• Corpus ID: 51797199

# On special partitions of Dedekind- and Russell-sets

@inproceedings{Herrlich2012OnSP,
title={On special partitions of Dedekind- and Russell-sets},
author={Horst Herrlich and Paul E. Howard and Eleftherios Tachtsis},
year={2012}
}
• Published 2012
• Mathematics
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…
3 Citations
Partition models, Permutations of infinite sets without fixed points, Variants of CAC, and weak forms of AC
• Amitayu Banerjee
• Mathematics
• 2021
We study new relations of the following statements with weak choice principles in ZF (ZermeloFraenkel set theory without the Axiom of Choice (AC)) and ZFA (ZF with the axiom of extensionality
On the Existence of Free Ultrafilters on ω and on Russell-sets in ZF
1. UF(ω) implies that there exists a free ultrafilter on every Russell-set. The implication is not reversible in ZF. 2. The statement “there exists a free ultrafilter on every Russell-set” is not
Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC
In set theory without the Axiom of Choice (AC), we observe new relations of the following statements with weak choice principles. 1. Every locally finite connected graph has a maximal independent

## References

SHOWING 1-10 OF 12 REFERENCES
Odd-sized partitions of Russell-sets
• Mathematics, Computer Science
Math. Log. Q.
• 2010
It is established that it is relative consistent with ZF that there exists a Russell-set X such that |X | is not divisible by any finite cardinal n > 1.
Divisibility of Dedekind finite Sets
• Mathematics, Computer Science
J. Math. Log.
• 2005
The results are that the divisors of a Dedekind-finite set can consistently be any set of natural numbers (containing 1 but not 0), that a Dedkind- finite power of 2 cannot be divisible by 3, and that a dedicated set can be congruent modulo 3, to all of 0, 1, and 2 simultaneously.
On the number of Russell’s socks or $2+2+2+\dots =\text{?}$
• Mathematics
• 2006
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
Binary Partitions in the Absence of Choice or Rearranging Russell's Socks
Given a sequence (Xn )n∈N of pairwise disjoint 2-element sets. Can ∪ n∈N Xn be rearranged as a union ∪ i∈I Yi of a family (Yi )i∈I of pairwise disjoint 2-element sets, indexed by an uncountable set
The cardinal inequality α2 < 2α
• Mathematics
• 2011
Abstract In ZFC set theory (i.e., Zermelo-Fraenkel set theory with the Axiom of Choice (AC)) any two cardinal numbers are comparable. However, this may not be valid in ZF (i.e., Zermelo-Fraenkel set
On Russell and Anti Russell-Cardinals
• Mathematics
• 2010
Abstract In the absence of the Axiom of Choice we study countable families of 2-element sets with no choice functions which either have infinite subfamilies with a choice function or no infinite
The Axiom of Choice in the Foundations of Mathematics
The principle of set theory known as the Axiom of Choice ( AC) has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second
A first course in abstract algebra
A Few Preliminaries. Mathematics and Proofs.Sets and Relations.Mathematical Induction.Complex and Matrix Algebra. 1. Groups and Subgroups. Binary Operations.Finite-State Machines
Consequences of the axiom of choice
• Mathematics
• 1998
Numerical list of forms Topical list of forms Models Notes References for relations between forms Bibliography Table 1 and Table 2 Subject index Author index Software.
On the number of Russell ’ s socks or 2 + 2 + 2 + · · · = ? , Comment
• Quaest . Math .
• 2010