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An example from elementary geometry is the very simple forcing argument to establish the non-axiomatizability (in infinitary logic) of Sperner spaces, due to Blass and Pambuccian: Blass, A.; Pambuccian …

Todorcevic, Notes on Forcing Axioms, Chapter 7. Itay Neeman, Forcing with side conditions. Oberwolfach, 2011. http://www.math.ucla.edu/~ineeman/ B. Velickovic, G. Venturi, Proper forcing remastered. …

This answer is the proof given by Ashutosh, but formulated in terms of the splitting number. Proposition If the splitting number $s$ is $\aleph_{1}$, then every nonseparable metric space contains a s …

If every nonseparable metric space contains a sequence of subsets with no convergent subsequence, does the Continuum Hypothesis hold? The answer is negative, and in the interests of self-contained it …

A forcing $\mathbb{P}$ is Jonsson-preserving if $(V, V^{\mathbb{P}})$ is Jonsson-friendly, where $V^{\mathbb{P}}$ is the family of $\mathbb{P}$-generic extensions of $V$. … It follows that for example every ccc forcing is Jonsson-preserving. …

, Forcing with ideals and simple forcing notions, Israel J. … And the same authors have more in: More on simple forcing notions and forcing with ideals, APAL, 59 (1993), 219-238. …

If two $\tau$--structures $A$ and $B$ in a vocabulary $\tau$ are partially isomorphic, then there is a forcing extension in which they are isomorphic. … Several interesting results first proved by forcing are also listed in the answers to the question: https://mathoverflow.net/a/53887/57583 …

In belated partial response to (A), canonical functions are certainly already defined in the Jech-Shelah paper: A note on canonical functions, where reference is made to the work F. Galvin and A. Hajn …