Corpus ID: 237492042

Partition models, Permutations of infinite sets without fixed points, Variants of CAC, and weak forms of AC

@inproceedings{Banerjee2021PartitionMP,
  title={Partition models, Permutations of infinite sets without fixed points, Variants of CAC, and weak forms of AC},
  author={Amitayu Banerjee},
  year={2021}
}
  • Amitayu Banerjee
  • Published 13 September 2021
  • Mathematics
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 weakened to allow the existence of atoms). • For every infinite set X, there exists a permutation of X without fixed points. • There is no Hausdorff space X such that every infinite subset of X contains an infinite compact subset. • If a field has an algebraic closure then it is unique up to isomorphism… Expand

References

SHOWING 1-10 OF 32 REFERENCES
On variants of the principle of consistent choices, the minimal cover property and the 2-compactness of generalized Cantor cubes
In set theory without the Axiom of Choice (AC), we study the deductive strength of variants of the Principle of Consistent Choices (PCC) and their relationship with the minimal cover property, theExpand
THE AXIOM OF CHOICE
We propose that failures of the axiom of choice, that is, surjective functions admitting no sections, can be reasonably classified by means of invariants borrowed from algebraic topology. We showExpand
A Permutation Model with Finite Partitions of the Set of Atoms as Supports
The method of permutation models was introduced by Fraenkel in 1922 to prove the independence of the axiom of choice in set theory with atoms. We present a variant of the basic Fraenkel model inExpand
On metrizability and compactness of certain products without the Axiom of Choice
Abstract We prove that there exists a model of ZF (Zermelo–Fraenkel set theory without the Axiom of Choice ( AC )) in which there is a compact, metrizable, non-second countable, Cantor cube. ThisExpand
ON RAMSEY’S THEOREM AND THE EXISTENCE OF INFINITE CHAINS OR INFINITE ANTI-CHAINS IN INFINITE POSETS
  • E. Tachtsis
  • Computer Science, Mathematics
  • The Journal of Symbolic Logic
  • 2016
TLDR
It is shown that there is a model of ZF which satisfies “every infinite partially ordered set has either an infinite chain or an infinite anti-chain” and the negation of Ramsey's Theorem, and it is proved that Ramsey’s Theorem (hence, the above principle for infinite partiallyordered sets) is true in Mostowski's linearly ordered model. Expand
On special partitions of Dedekind- and Russell-sets
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 numberExpand
$k$-spaces, sequential spaces and related topics in the absence of the axiom of choice
In the absence of the axiom of choice, new results concerning sequential, Fréchet-Urysohn, k-spaces, very k-spaces, Loeb and Cantor completely metrizable spaces are shown. New choice principles areExpand
Ramsey's Theorem in the Hierarchy of Choice Principles
  • A. Blass
  • Mathematics, Computer Science
  • J. Symb. Log.
  • 1977
TLDR
This paper proves or refute the provability of each of the possible implications between Ramsey's theorem and the weak axioms of choice mentioned in Appendix A.3 of Jech's book. Expand
On Martin's Axiom and Forms of Choice
  • E. Tachtsis
  • Mathematics, Computer Science
  • Math. Log. Q.
  • 2016
TLDR
This paper defines MA∗ to be the statement that for every (not necessarily well-ordered) cardinal n < 2א0 , the authors have that MAn holds, and investigates the set-theoretic strength of the principleMA∗. Expand
Spanning Graphs and the Axiom of Choice
A b s t r a c t. We show in set-theory ZF that the axiom of choice is equivalent to the statement every bipartite connected graph has a spanning sub-graph omitting some complete finite bipartiteExpand
...
1
2
3
4
...