# Tagged Questions

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

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### 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. ...
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### Forcing PFA with ccc forcing

Is it consistent (from suitable large cardinals) that there is a ccc poset which forces PFA? This seems quite implausible to me. If we could force PFA via ccc forcing, the ground model would have to ...
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### Why do we need a transitive model in forcing arguments?

One major approach to the theory of forcing is to assume that ZFC has a countable transitive model $M \in V$ (where $V$ is the "real" universe). In this approach, one takes a poset $\mathbb{P} \in M$, ...
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### What ccc forcings add a Suslin tree?

In a comment to Miha's question in Forcing PFA with ccc forcing, I suggested that if such situation is even possible, it might be achieved by screwing with PFA by some ccc forcing (e.g. adding a Cohen ...
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### Dropping “generic” from the definition of forcing

Back when I was first learning about forcing and trying to understand the need to consider generic filters, I came up with the following question. Suppose we have a countable transitive model $M$. ...
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### Why is Random forcing with $\mathbb{R}$, $2^\omega$, $\omega^\omega$ all the same?

By random forcing, I mean the partial order of Borel sets of the given space, modulo Lebesgue null sets, ordered by inclusion. I can not find a source proving that all these partial orders are forcing ...
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### Radin forcing and large cardinals

Assume $\kappa$ is a $(\kappa+2)$-strong cardinal and let $j: V \to M \simeq Ult(V, E) \supseteq V_{\kappa+2}$ witness this where $E$ is a $(\kappa, \kappa^{++})$-extender. Also let $u$ be the measure ...
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### Preservation of Baumgartner's I-ultrafilters under various forcings

For $I\subset \mathcal{P}(2^\omega)$, an ultrafilter $U$ on $\omega$ is said to be an I-ultrafilter if for all $f:\omega \to 2^\omega$, there exists $A\in U$ such that $f''[A]\in I$ [Baumgartner]. In ...
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### A kind of saturation property related to forcing notions

Forcing is typically done over well-founded models. There are lots of good reasons for this, but it can feel confining at times. Fortunately, we can equally well force over non-well-founded models! It ...
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### $\mathsf{AD}_\mathbb{R}$ and Elementary Embeddings

Suppose $\mathsf{AD}_\mathbb{R} + V = L(\mathscr{P}(\mathbb{R})) + \mathsf{DC}$ holds. (We can use more if it is helpful.) I believe under $\mathsf{AD}_\mathbb{R}$, every $A \subseteq \mathbb{R}$ is ...
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### Set-theoretic forcing over sites?

All texts I have read on set-theoretic independence proofs consider several different sorts of constructions separately, such as Boolean-valued models (equivalently, forcing over posets), permutation ...
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### A button for individual reals

Hamkins introduced the notion of a "button" in forcing. This is a set-theoretic statement that can be forced, and can never be made false by further forcing. An example is $V \not= L$. Another ...
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### 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 $j(\omega_1) = \omega_n^V$ ...
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### Is it consistent with ZFC that no nontrivial forcing notion has automatic mutual genericity?

A nontrivial forcing notion $\newcommand\Q{\mathbb{Q}}\Q$ exhibits automatic mutual genericity, if whenever $G,H\subseteq\Q$ are distinct $V$-generic filters (existing, say, in some forcing extension ...
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### Set theory and forcing from the point of view of a formal system $G^+$ of Gentzen type

There are four papers by Vladimir Alfeevich Kuznetsov, which discuss the above titled topic: (1) Some problems in set theory from the standpoint of a formal system G+ of Gentzen type. (Russian) Akad. ...
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### Statements that Could be Forced by Ultrapowers

Ultrapower of a structure is a very flexible mathematical creature in comparison with the ground structure and its ordinary products. Depending on the nature of ground structure and the good ...
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### Failure of GCH at a strongly compact cardinal

Does Con(ZFC+ there exists a strongly compact cardinal) imply Con(ZFC+ there exists a strongly compact cardinal $\kappa+ 2^\kappa > \kappa^+$)?
### Elementary chains in forcing extensions of $M_1$
Let $M_1$ be the canonical inner model with one Woodin cardinal $\delta$. Now suppose that $\mathbb{P}$ is a forcing notion of size $< \delta$, which preserves $\omega_1$ and that $G$ is a generic ...