All Questions
Tagged with axiom-of-choice ultrafilters
10
questions
12
votes
2
answers
814
views
Ultrafilter lemma for arbitrary lattice
Can someone kindly confirm whether the ultrafilter lemma for arbitrary (i.e., not necessarily Boolean) bounded lattices is equivalent to Zorn's lemma?
To be precise, if $\mathbf{L} = (L, \leq, \land, \...
- 273
5
votes
1
answer
266
views
Can Tychonoffs theorem for a countable number of spaces be proven with ZF plus the axiom of (countable) dependent choice?
It can be proven without any form of infinite choice that the product of two compact spaces (and thus any finite product) is compact, while on the other hand, it is well known that the general form of ...
- 1,843
9
votes
2
answers
934
views
SPOT as a conservative extension of Zermelo–Fraenkel
In Infinitesimal analysis without the Axiom of Choice, Hrbacek and Katz have shown that it is possible to formulate an axiomatic theory which provides a formalisation of calculus procedures which make ...
- 5,020
7
votes
0
answers
170
views
Non-wellorderable ultrafilters with wellorderable bases
There are some models in which $2^\omega$ is not wellorderable but there is a free ultrafilter over $\omega$. What about the consistency of: $2^\omega$ is not wellorderable + AC for countable sets of ...
- 2,241
3
votes
2
answers
241
views
What are the minimal requirements for the definable hyperreal field plus transfer?
It is interesting that to prove the transfer principle for the definable hyperreal field, one requires no more choice than for proving, for instance, the countable additivity of the Lebesgue measure. ...
- 15.8k
6
votes
0
answers
233
views
Does Łoś's theorem imply choice given a free ultrafilter?
In the paper "Łoś's theorem and the boolean prime ideal theorem imply axiom of choice" Howard has shown that Łoś's theorem and the boolean prime ideal theorem imply axiom of choice. At the ...
- 27.7k
8
votes
3
answers
714
views
In $L$, does there exist a definable non-principal ultrafilter on $\mathbb{N}$
The axiom of constructibility $V=L$ leads to some very interesting consequences, one of which is that it becomes possible to give explicit constructions of some of the "weird" results of AC. For ...
- 4,857
25
votes
2
answers
2k
views
Axiom of choice: ultrafilter vs. Vitali set
It is well known that from a free (non-principal) ultrafilter on $\omega$ one can define a non-measurable set of reals. The older example of a non-measurable set is the Vitali set,
a set of ...
- 16.1k
10
votes
3
answers
2k
views
Reference Request: Independence of the ultrafilter lemma from ZF
I'm looking for references for the following facts concerning the ultrafilter lemma (~ "there exist non-principal ultrafilters"):
The ultrafilter lemma is independent of ZF.
ZF + the ultrafilter ...
- 1,347
7
votes
3
answers
897
views
Construction of a maximal ideal
Hello,
Let R denote the ring of continuous functions defined on the real line, let I in R be the ideal consisting of functions with compact support. Obviously, I is not maximal, and by Zorn's Lemma ...
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