[FOM] Re: Re: The width of V (Roger Bishop Jones)

[Eric Steinhart]

On Tuesday 08 July 2003  7:25 pm, Eric Steinhart wrote:

> It would seem that the plain old power set axiom makes V as
> wide as possible, if you take V[n+1] to be the power set of
> V[n] for any ordinal n.

>Date: Wed, 9 Jul 2003 06:24:39 +0100
>From: Roger Bishop Jones <rbj at rbjones.com>

>The power set axiom says nothing about what subsets there
>are, it just says that such subsets as there are, are
>the members of the power set.

That's quite right, but the question is about the axiom that makes V wide; that will be the power set axiom regardless of how subsets are defined.  Other axioms may say what subsets there are, but the gathering of them together into the next successor stage is done by the power set (in standard definitions of V).   

I agree wholeheartedly that it's a fascinating and apparently unresolved problem as to which sets are subsets of any given set.