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The answer to the second question is negative. Take, for example, Sacks forcing over $L$. The result is $L[r]$ where $r$ is a real number and the following property is true: $$\forall x(x\in L\lor r\i …

Assume $\sf ZF+AD$. Is there some inner model $M$ containing all the ordinals such that $M\models\sf ZF+AD$ as well? What if we require $\omega_1$ and/or $\omega_2$ to be computed correctly? Can we sa …

Of course the axiom of choice is consistent with the failure of the axiom of foundation. To get Fraenkel's model with the atoms, you usually start with a model of $\sf ZFA+AC$. You can find the relev …

One of the way a person shows their wealth is by having many diamonds. The same can be said about models of $\sf ZFC$. We can add generic diamond sequences, while preserving the old ones, so in some s …

I cannot answer either formulation of Q1, although it did pop through my mind just yesterday. Funny. For the second question, I cannot give you an accurate answer, but I believe that the answer is ne …

This doesn't work. I'm leaving this as it might helpful to someone later on. I think that the answer is yes, but my idea has a gap. This happens for silly reasons: $V=L$ satisfies $\sf GCH$, and fake …

Say that an inner model $M$ of $V$ is generically saturated if for every forcing notion $\Bbb P\in M$, either there is an $M$-generic for $\Bbb P$ in $V$, or forcing with $\Bbb P$ over $V$ collapses c …

One of the most incredible results in modern set theory, due to Jensen, is that given any model of $\sf ZFC$, there is a class forcing which adds a real number $r$ and in the extension $V=L[r]$. Moreo …