[FOM] The Minimal Model of ZF

[Ilya Tsindlekht]

On Tue, Jan 01, 2008 at 12:49:43AM -0500, joeshipman at aol.com wrote:
[...]
> This is as parsimonious as a set theory can be. But there is a 
> confusing issue. When there IS a standard model, so that M is a set, 
> every element of M has a name. If there is no standard model, M is a 
> proper class, so there are way too many things in M to correspond to 
> countably many formulas. I'd like to ask what is the first set in M 
> which does not have a name, but this runs into the usual paradoxes. On 
> the other hand, if M is a set, then internally M satisfies V=M, and 
> there IS a first set in M which, externally, we can name, but which M 
> does not "know" has a name.
> 
For each element of M M has a formula naming it. It just doesn't have
the function which maps each such formula to the defined element.