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We assume that for every real $x$, $L[x]$ only contains countably many reals. Given a set $X$ of reals, then $L$-ideal generated by $X$ is the smallest set $I$ of reals so that For any reals $x\in ...

This is a crossposting of https://math.stackexchange.com/questions/1446602/anti-random-reals, which has not gotten any answers; after thinking about the problem, I've become more convinced that it ...