinal numbers is nonzero.
- 6.
Ramsey (1926).
- 7.
Quoted in Section 4.8 of Moore (1982).
- 8.
- 9.
This fact, according to Bernays, renders the usual objections against the principle of choice invalid, since these latter are based on the misapprehension that the principle “claims the possibility of a choice”
- 10.
Martin-Löf (2006).
- 11.
- 12.
For a proof see, e.g., Tait (1994).
- 13.
See Tait (1994).
- 14.
Here \(\displaystyle{\coprod_{i\in I}}A_i\) may be identified with \(\displaystyle{\bigcup_{i\in I}}(A_i\times\{i\})\).
- 15.
Diaconescu (1975).
- 16.
Note, however, that if the axiom of choice is formulated within set theory or topos theory in the “harmless”—indeed mathematically useless—way (+), it is perfectly compatible with intuitionistic logic.
- 17.
David Devidi has had the happy inspiration of calling ε α “the thing most likely to be α.”
- 18.
- 19.
Aczel (2001).
- 20.
Maietti (2005).
References
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Bell, J.L. (2011). The Axiom of Choice in the Foundations of Mathematics. In: Sommaruga, G. (eds) Foundational Theories of Classical and Constructive Mathematics. The Western Ontario Series in Philosophy of Science, vol 76. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0431-2_8
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