• Corpus ID: 17589687

Spanning Graphs and the Axiom of Choice

  title={Spanning Graphs and the Axiom of Choice},
  author={Christian Delhomm{\'e} and Marianne Morillon},
  journal={Reports Math. Log.},
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 bipartite graph K n,n , and in particular it is equivalent to the fact that every connected graph has a spanning cycle-free graph (possibly non connected). We consider simple undirected loop-free graphs. A forest is a graph with no cycles, a tree is a connected forest. A graph G is a sub-graph of G if all its… 
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
Linear forms and axioms of choice
We work in set-theory without choice ZF. Given a commutative field $\mathbb K$, we consider the statement $\mathbf D (\mathbb K)$: “On every non null $\mathbb K$-vector space there exists a non-null
König's Infinity Lemma and Beth's Tree Theorem
König, D. [1926. ‘Sur les correspondances multivoques des ensembles’, Fundamenta Mathematica, 8, 114–34] includes a result subsequently called König's Infinity Lemma. Konig, D. [1927. ‘Über eine
Commentationes Mathematicae Universitatis Carolinae
In set theory without the Axiom of Choice ZF, we prove that for every commutative field K, the following statement DK: “On every non null K-vector space, there exists a non null linear form” implies
  • Michael Ernst
  • Mathematics, Computer Science
    The Review of Symbolic Logic
  • 2015
It is shown that the answer is no by showing that unlimited category theory is inconsistent.
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
Free groups and 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 show
Equivalents of the Axiom of Choice II
Set Forms. The Well-Ordering Theorem. The Axiom of Choice. The Law of the Trichotomy. Maximal Principles. Forms Equivalent to the Axiom of Choice Under the Axioms of Extensionality and Foundation.
Equivalents of the axiom of choice
Parts of this note have been discussed elsewhere in the blog, sometimes in a different form (see here and here, for example), but I haven’t examined before the equivalence of 5–7 with choice. In the
The strength of Mac Lane set theory
  • A. Mathias
  • Mathematics, Computer Science
    Ann. Pure Appl. Log.
  • 2001
This paper shows that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke–Platek set theory and even the Axiom of Constructibility to Mac Lane’s axiOMs, and studies a simple set theoretic assertion.
Consequences of the axiom of choice
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.
Admissible sets and structures
Let's read! We will often find out this sentence everywhere. When still being a kid, mom used to order us to always read, so did the teacher. Some books are fully read in a week and we need the
Admissible sets and structures, Springer-Verlag, Berlin
  • 1975
The Axiom of Choice, Studies in Logic
  • The Axiom of Choice, Studies in Logic
  • 1973
Axioms of multiple choice