Timeline for Borel coloring of a graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$

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May 21, 2015 at 21:32	comment	added	Joel David Hamkins		But meanwhile, of course, the forcing-free proof is obviously easier to grasp for those who don't know forcing! I think of the situation as analogous to the difference in perspective between using non-standard analysis and arguing with actual nonstandard infinitesimals versus arguing with epsilons and deltas and sets in an ultrafilter. Sometimes it is clarifying to have the actual infinitesimal or the actual generic object, even in cases where the need for it is eliminable.
May 21, 2015 at 16:26	comment	added	Joel David Hamkins		+1 Very nice. Despite first appearances, this answer and mine are actually rather close, using the same underlying idea, and one can view this as an instance of the connection of Cohen genericity with comeager sets: a function is $V$-generic just in case it is in every ground-model comeager set. So the clopen set here corresponds to the condition $p$ in my answer, with $B$ the set of functions having the color that $g$ gets, and $B+e_k$ corresponds to my argument with $g'$, which made a change to $g$ outside $p$.
May 21, 2015 at 13:26	review	First posts
May 21, 2015 at 14:29
May 21, 2015 at 13:24	history	answered	Guest	CC BY-SA 3.0