By the way, I have posted on math stack exchange here : https://math.stackexchange.com/questions/3143616/monotone-limits-of-sets-do-not-exhaust-the-collection-defined-by-closure-by-thos

Regards.

]]>Anyway as your point applies then to my remark above for the case of for all (which was not so lame after all), as for it to be true for all which I though was simple, we would need in fact transfinite induction to get the result as the monotone limits of sets are not enough to conclude that for all . At last, I would really pleased to have a reference that shows that we have strict inclusion between the collection of monotone limits of sets in and the closure . Once again best regards ]]>

In particular “From ๐ฌ๐ (plus ยฌ๐ข๐ง) it follows that every set of reals is Lebesgue measurable”. This suggests that, to show that the projection of a co-analytic set is Lebesgue measurable, requires Martin’s Axiom and that the Continuum Hypothesis is false.

Also, you cannot go much further up the projective hierarchy without requiring theories which are stronger than ZFC — “Nevertheless, you cannot go much further by restricting to ๐น๐ฅ๐ข consistencywise: Shelah showed that the measurability of the sets implies the existence of inaccessible cardinals in ๐ฟ.”

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