On special partitions of Dedekind- and Russell-sets

Horst Herrlich; Paul Howard; Eleftherios Tachtsis

Commentationes Mathematicae Universitatis Carolinae (2012)

  • Volume: 53, Issue: 1, page 105-122
  • ISSN: 0010-2628

Abstract

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A Russell set is a set which can be written as the union of a countable pairwise disjoint set of pairs no infinite subset of which has a choice function and a Russell cardinal is the cardinal number of a Russell set. We show that if a Russell cardinal a has a ternary partition (see Section 1, Definition 2) then the Russell cardinal a + 2 fails to have such a partition. In fact, we prove that if a ZF-model contains a Russell set, then it contains Russell sets with ternary partitions as well as Russell sets without ternary partitions. We then consider generalizations of this result.

How to cite

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Herrlich, Horst, Howard, Paul, and Tachtsis, Eleftherios. "On special partitions of Dedekind- and Russell-sets." Commentationes Mathematicae Universitatis Carolinae 53.1 (2012): 105-122. <http://eudml.org/doc/246614>.

@article{Herrlich2012,
abstract = {A Russell set is a set which can be written as the union of a countable pairwise disjoint set of pairs no infinite subset of which has a choice function and a Russell cardinal is the cardinal number of a Russell set. We show that if a Russell cardinal $a$ has a ternary partition (see Section 1, Definition 2) then the Russell cardinal $a+2$ fails to have such a partition. In fact, we prove that if a ZF-model contains a Russell set, then it contains Russell sets with ternary partitions as well as Russell sets without ternary partitions. We then consider generalizations of this result.},
author = {Herrlich, Horst, Howard, Paul, Tachtsis, Eleftherios},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Axiom of Choice; Dedekind sets; Russell sets; generalizations of Russell sets; odd sized partitions; permutation models; Dedekind set; Russell set; odd sized partition; permutation models},
language = {eng},
number = {1},
pages = {105-122},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On special partitions of Dedekind- and Russell-sets},
url = {http://eudml.org/doc/246614},
volume = {53},
year = {2012},
}

TY - JOUR
AU - Herrlich, Horst
AU - Howard, Paul
AU - Tachtsis, Eleftherios
TI - On special partitions of Dedekind- and Russell-sets
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2012
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 53
IS - 1
SP - 105
EP - 122
AB - A Russell set is a set which can be written as the union of a countable pairwise disjoint set of pairs no infinite subset of which has a choice function and a Russell cardinal is the cardinal number of a Russell set. We show that if a Russell cardinal $a$ has a ternary partition (see Section 1, Definition 2) then the Russell cardinal $a+2$ fails to have such a partition. In fact, we prove that if a ZF-model contains a Russell set, then it contains Russell sets with ternary partitions as well as Russell sets without ternary partitions. We then consider generalizations of this result.
LA - eng
KW - Axiom of Choice; Dedekind sets; Russell sets; generalizations of Russell sets; odd sized partitions; permutation models; Dedekind set; Russell set; odd sized partition; permutation models
UR - http://eudml.org/doc/246614
ER -

References

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  1. Blair D., Blass A., Howard P., 10.1142/S0219061305000389, J. Math. Log. 5 (2005), no. 1, 49–85. Zbl1095.03043MR2151583DOI10.1142/S0219061305000389
  2. Fraleigh J.B., A First Course in Abstract Algebra, Addison-Wesley Publ. Co., Reading, Mass.-London-Don Mills, Ont., 1967. Zbl1060.00001MR0225619
  3. Herrlich H., Axiom of Choice, Springer Lecture Notes in Mathematics, 1876, Springer, New York, 2006. Zbl1102.03049MR2243715
  4. Herrlich H., 10.2989/16073600709486213, Quaest. Math. 30 (2007), no. 4, 465–470. Zbl1138.05003MR2368564DOI10.2989/16073600709486213
  5. Herrlich H., Howard P., Tachtsis E., 10.2989/16073606.2011.570293, Quaest. Math. 34 (2011), no. 1, 35–66. MR2810887DOI10.2989/16073606.2011.570293
  6. Herrlich H., Keremedis K., Tachtsis E., 10.2989/16073601003718222, Quaest. Math. 33 (2010), 1–9. MR2755503DOI10.2989/16073601003718222
  7. Herrlich H., Tachtsis E., On the number of Russell’s socks or 2 + 2 + 2 + ⋯ = ?, Comment. Math. Univ. Carolin. 47 (2006), 707–717. MR2337424
  8. Herrlich H., Tachtsis E., 10.1002/malq.200810049, Math. Logic Quart. 56 (2010), no. 2, 185–190. Zbl1201.03040MR2650236DOI10.1002/malq.200810049
  9. Howard P., Rubin J.E., Consequences of the Axiom of Choice, Mathematical Surveys and Monographs, 59, American Mathematical Society, Providence, RI, 1998; (http://consequences.emich.edu/conseq.htm). Zbl0947.03001MR1637107
  10. Jech T.J., The Axiom of Choice, Studies in Logic and the Foundations of Mathematics, 75, North-Holland, Amsterdam, 1973; Reprint: Dover Publications, Inc., New York, 2008. Zbl0259.02052MR0396271
  11. Tarski A., Cancellation laws in the arithmetic of cardinals, Fund. Math. 36 (1949), 77-92. Zbl0039.04804MR0032710

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