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Results tagged with set-theory
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user 61785
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
2
votes
Nice algebraic statements independent from ZF + V=L (constructibility)
We can use a theorem of Eklof and Mekler to get a statement about abelian groups that is independent of $ZFC + V=L$ under the additional assumption that it is consistent that weakly compact cardinals …
- 3,768
11
votes
Who needs Replacement anyway?
Here's an example of a published, nontrivial use of the the product of the sequence $\{ V, V^{*}, V^{**}, \ldots \}$ in a functional analysis paper, for the specific case of $V = \mathbb{R}^{\mathbb{N …
- 3,768
4
votes
Accepted
Baire category theorem for uncountable unions
The hyperstonean case can be dealt with using a result from Fremlin's Measure Theory. For every hyperstonean space $X$, we can find a semi-finite measure $\mu$ defined on the sets with the Baire prope …
- 3,768
5
votes
Existence of a strange measure
This can be proved without introducing ultrafilters by name, by doing "finitary measure theory" and using Zorn's lemma.
An algebra $A$ on a set $X$ is just a $\sigma$-algebra without the $\sigma$, i …
- 3,768
0
votes
Accepted
Infinite distributive laws in atomless free sigma-algebra
This holds because $\mathfrak{A}$ is a concrete $\sigma$-algebra, being the Baire $\sigma$-algebra of $2^{\omega_1}$. In fact, the cardinality of $\omega_1$ plays no role whatsoever and $\omega_1$ cou …
- 3,768
1
vote
Accepted
(non) separability of the power set
The question has a trivial negative answer, as long as an atomlessly measurable cardinal exists (if one doesn't it is vacuously true, of course). Given an atomless probability measure $\mu$ on $(Y, \m …
- 3,768
3
votes
Non-separable metric probability space
Iosif Pinelis has given an answer to question 1 and partial answers to 2 and 3. Since he advised me to turn my comments into an answer, here it is.
I will deal with the case where the axiom of choice …
- 3,768
3
votes
Accepted
Consistency of the size of the least real-valued measurable cardinal, vis-a-vis the continuum
All of them are equiconsistent with the existence of a measurable cardinal. I find that a nice reference for this stuff, in addition to Solovay's article, is Jech's Set Theory: Third Millennium Editio …
- 3,768
2
votes
Applications of set theory in physics
I don't know how important these papers are considered to be, but they are by a physicist and published in a (mathematical) physics journal:
http://scitation.aip.org/content/aip/journal/jmp/17/5/10.1 …
- 3,768