Difference between revisions of "Zermelo axiom"
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| − | The [[Axiom of choice|axiom of choice]] for an arbitrary (not necessarily disjoint) family of sets. E. Zermelo stated this axiom in 1904 in the form of the following assertion, which he called the principle of choice [[#References|[1]]]: For every family of non-empty sets | + | {{TEX|done}} |
| + | The [[Axiom of choice|axiom of choice]] for an arbitrary (not necessarily disjoint) family of sets. E. Zermelo stated this axiom in 1904 in the form of the following assertion, which he called the principle of choice [[#References|[1]]]: For every family of non-empty sets $t$ one can choose from each of its terms exactly one representative and combine all these in a single set. He was the first to give a proof, based on his principle of choice, of his well-ordering theorem (cf. [[Zermelo theorem|Zermelo theorem]]). In 1906, B. Russell stated the axiom of choice in a multiplicative form: If $t$ is a disjoint set of non-empty sets, then the direct product $\prod t$ is not empty. In 1908 Zermelo proved the equivalence of the multiplicative form of the axiom of choice and its usual statement. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Zermelo, " | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Zermelo, "Beweis, dass jede Menge wohlgeordnet werden kann" ''Math. Ann.'' , '''59''' (1904) pp. 514–516</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Fraenkel, Y. Bar-Hillel, "Foundations of set theory" , North-Holland (1958)</TD></TR></table> |
Latest revision as of 17:05, 31 March 2017
The axiom of choice for an arbitrary (not necessarily disjoint) family of sets. E. Zermelo stated this axiom in 1904 in the form of the following assertion, which he called the principle of choice [1]: For every family of non-empty sets $t$ one can choose from each of its terms exactly one representative and combine all these in a single set. He was the first to give a proof, based on his principle of choice, of his well-ordering theorem (cf. Zermelo theorem). In 1906, B. Russell stated the axiom of choice in a multiplicative form: If $t$ is a disjoint set of non-empty sets, then the direct product $\prod t$ is not empty. In 1908 Zermelo proved the equivalence of the multiplicative form of the axiom of choice and its usual statement.
References
| [1] | E. Zermelo, "Beweis, dass jede Menge wohlgeordnet werden kann" Math. Ann. , 59 (1904) pp. 514–516 |
| [2] | A.A. Fraenkel, Y. Bar-Hillel, "Foundations of set theory" , North-Holland (1958) |
Comments
References
| [a1] | G.H. Moore, "Zermelo's axiom of choice" , Springer (1982) |
| [a2] | J. Rubin, H. Rubin, "Equivalents of the axiom of choice" , 1–2 , North-Holland (1963–1985) |
How to Cite This Entry:
Zermelo axiom. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zermelo_axiom&oldid=19078
Zermelo axiom. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zermelo_axiom&oldid=19078
This article was adapted from an original article by V.I. Malykhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article