On the number of Russell’s socks or 2 + 2 + 2 + ⋯ = ?

Horst Herrlich; Eleftherios Tachtsis

Commentationes Mathematicae Universitatis Carolinae (2006)

  • Volume: 47, Issue: 4, page 707-717
  • ISSN: 0010-2628

Abstract

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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 ℵ α or 2 ℵ α ; • 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 ℵ 0 Russell-cardinals between a and 2 a ; • there exist antichains consisting of 2 ℵ 0 Russell-cardinals; • there are neither minimal nor maximal Russell-cardinals; • no Russell-graph has a chromatic number.

How to cite

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Herrlich, Horst, and Tachtsis, Eleftherios. "On the number of Russell’s socks or $2+2+2+\dots =\text{?}$." Commentationes Mathematicae Universitatis Carolinae 47.4 (2006): 707-717. <http://eudml.org/doc/249882>.

@article{Herrlich2006,
abstract = {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.},
author = {Herrlich, Horst, Tachtsis, Eleftherios},
journal = {Commentationes Mathematicae Universitatis Carolinae},
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; Axiom of Choice; Generalized Continuum Hypothesis; Dedekind-finite sets},
language = {eng},
number = {4},
pages = {707-717},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the number of Russell’s socks or $2+2+2+\dots =\text\{?\}$},
url = {http://eudml.org/doc/249882},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Herrlich, Horst
AU - Tachtsis, Eleftherios
TI - On the number of Russell’s socks or $2+2+2+\dots =\text{?}$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 4
SP - 707
EP - 717
AB - 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.
LA - eng
KW - 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; Axiom of Choice; Generalized Continuum Hypothesis; Dedekind-finite sets
UR - http://eudml.org/doc/249882
ER -

References

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  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. (1907) 
  7. Russell B., Sur les axiomes de l'infini et du transfini, Bull. Soc. France 39 (1911), 488-501. (1911) 
  8. Schechter E., Handbook of Analysis and its Foundations, Academic Press, San Diego, 1997. Zbl0952.26001MR1417259
  9. Sierpiński W., Sur l’egalité 2 m = 2 n pour les nombres cardinaux, Fund. Math. 3 (1922), 1-6. (1922) MR0078413
  10. Tarski A., On the existence of large sets of Dedekind cardinals, Notices Amer. Math. Soc. 12 (1965), 719 pp. (1965) 

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