Re: SUO: RE: A proposed SUO content outline
Pat wrote:
> >MW: No but what things are sets is. I only mean here that to be
> >a set you must be iteratively constructed,
>
> Oh no no. Consider the set of all noncomputable numbers, for example.
I think Matthew is using "iteratively constructed" in the metaphorical
sense found in intuitive descriptions of the so-called iterative (or
cumulative) hierarchy. This is pretty much universally taken to be the
natural model of ZF. In this sense (as you know), "iteratively
constructed" most definitely does not mean "computably generated".
> >and you can't for example have yourself as
> >a member. As far as I can make out this is common ground for what
> >a set is
> >consdered to be. But I'm prepared to hear that the contrary is true.
>
> Actually, there are "non-well-founded" set theories where a set can
> be a member of itself, but they are new and not considered infra-dig
> in many mathematical circles.
True, of course, but again, if Matthew is appealing implicitly to theories
that are motivated intuitively by the iterative hierarchy here, then he is
quite right (as you know) that sets can't be self-membered. You have to
move to a different picture (e.g., Barwise's idea of sets as highly
attenuated situations) to motivate the non-well-founded universe.
-chris
--
Christopher Menzel # web: philebus.tamu.edu/~cmenzel
Philosophy, Texas A&M University # net: chris.menzel@tamu.edu
College Station, TX 77843-4237 # vox: (979) 845-8764