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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.
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 …
1
vote
Is the collection of isomorphism classes of groups a proper class?
All the other answers are more than satisfactory. I have some lecture notes on basic set theory which also answer these questions, so I might as well post links to them:
To see that the cardinals do …
- 65.4k
18
votes
Set theory in practice
If you are being, say, at least semiformal in your approach to set theory, whether or not objects which are not sets exist depends upon the particular brand of set theory you choose. The most common …
- 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
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
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 …
- 65.4k
5
votes
Cardinality of Equivalence Classes of Cauchy Sequences
The answer is that the cardinality is equal to that of the continuum: i.e., that of the real numbers, which is (independent of the continuum hypothesis!) also equal to the the cardinality of the power …
- 65.4k
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
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
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
7
votes
Accepted
Ultrafilters containing a principal filter
If a filter $X$ contains any set $A$, then it contains the principal filter of $A$. Thus you are really asking: for which subsets $A$ of a set $X$ can a free ultrafilter contain $A$?
It is a standard …
- 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 …
- 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} \ …
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
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 …
- 65.4k