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Results tagged with set-theory
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user 11145
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.
6
votes
Accepted
Consistency of the nonrigidity of $P(\omega_1)/NS$
It is consistent relative to large cardinals that $P(\omega_1)/NS \cong \mathcal B(\mathrm{Col}(\omega,\omega_1))$ and many other homogeneous algebras.
It almost looks like the answer is yes in $L$, …
- 18.7k
3
votes
Accepted
A conjecture about certain relations
The answer is no. Let $\mathscr{U}$ be an ultrafilter on $\mathbb{N}$. Let $A,B \in \mathscr{U}$ be such that $A \cap B$ is a proper subset of both $A$ and $B$. Let $\mathscr{F}$ be the collection …
- 18.7k
11
votes
Accepted
Cardinality of an ultraproduct
It depends on the function $f : i \mapsto |A_i|$. The ultraproduct $\prod A_i/ U$ has the same cardinality as $\prod f(i)/U$. $f$ represents an ordinal $\alpha <j(\kappa)$ in the ultrapower $M$ of t …
- 18.7k
4
votes
$\kappa$-dense ideals on successor $\kappa$
This is answered in chapter 2 of my thesis. As far as I know, this is essentially the only method for obtaining such ideals. I am very interested in finding alternative constructions. Please contac …
- 18.7k
6
votes
Formalizing Elementary Embeddings and Substructures in ZFC
(1) One approach to large cardinal embeddings is to view literal talk about them as "from the outside" about models of set theory, while knowing that particular properties of particular embeddings can …
- 18.7k
2
votes
1
answer
909
views
saturated ideals
Is it possible to have a saturated ideal on a successor cardinal which does not extend the nonstationary ideal? (i.e. some nonstationary set is positive for this ideal)
- 18.7k
10
votes
1
answer
250
views
$\kappa$-dense ideals on successor $\kappa$
Woodin gave a consistency proof of a normal $\omega_1$-dense ideal on $\omega_1$ from an almost-huge cardinal. He never published this argument, but it is written up by Foreman in the Handbook of Set …
- 18.7k
14
votes
1
answer
1k
views
Coherent trees: Is this result of Todorcevic correct?
A family of functions $F$ is coherent when for every $f,g \in F$, $\{ x \in dom(f) \cap dom(g) : f(x) \not= g(x) \}$ is finite. A tree on $\omega_1$ is coherent if it is a coherent collection of func …
- 18.7k
9
votes
Accepted
cofinality of $(P(\kappa)/NS,\subseteq)$
I'm going to assume that when you write $\subseteq$, you really mean $\subseteq_{NS}$. We say $A \subseteq_{NS} B$ when $A$ is contained in $B$ except for a nonstationary set, i.e. $A \setminus B \in …
- 18.7k
9
votes
What is the best way to construct an Aronszajn Tree?
Perhaps the easiest argument is given here, in Lemmas 1.1 and 1.2. The argument for $\kappa = \omega$ is due to Koszmider and I give a generalization.
Like the classical construction, we get a syste …
- 18.7k
13
votes
1
answer
793
views
elementary embeddings
Fact 1: If $M$ and $N$ are transitive models of ZF with the same ordinals, and $M \prec N$, then $M = N$.
Fact 2: If $M$ and $N$ are transitive models of ZFC with the same ordinals, and $j: M \to N$ …
- 18.7k
6
votes
2
answers
294
views
larger coherent family of functions
One way to construct an Aronszajn tree is to build a sequence of functions $\langle e_\alpha : \alpha < \omega_1 \rangle$ such that $e_\alpha$ is an injection from $\alpha$ to $\omega$, and for any $( …
- 18.7k
6
votes
2
answers
302
views
preservation of $\aleph_2$-c.c. in CS iterations
It seems that considerable care is taken in the literature to ensure that a countable support (CS) iteration of proper forcings preserves $\aleph_2$. Can you give an example, assuming CH, of a CS ite …
- 18.7k
7
votes
1
answer
399
views
continuity points of elementary embeddings from $0^\sharp$
Suppose $0^\sharp$ exists, and let $\langle \alpha_i : i \in \text{Ord}\rangle$ be the Silver indiscernibles for $L$. Let $j : L \to L$ be the embedding generated by mapping $\alpha_n$ to $\alpha_{n+ …
- 18.7k
7
votes
For which cardinal numbers $\kappa$ is it consistent with ZFC that $\kappa^{\mathrm{cf}(\kap...
Start with a model of GCH. Let $\kappa$ be singular. Add $\kappa^{++}$ Cohen subsets of $\mathrm{cf}(\kappa)^+$-- the forcing is $\mathrm{cf}(\kappa)^+$-closed and $\mathrm{cf}(\kappa)^{++}$-c.c. T …
- 18.7k