# Article

Full entry | PDF   (0.2 MB)
Keywords:
Bertrand Russell; Axiom of Choice; Generalized Continuum Hypothesis; Dedekind-finite sets; Dedekind-cardinals; Russell-cardinals; odd and (almost) even cardinals; cardinal arithmetic; coloring of graphs; chromatic number; socks
Summary:
The following question is analyzed under the assumption that the Axiom of Choice fails badly: Given a countable number of pairs of socks, then how many socks are there? Surprisingly this number is not uniquely determined by the above information, thus giving rise to the concept of Russell-cardinals. It will be shown that: • some Russell-cardinals are even, but others fail to be so; • no Russell-cardinal is odd; • no Russell-cardinal is comparable with any cardinal of the form $\aleph_\alpha$ or $2^{\aleph_\alpha}$; • finite sums of Russell-cardinals are Russell-cardinals, but finite products — even squares — of Russell-cardinals may fail to be so; • some countable unions of pairwise disjoint Russell-sets are Russell-sets, but others fail to be so; • for each Russell-cardinal $a$ there exists a chain consisting of $2^{\aleph_0}$ Russell-cardinals between $a$ and $2^a$; • there exist antichains consisting of $2^{\aleph_0}$ Russell-cardinals; • there are neither minimal nor maximal Russell-cardinals; • no Russell-graph has a chromatic number.
References:
[1] Brunner N.: Realisierung und Auswahlaxiom. Arch. Math. (Brno) 20 (1984), 39-42. MR 0785045 | Zbl 0551.54004
[2] Galvin F., Komjáth P.: Graph colorings and the axiom of choice. Period. Math. Hungar. 22 (1991), 71-75. MR 1145937
[3] Herrlich H.: Axiom of Choice. Lecture Notes in Math. 1876, Springer, Berlin, 2006. MR 2243715 | Zbl 1102.03049
[4] Howard P., Rubin J.E.: Consequences of the axiom of choice. Mathematical Surveys and Monographs 59, American Math. Society, Providence, 1998. MR 1637107 | Zbl 0947.03001
[5] Jech T.J.: The Axiom of Choice. Studies in Logic and the Foundations of Math. 75, North Holland, Amsterdam, 1973. MR 0396271 | Zbl 0259.02052
[6] Russell B.: On some difficulties in the theory of transfinite numbers and order types. Proc. London Math. Soc. Sec. Sci. 4 (1907), 29-53.
[7] Russell B.: Sur les axiomes de l'infini et du transfini. Bull. Soc. France 39 (1911), 488-501.
[8] Schechter E.: Handbook of Analysis and its Foundations. Academic Press, San Diego, 1997. MR 1417259 | Zbl 0952.26001
[9] Sierpiński W.: Sur l'egalité $2 m = 2 n$ pour les nombres cardinaux. Fund. Math. 3 (1922), 1-6. MR 0078413
[10] Tarski A.: On the existence of large sets of Dedekind cardinals. Notices Amer. Math. Soc. 12 (1965), 719 pp.

Partner of