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Let us say a forcing $P$ is proper for a class of models $\mathcal C$, if for large enough regular $\theta$ and $M \prec H_\theta$ in $\mathcal C$ with $P \in M$, every $p \in P \cap M$ can be extended …

Note that I am not asking whether $P(\omega_1)/I$ and $P(\omega)/\fin$ are forcing-equivalent, but rather if they are literally isomorphic. …

;-P) Finally, the tail of the forcing, $\mathrm{Col}(\omega_1,{<}\kappa)/ (G \restriction \mu)$, is homogeneous. … Since $\mathcal C(\mu)$ is a subset of $\mu$ definable from a parameter from $V[G \restriction \mu]$, general facts about homogeneous forcing show that it is an element of $V[G \restriction \mu]$. …

We also have $M^\kappa \subseteq M$, and this is preserved by the forcing so that $M[G*H]^\kappa \subseteq M[G*H]$ in $V[G*H]$. … So by forcing with $\mathbb P * \dot{\mathrm{Add}}(\kappa,\kappa^+)$, we get an elementary embedding lifting the ultrapower embedding. …

In my experience, each of the various approaches to forcing and subforcing has some context where it looks like the most useful and elegant one. …

The statement implies that for any $f : [\omega_1]^2 \to 2$, there is an uncountable $S$ such that $f$ is constant on $[S]^2$. This is impossible since $\omega_1$ is not weakly compact. See Jech, Le …

The Ground Axiom states that the set-theoretic universe is not a set-forcing extension of an inner model. … In contrast, forcing axioms do not imply the ground axiom because of preservation theorems. For example, PFA is preserved by $\omega_2$-closed forcing. …

Let us say that a partial order is "countably closed with infima" if every descending $\omega$-sequence has an infimum. Suppose $P$ and $Q$ are posets that are countably closed with infima, and for so …

(We can find such an antichain because the forcing is nowhere $|P|$-c.c.) …

You’re right; this kind of strong master condition cannot exist in the usual context where $j : M \to N$ is a class of $M$. The relevance is when we’re in an intermediate stage of lifting an embeddin …

$\kappa$ is preserved, and moreover all reals are added by the small generics. Let $(P_0,p_0)$ be a condition and let $\sigma$ be a name for a real. First, enumerate the elements of $P_0$ below $p_0$ …

Here is a naive idea for a forcing $\mathbb A(\kappa)$, for an inaccessible cardinal $\kappa$. Conditions are pairs $(P,p)$, where $P \in V_\kappa$ is a partial order and $p \in P$. …

The Banach-Mazur game on a poset $\mathbb P$ is the $\omega$-length game where the players alternate choosing a descending sequence $a_0 \geq b_0 \geq a_1 \geq b_1 \geq \dots$. Player II wins when th …

I told Neeman about this work, and he said that a modification of the forcing that directly avoids my counterexample should work. …

It follows from the arguments in their paper that Mitchell's forcing adds a disjoint stationary sequence. … By a theorem of Gitik, this is stationary, and its stationarity is preserved by the tail forcing. …