Timeline for What is the descriptive complexity of a set added by Cohen forcing?

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when toggle format	what		by	license	comment
Dec 1, 2016 at 7:08	vote	accept	fhyve
Nov 30, 2016 at 22:03	comment	added	Noah Schweber		But $V[H]$ satisfies "The unique real satisfying $\varphi$ does not contain $k$" - note that since the ordinals of $V[G]$ and $V[H]$ agree (being those of $V$), this is really the same sentence as the opposite one satisfied by $V[G]$. And that's a contradiction. So we can define the real $\nu$ names as "The set of all $k$ such that some condition forces $k\in\nu$" (this uses that the forcing relation is appropriately definable), and so the real named by $\nu$ in fact exists in $V$ already.
Nov 30, 2016 at 22:02	comment	added	Noah Schweber		+1. For the OP, here's an outline of how this is shown: suppose $\nu$ was a name for an OD real in the generic extension. (That is, it is forced that $\nu$ is defined by some formula $\varphi$ with ordinal parameters.) Suppose $p$ is some condition forcing $k\in \nu$ ($k$ some natural number). Then $p$ forces "The unique real satisfying $\varphi$ contains $k$." Now suppose $q$ were any other condition which forced $k\not\in \nu$. By homogeneity, we can find generics $G\ni p, H\ni q$ yielding the same forcing extension: $V[G]=V[H]$. (cont'd)
Nov 30, 2016 at 20:54	history	answered	Andreas Blass	CC BY-SA 3.0