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When is there an unbounded tower in $[\mathbb{N}]^\infty$?
A model for Question 1: Let $V \models$ MA + $2^{\aleph_0} = \kappa \geq \aleph_2$. Using MA, construct a $\supseteq^{\star}$-chain $\overline{A} = \langle A_i : i < \kappa \rangle$ such that $\mathca …