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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.
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
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?
- 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
10
votes
2
answers
1k
views
$\aleph$ looks like $\mathbb N$?
We all know the notation $\aleph_\lambda$ for the $\lambda$th (or, I guess, $\lambda+1$st) infinite cardinal number; in particular $\aleph_0$ is the cardinality of the the set of natural numbers $\mat …
- 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
10
votes
4
answers
2k
views
Axiom of Infinity needed in Cantor-Bernstein?
Can one prove the Cantor-Bernstein (or Schröder-Bernstein) theorem without using the Axiom of Infinity?
- 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
9
votes
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 …
- 24.8k
8
votes
Is it consistent with ZFC (or ZF) that every definable family of sets has at least one defin...
Yes,
fix a definable relation $\le_L$ that well-orders all of $L$.
If $V=L$ then every definable nonempty set $A$ has a definable member $a$, namely:
$a :=$ the $\le_L$-least element of $A$.
- 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 …
- 24.8k
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 …
- 24.8k
7
votes
Accepted
Is there a name for this equivalence relation?
$\mathscr F$-indistinguishability.
In analogy with Topological indistinguishability.
- 24.8k
7
votes
Accepted
The consistency of ZFC + CH gives the ability to travel to a universe which models ZFC + \ne...
The empty set $\emptyset$ here is the forcing condition that has the least amount of information about the generic object being constructed.
Since it has the least information, you might think it sh …
- 24.8k
6
votes
Accepted
Translates of meager sets
No, there is no such set.
The situation for meager sets is dual to that described by Pietro Majer in a comment on Translates of null sets,
"I was vaguely thinking to Hausdorff measures w.r.to gaug …
- 24.8k