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I am new to set theory.

As I understand it, complex sets are constructed by set constructors from simpler sets (originally atoms).

I learned that such a set is called a constructible set.

I do not know of any set that is not constructible. Are there any examples?

Asaf Karagila
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BonBon
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    The answer to this question depends on exactly what axioms of set theory one starts from (and the exact definition of "constructible"). In some formulations of set theory, every set is constructible; in other formulations, the second sentence is not true. – Greg Martin Aug 27 '24 at 08:50
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    Relevant: Constructible universe. – Mauro ALLEGRANZA Aug 27 '24 at 09:08
  • Compare with this post and others here. – Dietrich Burde Aug 27 '24 at 11:40
  • @GregMartin Do you have a concrete example of which set is not constructible in which formulation? – BonBon Aug 27 '24 at 11:54
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    Set of uncomputable numbers – jimjim Aug 27 '24 at 12:19
  • @jimjim I do not know the exact definition of computability, but if it can be written in the language of set theory, then the set can be constructed by the axiom schema of specification. – BonBon Aug 27 '24 at 12:48
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    The term doesn't mean what you seem to expect it to mean. In particular, you can't prove that the set of real numbers, or complex numbers, etc. is constructible in the technical meaning of the word in set theory. Moreover, the two tags you're using should, in theory, be disjoint. – Asaf Karagila Aug 27 '24 at 13:14

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