| Title:
|
On special partitions of Dedekind- and Russell-sets (English) |
| Author:
|
Herrlich, Horst |
| Author:
|
Howard, Paul |
| Author:
|
Tachtsis, Eleftherios |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
53 |
| Issue:
|
1 |
| Year:
|
2012 |
| Pages:
|
105-122 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Axiom of Choice |
| Keyword:
|
Dedekind sets |
| Keyword:
|
Russell sets |
| Keyword:
|
generalizations of Russell sets |
| Keyword:
|
odd sized partitions |
| Keyword:
|
permutation models |
| MSC:
|
03E10 |
| MSC:
|
03E25 |
| MSC:
|
03E35 |
| MSC:
|
05A18 |
| idZBL:
|
Zbl 1249.05018 |
| idMR:
|
MR2880914 |
| . |
| Date available:
|
2012-02-07T10:27:15Z |
| Last updated:
|
2014-04-07 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141829 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
[3] Herrlich H.: Axiom of Choice.Springer Lecture Notes in Mathematics, 1876, Springer, New York, 2006. Zbl 1102.03049, MR 2243715 |
| Reference:
|
[4] Herrlich H.: Binary partitions in the absence of choice or rearranging Russell's socks.Quaest. Math. 30 (2007), no. 4, 465–470. Zbl 1138.05003, MR 2368564, 10.2989/16073600709486213 |
| Reference:
|
[5] Herrlich H., Howard P., Tachtsis E.: The cardinal inequality $\alpha^2< 2^\alpha $.Quaest. Math. 34 (2011), no. 1, 35–66. MR 2810887, 10.2989/16073606.2011.570293 |
| Reference:
|
[6] Herrlich H., Keremedis K., Tachtsis E.: On Russell and anti Russell–cardinals.Quaest. Math. 33 (2010), 1–9. MR 2755503, 10.2989/16073601003718222 |
| Reference:
|
[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 |
| Reference:
|
[8] Herrlich H., Tachtsis E.: Odd-sized partitions of Russell-sets.Math. Logic Quart. 56 (2010), no. 2, 185–190. Zbl 1201.03040, MR 2650236, 10.1002/malq.200810049 |
| Reference:
|
[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). Zbl 0947.03001, MR 1637107 |
| Reference:
|
[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. Zbl 0259.02052, MR 0396271 |
| Reference:
|
[11] Tarski A.: Cancellation laws in the arithmetic of cardinals.Fund. Math. 36 (1949), 77-92. Zbl 0039.04804, MR 0032710 |
| . |
