# [FOM] Definable sets in ZFC

Monroe Eskew meskew at math.uci.edu
Fri Sep 17 18:41:01 EDT 2010

On Thu, Sep 16, 2010 at 3:21 PM, Max Weiss <30f0fn at gmail.com> wrote:
> 2) Every ordinal is constructible, but there are only countably many
> definable sets.  So not every constructible set is definable.
> Conversely, the set of countable sets is definable.  But, the question
> whether or not it is constructible depends on M.

It's worth noting that, since the notion of "definable" is not
formalizable within first order set theory, the argument that there
are only countably many definable sets takes place "from the outside."
There is no way to translate this argument to the "inside"
perspective, because it is actually possible to have a model of ZFC
where all sets are definable.  If \alpha is the least ordinal such
that L_{\alpha} satisfies ZFC, then L_{\alpha} is a countable
transitive model in which all sets have a definition.

Monroe