# Tagged Questions

Forcing is a method first used to prove the continuum hypothesis is independent of the classical axioms of set theory

589 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 ...
476 views

### Can you have many independent reals?

Working in $\sf ZFC$, is it provable, or at least consistent (say, over $L$), that you have $\aleph_1$ forcings, $\Bbb P_\alpha$ such that: $\Bbb P_\alpha$ is c.c.c. $\Bbb P_\alpha$ adds a real ...
162 views

### Idea behind the proof of consistency of club filter of $\omega_1$ is ultrafilter + ZF + DC

I've been trying to understand Radin Forcing and some of its applications, one of which is the use of it to prove the consistency of ''Club filter of $\omega_1$ is an ultrafilter + ZF + DC''. However, ...
253 views

414 views

### tree properties on $\omega_1$ and $\omega_2$

Are the following mutually consistent (relative to large cardinals)? (1) There are no $\omega_2$-Aronszajn trees. (2) There is an $\omega_1$-Kurepa tree. In the models I know of the tree property ...
179 views

### tree property at $\aleph_2$ and $\aleph_4$

It is claimed that if there are two weakly compact cardinals, then there is a generic extension in which $\aleph_2$ and $\aleph_4$ have the tree property. Assuming one knows Mitchell's forcing, what ...
241 views

### The least admissible above a dominating real

Let $\mathbb{P}$ be the usual forcing which adds a dominating real: conditions in $\mathbb{P}$ are pairs $(p, f)$ with $p:\omega\rightarrow\omega$ finite partial and $f:\omega\rightarrow\omega$ total, ...
135 views

283 views

### On generic forcing conditions

Let $P$ be a forcing poset, and $Q \in V^P$ a forcing poset in $V^P$. Let $M \prec H(\lambda)$ ($\lambda$ sufficiently large) countable with $P,Q \in M$. What I want to know is if then the following ...
183 views

### Large Cardinal Principles that Imply $\Sigma_3^1$-Generic Absoluteness

It is known that (light-face) $\Sigma_3^1$ generic absoluteness is consistent with $\mathsf{ZFC}$: Friedman and Bagaria showed that it holds in the $\text{Coll}(\omega, < \kappa)$ extension of $V$ ...
129 views

189 views

### Can we always add sets without collapsing cardinals or adding [very] bounded sets?

Given a model of $\sf ZFC$, and an infinite ordinal $\alpha$. Can we prove that there is always a cardinal $\kappa$, and a forcing $\Bbb P$, such that: $\Bbb P$ does not add sets of rank $\leq\alpha$...
432 views

### Collapsing the cardinals between two singular cardinals

Question 1: Is it consistent that there is a forcing notion collapsing $\aleph_{\omega\cdot 2}$ to $\aleph_\omega$ without collapsing $\aleph_\omega$ or $\aleph_{\omega\cdot 2 + 1}$? If the answer ...
266 views

### Class forcings and elementary embeddings

In the Hamkins-Kirmayer-Perlmutter paper "Generalizations Of The Kunen Inconsistency", they prove the following theorem: "Theorem 7: In any set forcing extension $V[G]$, there is no nontrivial ...
340 views

### A question related to Woodin's $HOD$ conjecture

Recall that an $\kappa$ is $\omega$-strongly measurable in $\text{HOD}$ if there exists $\lambda < \kappa$ such that $(2^{\lambda})^{\text{HOD}} < \kappa$ and such that there is no partition of ...
259 views

### Posets preserving stationary subsets of $\omega_1$ and no new $\text{cof}(\omega)$ ordinals, but without countable covering property

What are some examples of posets $\mathbb{P}$ which have the following properties? It's OK if the definition uses large cardinals or some other hypothesis. $\mathbb{P}$ preserves stationary subsets ...
228 views

### References for Forcing with Side Conditions

I'm looking for some good references about Forcing with Side Conditions, including expository papers that explain the main ideas with some details in order to give me a fairly clear insight of those ...
271 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 ...
152 views

203 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 ...
409 views

### Just a little absoluteness might be cheaper?

Absoluteness is a wonderful thing, but expensive consistency-strength wise. My question is, when can we get large amounts of absoluteness in specific situations for much cheaper? Specifically, fix a ...
289 views

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 ...
316 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 ...
339 views

### Details for Woodin's forcing argument for a saturated ideal from the Levy collapse

Theorem 2.65 in Woodin's book shows that a saturated ideal on $\omega_1$ exists after Levy-collapsing a Woodin cardinal $\delta$ to $\omega_2$. I am confused about the part of the argument where he ...
200 views

254 views

### $\text{cov}(\mathcal{M})$ vs. $\mathfrak{b}$ vs. $\mathfrak{s}$

Let me first recall some pretty standard notations: $\text{cov}(\mathcal{M})$ is the covering number of the ideal $\mathcal{M}$ of all meager subsets of $\mathbb{R}$; $\mathfrak{b}$ is the bounding ...
211 views

### Universally Baire Tree Representation of Projective Sets

In Feng, Magidor, and Woodin "Universally Baire Sets of Reals", they show that if $A$ is a $\mathbf{\Pi}_2^1$ set and $U$ and $V$ are any pair of trees witnessing the universal baireness of $A$, then ...
136 views

248 views

### “Namba forcing adds reals” independent of $ZFC + \neg CH$?
I know that, in the presence of $CH$, Namba forcing does not add reals. But when $CH$ fails, is it consistent that it still does not add reals?
### Scott-Solovay unpublished paper on Boolean valued models of set theory''
I have read some papers from 1970$^{th}$, and in some of them, the paper of Scott and Solovay on Boolean valued models of set theory'' is given as a main reference, with many references to the ...