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Results tagged with set-theory
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user 1149
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.
5
votes
Cardinality of a set of countable connected Hausdorff spaces
That there are $2^{\mathfrak{c}} = 2^{2^{\aleph_0}}$ homeomorphism types of countably infinite connected Hausdorff spaces is already proved in
Kannan, V.; Rajagopalan, M.
Regularity and dispersi …
- 65.4k
31
votes
Is Monsky's theorem dependent on the axiom of choice?
I was recently looking over Monsky's Theorem as supplementary material for my course notes on local fields, and I noticed that his original article (available here) ends by addressing your question:
…
- 65.4k
0
votes
How much larger is the powerset of a transfinite set?
Here I am collecting some comments I made on this question into a CW answer, which should be easier to read and respond to.
Is there a better argument for showing that the powerset of a transfini …
4
votes
Seemingly complex logic/set-theoretic puzzle
I wonder why Adam cannot reason as follows:
If the answer to the question is yes, then I will answer truthfully and use the high note to mean "yes". Thus I will play the high note.
If the answer to …
- 65.4k
4
votes
A question about Transfinite Induction
An answer from a non-set theorist (beware!):
It is not necessary to talk about ordinals or cardinals at all to discuss transfinite induction. It is something that makes sense with respect to any wel …
- 65.4k
3
votes
A principle of mathematical induction for partially ordered sets with infima?
This is a reply to Cam's question.
No, I did not base my talk directly on Kalantari's article, although I think it would be possible to do so. Instead I wrote up some lecture notes before the talk …
43
votes
4
answers
4k
views
A principle of mathematical induction for partially ordered sets with infima?
Recently I learned that there is a useful analogue of mathematical induction over $\mathbb{R}$ (more precisely, over intervals of the form $[a,\infty)$ or $[a,b]$). It turns out that this is an old i …
- 65.4k
90
votes
Accepted
Inaccessible cardinals and Andrew Wiles's proof
The basic contention here is that Wiles' work uses cohomology of sheaves on certain Grothendieck topologies, the general theory of which was first developed in Grothendieck's SGAIV and which requires …
- 65.4k
30
votes
Accepted
Naturally occuring groups with cardinality greater than the reals.
In line with Joel's answer, my favorite "outrageously large group" is the group $G = \operatorname{Aut}(\mathbb{C})$ of field automorphisms of the complex numbers. It has cardinality $2^{2^{\aleph_0} …
- 65.4k
4
votes
Is {Ø,{Ø},{Ø,{Ø}}, ... } the only known universe?
Just to make perfectly sure: Grothendieck is absolutely not questioning the existence of infinite sets in this quotation. (He had, and has, some eccentricities, but not in this direction!)
Remember …
- 65.4k
3
votes
Easy proof of the uncountability of bijections on natural numbers
Consider the following map $\Phi$ from bijections of the natural numbers to subsets of the natural numbers: to each bijection $\sigma$, associate its fixed point set $S(\sigma) = \{x \in \mathbb{N} \ …
6
votes
Terminology for relation on sets
I agree that this is a natural and interesting property of a family of sets.
From my (arithmetic-geometric) perspective, the most prominent examples of such families are the disks in an ultrametric s …
- 65.4k
12
votes
Accepted
Products of Baire spaces
See:
Cohen, Paul E.
Products of Baire spaces.
Proc. Amer. Math. Soc. 55 (1976), no. 1, 119--124.
$ $
MathSciNet review by Douglas Censer: A topological space is said to be Baire if any countable …
- 65.4k
5
votes
Choice vs. countable choice
I wish to note that in the comment that prompted this question, I was paraphrasing Kaplansky from memory, not quoting him. Since there is now quite a discussion, people may be interested in the exact …
47
votes
Why worry about the axiom of choice?
I would like to point out that a lot of the people who are interested in the Axiom of Choice (AC) are not worried about it in any mathematical way.
In general, I think "Why do people worry about X?" …