# Article

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Keywords:
Axiom of Choice; Dedekind sets; Russell sets; generalizations of Russell sets; odd sized partitions; permutation models
Summary:
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.
References:
[1] Blair D., Blass A., Howard P.: Divisibility of Dedekind finite sets. J. Math. Log. 5 (2005), no. 1, 49–85. DOI 10.1142/S0219061305000389 | MR 2151583 | Zbl 1095.03043
[2] Fraleigh J.B.: A First Course in Abstract Algebra. Addison-Wesley Publ. Co., Reading, Mass.-London-Don Mills, Ont., 1967. MR 0225619 | Zbl 1060.00001
[3] Herrlich H.: Axiom of Choice. Springer Lecture Notes in Mathematics, 1876, Springer, New York, 2006. MR 2243715 | Zbl 1102.03049
[4] Herrlich H.: Binary partitions in the absence of choice or rearranging Russell's socks. Quaest. Math. 30 (2007), no. 4, 465–470. DOI 10.2989/16073600709486213 | MR 2368564 | Zbl 1138.05003
[5] Herrlich H., Howard P., Tachtsis E.: The cardinal inequality $\alpha^2< 2^\alpha$. Quaest. Math. 34 (2011), no. 1, 35–66. DOI 10.2989/16073606.2011.570293 | MR 2810887
[6] Herrlich H., Keremedis K., Tachtsis E.: On Russell and anti Russell–cardinals. Quaest. Math. 33 (2010), 1–9. DOI 10.2989/16073601003718222 | MR 2755503
[7] Herrlich H., Tachtsis E.: On the number of Russell's socks or $2+2+2+\cdots=$?. Comment. Math. Univ. Carolin. 47 (2006), 707–717. MR 2337424
[8] Herrlich H., Tachtsis E.: Odd-sized partitions of Russell-sets. Math. Logic Quart. 56 (2010), no. 2, 185–190. DOI 10.1002/malq.200810049 | MR 2650236 | Zbl 1201.03040
[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) MR 1637107 | Zbl 0947.03001
[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. MR 0396271 | Zbl 0259.02052
[11] Tarski A.: Cancellation laws in the arithmetic of cardinals. Fund. Math. 36 (1949), 77-92. MR 0032710 | Zbl 0039.04804

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