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Results tagged with set-theory
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user 4600
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.
9
votes
1
answer
396
views
VC dimension of Borel sets [duplicate]
Can there be an uncountable set $S\subseteq\mathbb R$ such that for each subset $D\subseteq S$, there is a Borel set $U$ with $D=S\cap U$?
I'm asking merely out of curiosity, but I'll mention that thi …
- 18.6k
8
votes
1
answer
479
views
VC dimension of standard topology on the reals
Can there be an uncountable set $S\subseteq\mathbb R$ such that for each subset $D\subseteq S$, there is an open set $U$ with $D=S\cap U$?
I'm asking merely out of curiosity, but I'll mention that thi …
- 6,101
5
votes
Accepted
Partitioning an infinite cardinal $\kappa$ into pairwise neighboring subsets
Yes. List the pairs $(\alpha,\beta)$ with $\alpha<\beta<\kappa$ as $(\alpha_\lambda,\beta_\lambda), \lambda<\kappa$.
Then construct the sets $B_\alpha\in\mathcal B, \alpha<\kappa$ as follows:
At stage …
- 24.8k
2
votes
Source on smooth equivalence relations under continuous reducibility?
Seems like this structure must be pretty complicated. For example, consider Brownian motion $\{W_t\}_{t\ge 0}$ with the equivalence relations
$$t\sim_\omega s\iff W_t(\omega)=W_s(\omega).$$
Here $\ome …
- 24.8k
7
votes
Accepted
Is there a name for this equivalence relation?
$\mathscr F$-indistinguishability.
In analogy with Topological indistinguishability.
- 24.8k
1
vote
A stronger notion of injectivity
If $A$ and $B$ are countable and equipped with probability measures $\mu_A$, $\mu_B$ that give each element positive probability then any measure-preserving map is a bijection. (I guess per @MonroeEsk …
- 24.8k
10
votes
Accepted
What non-standard model of arithmetic does Hofstadter reference in GEB?
My first guess is that the triples come from the fact that nonstandard countable models of PA look like
$$\mathbb N + \mathbb Z\times\mathbb Q$$
and elements of $\mathbb Q$ can be represented by pairs …
- 24.8k
8
votes
1
answer
252
views
Automorphisms of power set lattice mod finite
Let $N$ be a countably infinite set and let $\mathcal P$ denote power set.
I get that the automorphisms of $(\mathcal P(N),\subseteq)$ are all induced by permutations of $N$.
But what can be said abo …
- 63.9k
18
votes
4
answers
2k
views
Complete Boolean algebra not isomorphic to a $\sigma$-algebra
Does there exist a complete Boolean algebra that is not isomorphic to any $\sigma$-algebra? If so, what is an easy or canonical example or construction?
- 4,713
1
vote
Choice sets in "thick" sets of sets
Let $\kappa$ be any infinite cardinal.
Let $\cal S=\{\kappa\setminus\{x\}:0<x<\kappa\}$ and $C=\{0\}$.
Then all the given conditions are satisfied.
- 24.8k
9
votes
Accepted
Choice sets from above and below
Let $\cal S=\{\{1,2\},\{2,3\},\{3,1\}\}$.
Then $\cal S$ has no choice set, whatsoever.
So there is no asymmetry -- not every shy set is contained in a choice set, and not every gregarious set contai …
- 24.8k
4
votes
Intuition for the infinite cardinals p and t (now that p = t)?
A couple of simple examples; they don't have cardinality $\mathfrak p$ or $\mathfrak t$ but depending on your knowledge they may or may not help.
They are families of sets of natural numbers that I'l …
- 35.5k
1
vote
Accepted
Models of ZFC and the Borel hierarchy
@FrançoisG.Dorais commented:
By reading the axioms (i.e. every axiom is satisfied, as a formalized statement) it's a $\Pi^0_{\omega+1}$ set (since satisfying a first-order sentence is $\Pi^0_n$ fo …
- 24.8k
19
votes
1
answer
665
views
Models of ZFC and the Borel hierarchy
The collection of
binary relations $R$ on the natural numbers such that $(\mathbb{N},R) \models ZFC$
forms a Borel set, neither closed nor open -- assuming Con(ZFC).
Can you show it's not $F_\sigm …
- 24.8k
11
votes
Accepted
Usual technical term for replacing a set by the set of singletons of its members?
$A'$ is the discrete partition of $A$.
That is, we think of it as a partition of $A$ induced by the finest equivalence relation, the identity relation.
- 24.8k