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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.
8
votes
Implicit uses of Countable or Dependent Choice
Rudin's Principles of Mathematical Analysis (and most every book on Mathematical Analysis) in the proof that $\lim_{x\to p}f(x)=q$ is equivalent to "$\lim_{n\to\infty}f(p_n)=q$ for every sequence $\la …
- 35.5k
3
votes
Questions on continuum hypothesis
As the comment says: the roles of Gödel and Cohen are reversed.
One cannot disprove CH using ZF(C), so I take it you simply assume the negation of CH and ask about fields between $\mathbb{Q}$ and $\m …
- 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
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
1
vote
Modification of Lemma 0 in Hajnal's paper "Embedding finite graphs into graphs colored with ...
A construction that uses the equality $\kappa^\lambda=\kappa$ directly runs as follows. By that equality the set $H$ has cardinality $\kappa$, so we can find a surjection $f:\kappa\to H$ such that for …
- 11.4k
3
votes
Accepted
Sieve for an infinite array of sets, resulting in an array of the same size of pairwise disj...
Consider the reverse lexicographic order $\prec$ on $\lambda\times\lambda^+$ ($(\alpha,\beta)\prec(\gamma,\delta)$ iff $\beta<\delta$ or $\beta=\delta$ and $\alpha<\gamma$); its order type is equal to …
- 11.4k
4
votes
Accepted
Copy of $P(\omega)/\mathrm{fin}$ on $\omega_1$
To expand my comment into an answer: take, for each $n\in\omega$, a uniform ultrafilter $u_n$ on $\omega_1$ that contains the set $\{\lambda+n:\lambda$ is a limit or $0\}$.
The set $U=\{u_n:n\in\omega …
- 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
4
votes
Accepted
Simplified method of building an Aronszajn tree
Your argument is basically Kurepa's proof from his thesis Ensembles ordonnées et ramifiés, see page 96 (a footnote has Aronszajn's construction).
As noted in the comments you need to show that what yo …
- 11.4k
5
votes
Accepted
Is there a metric separable space with the following properties...?
Let $X$ be a Bernstein subset of $\mathbb{R}$, so $X$ and its complement intersect every uncountable closed set in $\mathbb{R}$.
Let $f:X\to\mathbb{R}$ be continuous and assume $f[X]$ is uncountable. …
- 11.4k
7
votes
Accepted
Can any poset of cardinality $\leq 2^{\aleph_0}$ be embedded in ${\cal P}(\omega)/(\text{fin...
Here is an attempt at a 'definitive summary'.
To begin with positive results: $\mathsf{CH}$ implies a “yes” answer to
this question. The fastest way to see this is to first embed a given partial
order …
- 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
3
votes
Posets such that the collection of principal down-sets does not have property ${\bf B}$
The axiom of choice implies that for every partial order $P$ the
hypergraph $H_P$ has property $B$.
Let $(P,\le)$ be a partial order.
We first claim the following: for every $p\in P$ there is a $q\le …
- 11.4k
4
votes
Accepted
Posets such that the collection of principal down-sets does not have property ${\bf B}$
Let $M$ be the ordered Mostowski model (T. Jech, The Axiom of Choice, Section 4.5). Its set of atoms, $A$, has a linear order $\prec$ that makes it isomorphic to the rationals. Let $S\in M$ be a subse …
- 11.4k
6
votes
Who was the first to propose a formal definition of infinity?
In the preface of the second edition of Was sind und was sollen die Zahlen Dedekind mentions another definition of `finite': a set $S$ is called finite if there is a map $\varphi$ from $S$ to itself s …
- 11.4k