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Results tagged with axiom-of-choice
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user 5903
An important and fundamental axiom in set theory sometimes called Zermelo's axiom of choice. It was formulated by Zermelo in 1904 and states that, given any set of mutually disjoint nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets. The axiom of choice is related to the first of Hilbert's problems.
6
votes
Axiom of Choice and Vitali's theorem
In this paper (PDF) Sierpinski constructed a non-measurable set from an ultrafilter on $\mathbb{N}$. The existence of ultrafilters on $\mathbb{N}$ is weaker than the Axiom of Choice.
- 11.4k
8
votes
Implicit uses of Countable or Dependent Choice
Rudin's Principles of Mathematical Analysis (and most every book on Mathematical Analysis) in the proof that $\lim_{x\to p}f(x)=q$ is equivalent to "$\lim_{n\to\infty}f(p_n)=q$ for every sequence $\la …
12
votes
Does Urysohn's Lemma imply Dependent Choice?
In Versions of normality and some weak forms of the axiom of choice Paul Howard et al exhibit a model of MC (Multiple Choice) and not-DC, see page 381.
In that model Urysohn's Lemma (NU) holds, so it …
- 11.4k
16
votes
Accepted
Partition of unity without AC
The proofs rely, in the background, on Urysohn's Lemma, which follows from the Principle of Dependent Choices but is not provable without some Choice. It is false in the ordered Mostowski model, see
G …
- 11.4k
4
votes
Accepted
Posets such that the collection of principal down-sets does not have property ${\bf B}$
Let $M$ be the ordered Mostowski model (T. Jech, The Axiom of Choice, Section 4.5). Its set of atoms, $A$, has a linear order $\prec$ that makes it isomorphic to the rationals. Let $S\in M$ be a subse …
- 11.4k
3
votes
Posets such that the collection of principal down-sets does not have property ${\bf B}$
The axiom of choice implies that for every partial order $P$ the
hypergraph $H_P$ has property $B$.
Let $(P,\le)$ be a partial order.
We first claim the following: for every $p\in P$ there is a $q\le …
- 11.4k