All Questions
97 questions with no upvoted or accepted answers
26
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0
answers
1k
views
Where do uncountable models collapse to?
Suppose $T$ is a complete first-order theory (in an finite, or at worst countable, language). Given any model $\mathcal{M}\models T$ of cardinality $\kappa$, we can ask whether $\mathcal{M}$ can be ...
- 21.2k
19
votes
0
answers
905
views
What examples of existence forcing proofs are there?
Forcing proofs tend to be fairly constructive, in the sense that if I claim that there is a forcing that does something, I usually prove this by constructing that forcing.
There are only a handful of ...
- 39.7k
17
votes
0
answers
558
views
Gitik's work on Shelah's weak hypothesis
It seems that Moti Gitik has recently refuted some variants of Shelah's weak hypothesis. For this see the title and abstract of his talk at the Set Theory, Model Theory and Applications conference.
I ...
- 32.1k
17
votes
0
answers
908
views
Souslin trees and weakly compact cardinals
In Souslin trees on the first inaccessible cardinal it is asked if it is consistent that there are no $\kappa-$Souslin trees at the least inaccessible cardinal $\kappa$.
In this question I would like ...
- 32.1k
14
votes
0
answers
404
views
O-minimality and forcing
It is well-known that the structure $(\mathbb{R}, +, \cdot, <, 0, 1)$ is an o-minimal structure and hence the set of integers $\mathbb{Z}$ is not definable in it.
In an ongoing project with Will ...
- 32.1k
14
votes
0
answers
353
views
The failure of GCH al $\aleph_\omega$ by nice forcing
There are several ways to force $GCH$ below $\aleph_\omega$ and $2^{\aleph_\omega}> \aleph_{\omega+1},$ say:
1) The Gitik-Magidor's extender based forcing, see Prikry type Forcings.
2) Woodin's ...
- 32.1k
12
votes
0
answers
506
views
Can a generic $\mathbb{R}$ have a new cardinality?
This question was asked and bountied at MSE, without success.
My main question is whether, starting with a model of determinacy, a "generic $\mathbb{R}$" could be different in cardinality ...
- 21.2k
12
votes
0
answers
538
views
What happens when you iterate Cohen reals?
There are a few classical theorems in set theory:
The finite support iteration of ccc forcing is ccc.
The countable support iteration of proper forcing is proper.
The finite support iteration of ...
- 39.7k
12
votes
0
answers
263
views
Generic $\mathbf{\Sigma}_3^1$-absoluteness for class forcings
In the paper "Generic Absoluteness" by Bagaria and Friedman (http://www.logic.univie.ac.at/~sdf/papers/bagfried.pdf) it is shown that in ZFC generic $\mathbf{\Sigma_3^1}$-absoluteness is false for ...
- 121
12
votes
0
answers
458
views
Adding a saturated ideal
Is it consistent that there is no $\omega_2$-saturated ideal on $\omega_1$, but one is introduced by an $\omega_2$-closed forcing?
Some motivation:
If $\delta$ is a Woodin cardinal, then it remains ...
- 18.6k
11
votes
0
answers
314
views
Can we bound $2^{\aleph_\omega}$ without pcf theory?
One of the famous applications of pcf theory is that if $\aleph_\omega$ is a strong limit cardinal then $2^{\aleph_\omega}<\aleph_{\omega_4}$. I'm curious whether any weaker result with the same ...
- 21.2k
11
votes
0
answers
272
views
Preservation of chain condition under strategically closed forcing
It is well-known that $\kappa$-closed forcing preserves $\kappa$-c.c. posets. The same argument works for $\kappa$-strategically closed forcing. Here is the definition:
A poset $\mathbb P$ is $\...
- 18.6k
11
votes
0
answers
442
views
c.c.c forcing notions and adding minimal generic reals
Is the following statement consistent:
``There is no non-trivial c.c.c forcing notion adding a minimal generic real''?
The question is related to Prikry's question: Is it consistent that any non-...
- 32.1k
10
votes
0
answers
283
views
Martin's Maximum implies stationary/club Chang's conjecture?
Chang's Conjecture (CC) states: for any $f: [\omega_2]^{<\omega} \to \omega_1$, there exists a set $X\subset \omega_2$ of order type $\omega_1$ such that $|f''[X]^{<\omega}|\leq \aleph_0$.
...
- 3,038
10
votes
0
answers
333
views
What kind of objects can code a universe?
Jensen proved that given $V\models\sf ZFC+GCH$, there is a class generic real $r$, such that $V[r]=L[r]$, and no cardinals are collapsed.
We know that this can be modified such that $r$ is minimal, i....
- 39.7k
10
votes
0
answers
169
views
Isomorphisms mod nonstationary
Suppose $G \subseteq \mathrm{Add}(\omega_1)$ is generic over $V$. Let $X_i = \{ \alpha : G(\alpha) = i \}$. Is it true that $P(X_0)/\mathrm{NS} \cong P(X_1)/\mathrm{NS}$?
- 18.6k
10
votes
0
answers
561
views
stationary reflection in $[\kappa]^\omega$
It is well-known that the following reflection principle is consistent relative to a supercompact:
For all $\kappa \geq \omega_2$ and all stationary $S \subseteq [\kappa]^\omega$, there is $X \...
- 18.6k
10
votes
0
answers
438
views
On the Number of Parallel Automorphism Lines
Given a group $G$, one can define the transfinite line of iterative automorphisms of $G$ to be the following chain of the groups where $G_{\alpha+1}=Aut(G_{\alpha})$ for each ordinal $\alpha$ and the ...
10
votes
0
answers
276
views
A variant of strong ideals, is it consistent?
Is it consistent relative to large cardinals that there is a precipitous ideal on $\omega_1$ forcing a generic elementary embedding $j : V \to M \subseteq V[G]$, such that $j(\omega_1) = \omega_n^V$ ...
- 18.6k
10
votes
0
answers
292
views
When does $HOD^{V[G]} \subseteq V$?
Assume that $\mathbb{P}\in HOD$ is non-trivial. It is well-known that if $\mathbb{P}$
satisfies some homogeneity properties, then $HOD^{V[G]} \subseteq V$, where $G$ is $\mathbb{P}$-generic over $V$.
...
- 32.1k
10
votes
0
answers
331
views
preserving saturated ideals
A reliable source made the following claim:
Suppose CH there is an $\omega_2$-saturated ideal on $\omega_1$. Then this is preserved by $\mathrm{Add}(\omega_1,\omega_2)$.
Question 1: How do you ...
- 18.6k
10
votes
0
answers
514
views
Existence of a regular subposet which collapses everything except the top cardinal
Suppose $\delta$ is an inaccessible cardinal, and $\mathbb{P}$ is the Levy Collapse $\text{Col}(\kappa, \delta)$ which adds a surjection from $\kappa \to \delta$ (for some regular $\kappa < \delta$)...
- 2,231
9
votes
0
answers
317
views
Non-closed Neeman forcing
This question is something of a follow-up to this one:
Iterating Neeman's forcing
It regards the work of Itay Neeman, MR3201836.
Neeman formulates his two-type models forcing seemingly in greater ...
- 18.6k
9
votes
0
answers
250
views
Distributivity of certain infinite products
Suppose we have a sequence of posets $\{\mathbb P_n : n\in\omega\}$ such that for each $n$, $\mathbb P_{n+1}$ is $|\mathbb P_n|^+$-distributive. Is $\prod_{n>0} \mathbb P_n$ necessarily $|\mathbb ...
- 18.6k
9
votes
0
answers
267
views
Analogue of strong stationary reflection from MM
Foreman-Magidor-Shelah proved that if Martin’s Maximum holds, then for every regular $\kappa>\omega_1$, every stationary $S \subseteq S^\kappa_\omega$ contains a club of ordertype $\omega_1$. This ...
- 18.6k
9
votes
0
answers
440
views
A new maximality principle and its consequences
Let us consider the following maximality principle:
$(MP_*):$ For all uncountable regular cardinals $\kappa, 2^{<\kappa}=\kappa^{+}$
and all trees of height and size $\kappa$ are specialized.
It ...
- 32.1k
9
votes
0
answers
240
views
Homogeneity of a variant of Prikry forcing
Prikry forcing is easily seen to be cone homogeneous (for any $p, q \in \mathbb{P}$, there are $p' \leq p, q' \leq q$ and an isomorphism $\Phi: \mathbb{P}/p' \simeq \mathbb{P}/q'$); in particular for ...
- 32.1k
9
votes
0
answers
299
views
On an unpublished result of Magidor
In 1970th, Magidor proved the following important results:
(1) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$
is strong limit and $2^{\aleph_\omega}=\aleph_{\...
- 32.1k
9
votes
0
answers
271
views
Which forcing types preserve the axiom of determinacy?
Do we have some rudimentary understanding of some properties that a forcing can have in order to guarantee that it doesn't violate the axiom of determinacy?
To be more specific, in Which forcings ...
- 39.7k
8
votes
0
answers
206
views
ladder system uniformization at successors of singulars
Shelah proved (paper 667) that if GCH holds and $\lambda$ is singular, then for every stationary $S \subseteq \{ \alpha < \lambda^+ : \text{cf}(\alpha) = \text{cf}(\lambda) \}$, there is a ladder ...
- 18.6k
8
votes
0
answers
241
views
Topological applications of $\mathfrak{p}=\mathfrak{t}$
I'm working on the Malliaris-Shelah's recent result of $\mathfrak{p}=\mathfrak{t}$, but I'm more interested in what possible topological applications can derivate from this equality.
Searching in ...
- 181
8
votes
0
answers
191
views
Specializing fat trees
The discussion is about trees of height $\omega_1$ that are not necessarily thin, namely, no cardinality constraints on the size of each level. A classcial theorem of Baumgartner states that it is ...
- 1,006
8
votes
0
answers
226
views
When can we force two frames to be homeomorphic?
Recall that if $M,N$ are two structures of the same type, then
$M$ is $\mathcal{L}_{\infty,\omega}$ elementarily equivalent to $N$ precisely when $M$ and $N$ are isomorphic in some forcing extension. ...
8
votes
0
answers
389
views
PCF conjecture and fixed points of the $\aleph$-function
Recently Moti Gitik refuted Shelah's PCF conjecture, by producing a countable set $a$ of regular cardinals with $|\operatorname{pcf}(a)| \geq \aleph_1.$ See his papers
Short extenders forcings I and ...
- 32.1k
8
votes
0
answers
275
views
Proving regularity properties from forcing axioms
It's well known that PFA implies projective determinacy. It's also well known that PD implies that all projective sets are Lebesgue measurable, have the Baire property, etc.
Is there a direct proof ...
- 391
8
votes
0
answers
394
views
Silver's unpublished work on reverse Easton iteration
Silver was the first person who used the method of reverse Easton iterations in connection with large cardinals, and used it to force the failure of $GCH$ at some measurable cardinal.
At most papers ...
- 32.1k
7
votes
0
answers
260
views
A version of determinacy for all sets
Under ZF + AD, some games are undetermined because of lack of choice. In fact, the axiom of choice is equivalent to determinacy of games of length 2. However, we can ask whether the lack of choice ...
- 7,545
7
votes
0
answers
234
views
Is this equivalent to (some version of) Hechler forcing?
Let $\omega^{<\omega}$ be the set of finite strings of naturals, and let $\omega^{<\omega}_{\not=\emptyset}$ be the set of nonempty finite strings of naturals. Consider the following forcing ...
- 21.2k
7
votes
0
answers
313
views
An uncountable structure with unusual "relatively-computable shadow"
Below, all structures are infinite and in a finite language. Given a structure $\mathcal{A}$ with domain $\omega$, we conflate $\mathcal{A}$ with some reasonable encoding of its atomic diagram for ...
- 21.2k
7
votes
0
answers
468
views
"Relative plausibility" of some infinitary theories
We work in $\mathsf{ZFC+V=L}$.
Define a plausible theory to be a theory $T\subseteq\mathcal{L}_{\omega_1,\omega}$ in an $\omega_1$-finite language which is $\omega_1$-c.e. and $\omega_1$-finitely ...
- 21.2k
7
votes
0
answers
197
views
Query about iterated collapse forcing
I have heard it said that it is possible to use Woodin's iterated collapse forcing to prove that the theory $\textsf{ZF}$+"there exists a non-trivial elementary embedding $j:V_{\lambda+2} \rightarrow ...
- 2,125
7
votes
0
answers
202
views
Does Solovay's $\Sigma$-construction preserve "niceness"?
Suppose I have two forcing notions $\mathbb{P}$ and $\mathbb{Q}$, and a $\mathbb{Q}$-name $\nu$ for a real. Let $G$ be $\mathbb{P}$-generic, and suppose $r$ is a real in $M[G]$ such that for some $H$ ...
- 21.2k
7
votes
0
answers
173
views
Sacks minimality without choice
The usual argument for the minimality result for Sacks forcing uses choice.
Theorem (Sacks): Let $s \subseteq \mathbb S_\kappa$ be generic for the forcing to add a Sacks subset to $\kappa$, where $\...
- 2,167
7
votes
0
answers
262
views
$V$ as a $HOD$ of its class generic extension
By an old result of Roguski, The theory of the class $HOD$, any model $V$ of $ZFC$ has a class generic extension $V[G]$ such that $HOD$ of $V[G]$ equals $V$.
This result is also stated and generalized ...
- 32.1k
7
votes
0
answers
378
views
Sets of reals amenable to each L[x]
If $A$ is a set of reals such that $A \cap L[x] \in L[x]$ for each real $x$, is there a real $z$ such that $A \cap L[x] \in OD_z^{L[x]}$ for a cone of $x$?
This can be proved under the Axiom of ...
- 5,471
6
votes
0
answers
179
views
$Δ^1_3$ reals in transitive models
Every real number is $Δ^1_2$ in some $ω$-model of ZFC\P. I am looking for analogous statements for $Δ^1_3$ definability and transitive models, where the situation is more subtle.
What is the ...
- 7,545
6
votes
0
answers
144
views
Alternative proofs of the countable chain condition in forcing
Advance warning: This question is more about history and pedagogy than "hard" mathematics.
I am studying Cohen forcing with the forcing poset $(\operatorname{Fin}(E,2),\supseteq,0)$, and I ...
6
votes
0
answers
125
views
From HODs to corresponding models of AD
If $M$ is HOD of a model $N$ of $\text{AD}^+ + V=L(P(ℝ))$, what kind of forcing construction in $M$ gives back such an $N$?
HODs for $\text{AD}^+ + V=L(P(ℝ))$ are conjectured (and under anti-large ...
- 7,545
6
votes
0
answers
243
views
Tree property using side conditions
The following problems were asked during the high and low forcing workshop:
Question 1. Can one force tree property at $\kappa^{++}$ for $\kappa$ singular using side conditions?
Question 2. Can one ...
- 32.1k
6
votes
0
answers
242
views
Adding minimal subsets to $\aleph_\omega$
Given a cardinal $\kappa,$ recall that $X \subset \kappa$ is called fresh (over $V$), if $X \notin V,$ but $X \cap \alpha \in V$ for all
$\alpha < \kappa.$
Question. Is it consistent that there ...
- 32.1k