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Can someone please give me an example of a set that lies in a $\sigma$-algebra generated by some set other then the $\sigma$-algebra itself, such that this (the first) set can't be obtained by performing countably many set theoretical operations ? (Obviously with "example of a set" I mean "prove that such a set exist", since this set probably isn't obtainable in a constructive way.)

I'm asking this since I've heard countless times, that generally the elements of a $\sigma$-algebra are not really constructively reachable.

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    This is hardly worth an answer, so it'll just have to do as a comment: Have you looked into [descriptive set theory](https://en.wikipedia.org/wiki/Descriptive_set_theory)?2012-11-04
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    @kahen outch...what an derogative answer. I looked at the link you provided, but that was way above my level :(.2012-11-04
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    @user19758: I think kahen was referring to his comment, not to your question!2012-11-04
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    @user19758: a link by itself is usually considered a poor answer, so to give you the link, kahen gave it in a comment. I don't think that kahen was implying that your question was "hardly worth an answer".2012-11-04

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