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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.
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 …
- 4,713
34
votes
3
answers
2k
views
How much choice is needed to show that formally real fields can be ordered?
Background: a field is formally real if -1 is not a sum of squares of elements in that field. An ordering on a field is a linear ordering which is (in exactly the sense that you would guess if you ha …
- 1,084
29
votes
Accepted
Were Bourbaki committed to set-theoretical reductionism?
First, most mathematicians don't really care whether all sets are "pure" -- i.e., only contain sets as elements -- or not. The theoretical justification for this is that, assuming the Axiom of Choice …
- 3,738
9
votes
Does $\operatorname{Con}\sf(ZF)$ imply $\operatorname{Con}\sf(ZF + \operatorname{Aut}{\bf C ...
First things first: assuming AC, it is indeed true that for any algebraically closed field $F$, $\# \operatorname{Aut}(F) = 2^{\# F}$. The main idea for this is that we can choose a transcendence bas …
- 12.9k
5
votes
2
answers
1k
views
Improvements of the Baire Category Theorem under (not CH)?
The Baire category theorem implies that a nonempty complete metric space without isolated points must be uncountable. In many situations I have encountered, the "natural examples" of complete metric …
CommunityBot
- 1
22
votes
Find a "natural" group that contains the quotient of the infinite symmetric group by the alt...
This is not an answer per se [Edit: OK, maybe it is! I was a little fuzzy on exactly what was being asked for when I wrote this, and in the past Martin has expressed unhappiness with responses which h …
- 4,713
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 …
- 6,237
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 …
- 6,237
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
32
votes
4
answers
2k
views
Do there exist non-PIDs in which every countably generated ideal is principal?
The title pretty much says it all: suppose $R$ is a commutative integral domain such that every countably generated ideal is principal. Must $R$ be a principal ideal domain?
More generally: for whic …
CommunityBot
- 1
5
votes
Hausdorff dimension vs. cardinality
The part of the question about the continuum hypothesis (CH) seems confused: without assuming (CH) (but assuming axiom of choice so that cardinals work as they should), $\aleph_1$ is by definition the …
- 4,713
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:
…
CommunityBot
- 1
10
votes
How far is Lindelöf from compactness?
I've never heard of that result (which is not to say that I doubt its truth -- I have no opinion either way), but it reminds me of the following
Theorem (N. Noble): If each power of a $T_1$-space is …
- 30.3k
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 …
- 65.4k
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