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Andrej Bauer
Andrej Bauer
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I consider definability to mean one of either cases:

(a) Definability without parameters (in the language of set theory), or

(b) Definability from ordinals and a real (in the same language).

  1. Definability without parameters (in the language of set theory), or

  2. Definability from ordinals and a real (in the same language).

So my question is: Is there a model $M$ of ZFC (or at least of ZF) such that every definable family of sets (not necessarily of reals) contains at least one definable member (in the sense (a)1. or (b)2. respectively) but such that $M$ contains nonetheless many non-definable members (i.e. $M$ is not pointwise definable)?

I consider definability to mean one of either cases:

(a) Definability without parameters (in the language of set theory), or

(b) Definability from ordinals and a real (in the same language).

So my question is: Is there a model $M$ of ZFC (or at least of ZF) such that every definable family of sets (not necessarily of reals) contains at least one definable member (in the sense (a) or (b) respectively) but such that $M$ contains nonetheless many non-definable members (i.e. $M$ is not pointwise definable)?

I consider definability to mean one of either cases:

  1. Definability without parameters (in the language of set theory), or

  2. Definability from ordinals and a real (in the same language).

So my question is: Is there a model $M$ of ZFC (or at least of ZF) such that every definable family of sets (not necessarily of reals) contains at least one definable member (in the sense 1. or 2. respectively) but such that $M$ contains nonetheless many non-definable members (i.e. $M$ is not pointwise definable)?

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user38200
user38200
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Is it consistent with ZFC (or ZF) that every definable family of sets has at least one definable member?

I consider definability to mean one of either cases:

(a) Definability without parameters (in the language of set theory), or

(b) Definability from ordinals and a real (in the same language).

So my question is: Is there a model $M$ of ZFC (or at least of ZF) such that every definable family of sets (not necessarily of reals) contains at least one definable member (in the sense (a) or (b) respectively) but such that $M$ contains nonetheless many non-definable members (i.e. $M$ is not pointwise definable)?