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Results tagged with set-theory
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user 5903
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.
15
votes
Is "compact implies sequentially compact" consistent with ZF?
The sequential compactness of $[0,1]^{\omega_1}$ is undecidable in ZFC: as noted above $[0,1]^{[0,1]}$ is not, so under CH $[0,1]^{\omega_1}$is not sequentially compact; on the other hand $\mathrm{MA} …
- 11.4k
14
votes
Accepted
Is "All hyperreal fields $C(\mathbb{R})/M$ are isomorphic" independent of ZFC+$\lnot$CH?
In On ultra powers of Boolean algebras (Topology Proceedings 9 (1984) 269-291) Alan Dow proved (Corollary 2.3) that $\neg\mathsf{CH}$ implies there are two fields of the form $C(\mathbb{N})/M$ that ar …
- 11.4k
12
votes
Does Urysohn's Lemma imply Dependent Choice?
In Versions of normality and some weak forms of the axiom of choice Paul Howard et al exhibit a model of MC (Multiple Choice) and not-DC, see page 381.
In that model Urysohn's Lemma (NU) holds, so it …
- 11.4k
12
votes
Independent families of subsets of $\mathbb N$ of size continuum
Let $2^{<\omega}$ be the binary tree and assign to each branch $x$ the family $F_x$ of finite sets that intersect it. If $x_1$, $x_2$, $\ldots$ $x_k$ is a finite set of (distinct) branches then there …
- 11.4k
11
votes
Which topological spaces admit a nonstandard metric?
These are $\omega_\mu$-metrizable spaces, where $\omega_\mu$ is the cofinality of ${}^*\mathbb{R}$. Take a decreasing sequence $\langle x_\alpha:\alpha<\omega_\mu\rangle$ of positive elements that con …
- 11.4k
11
votes
4
answers
2k
views
Earliest diagonal proof of the uncountability of the reals.
I cited the diagonal proof of the uncountability of the reals as an example of a `common false belief' in mathematics, not because there is anything wrong with the proof but because it is commonly bel …
- 11.4k
10
votes
Accepted
What is the extent of a $\Sigma$-product of a (uncountable) power of a (countable) discrete ...
To answer the explicit question: the extent of every $\Sigma$-product of $\mathbb{N}$ is countable. H. H. Corson showed in Normality in subsets of product spaces, Amer. J. Math 81(1959), 785–796 that …
- 11.4k
9
votes
"Mächtigkeit" versus "Kardinalität"?
Here is Cantor's Beiträge zur Begründung der transfiniten Mengenlehre (Erster Artikel). Read the bottom four lines on the first page: ",Mächtigkeit' oder ,Cardinalzahl' von $M$ nennen wir $\ldots$". T …
- 11.4k
9
votes
Products of Baire spaces
There are even two normed Baire spaces whose product is not Baire. See J. van Mill and R Pol, The Baire category theorem in products of linear spaces and topological groupsn, Topology Appl. 22 (1986) …
- 11.4k
9
votes
Accepted
Is the set of $\kappa$-complete ultrafilters closed in $\beta X$?
If $\kappa=\aleph_0$ then yes: every ultrafilter is $\aleph_0$-complete.
If $\kappa>\aleph_0$ then no, if $\lambda X$ is nonempty. Split $X$ into countably many sets $\{X_n:n\in\mathbb{N}\}$, of the s …
- 11.4k
9
votes
Accepted
Is it true that $\mathit{MA}(\omega_1)$ iff $\omega_1<\mathfrak{p}$?
Q1: No, see Between Martin's Axiom and Souslin's Hypothesis by Kunen and Tall. Note: Bell proved in The combinatorial principle $P(\mathfrak{c})$ that $\mathfrak{p}>\aleph_1$ is equivalent to $\mathsf …
- 11.4k
9
votes
Accepted
A strictly descending chain of subalgebras of $P(\omega)/_{\mathrm{fin}}$
There is a family $\{K_X:X\subseteq\mathfrak{c}\}$ of separable compact
zero-dimensional spaces such that there is a continuous surjection of $K_X$
onto $K_Y$ if and only if $X\subseteq Y$.
These spac …
- 11.4k
9
votes
Construction of nonmeasurable sets
Here is an earlier effort of Sierpiński: Sur une propriété de la décomposition de M. Vitali, Mathematica 3, 30-32 (1930).
He took "Vitali's Decomposition", that is, the family of cosets of $\mathbb{Q} …
- 11.4k
8
votes
Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ and distinct pairwise distances?
You can actually make $S$ a (copy of the) Cantor set.
First a claim: if $[a_1,b_1]$, $[a_2,b_2]$, ..., $[a_n,b_n]$ is a finite set of intervals, ordered left-to-right ($b_1<a_2$, $b_2<a_3$, etc) then …
- 11.4k
8
votes
Accepted
Can totally inhomogeneous sets of reals coexist with determinacy?
In Rigid Borel sets and better quasiorder theory (Logic and combinatorics, Proc. AMS-IMS-SIAM Conf., Arcata/Calif. 1985, Contemp. Math. 65, 199-222 (1987), zbMath review here) Fons van Engelen, Arnold …
- 11.4k