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No. Suppose that there is a MAD family of size $\aleph_1$ and $\sf CH$ fails. Let $\mathcal E=\{E_\alpha\mid\alpha<\omega_1\}$ be a MAD family on the even integers. Suppose now that $\cal A$, your AD …

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 …

Is it acceptable? Sure. In some sense, it is an Aronszajn tree. The condition of being well-founded, which in the presence of $\sf DC$ is the same as saying there are no decreasing sequences, is equi …

Recall that a tree on $\omega_1$ is called Suslin if every chain and antichain are countable. If every level is countable and there are no cofinal branches, then it is called Aronszajn (in particular …

Let $\lambda$ be a singular cardinal of countable cofinality. Is there necessarily a sequence $\{A_\alpha\mid\alpha<\lambda^+\}$ of countable subsets of $\lambda$, such that $\alpha<\beta$ if and onl …

(Pierre Gillibert asked me this question and I post it with his permission.) Let $X$ be an infinite set, and $f\colon[X]^\omega\to[X]^\omega$. We say that $\{x_n\mid n<\omega\}\subseteq X$ is a free …

Assume $\sf GCH$. Let $\kappa$ be a regular cardinal, we say that $\{A_\alpha\mid\alpha<\kappa^+\}\subseteq\mathcal P(\kappa)$ is an almost disjoint family, if whenever $\alpha\neq\beta$, $A_\alpha\c …

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 …

If we start with $V\models\lozenge$, it is not hard to force the failure of diamond. You can blow up the continuum, or destroy all the Suslin trees. You can blow up the continuum of $\aleph_1$, and th …

Suppose that $V$ is a model of $\sf ZFC$, and for concreteness I should point that at this point I am interested in $V=L$ as a ground model. Suppose that $V[c]$ is a Cohen extension of $V$ where $c$ …

Let $A,B\subseteq\omega$. We write $A\subseteq^*B$ if $A\setminus B$ is finite, if additionally $B\setminus A$ is infinite then we write $A\subsetneq^*B$, otherwise we write $A=^*B$. We say that a $\ …