Re: Getting rid of cardinality as an issue

[Mike Oliver <moliver@unt.edu>]

Vaughan Pratt wrote:

> While I'm happy to field objections like "too flippant", I'm more concerned as
> to whether there are any technical flaws, and to a lesser extent philosophical
> or religious concerns.  (I would not want to be held responsible for guns
> being brought to the next UACT meeting if ever there is one.)
> [...]
> Now the Grothendieck hierarchy is stepped through via ZF rather than Z, with
> Fraenkel's Replacement axiom doing the heavy hitting.  This creates
gaps
> mind-bogglingly larger than my teensy exponential gaps above.  The general
> idea seems to be that these gaps ought to be large enough to take care of
> Russell while still not running headlong into inconsistency.  However gaps
> this large do entail a certain amount of finger-crossing, and one might
> question the logic of hitting Russell with a nuclear weapon that might send
> some fallout your way when a harmless little tack-hammer will take him out.

I'm not entirely sure I follow what Vaughan's project is here, so this
may come out as a non sequitur, but:  Surely, from time to time,
categorists must care about genuinely ultra-first-order notions, such as
(say) the metric completeness of the real numbers?  To me the natural
way of getting such notions right is to make sure that each of your
universes is closed under the (true) powerset operation.  That would
require the cardinality of your universes to be, at least, strong limit
cardinals.

Having them closed under ranges of functions also seems natural enough;
at that point you need inaccessibles.

It's by no means clear that inaccessibles are sufficient.  What happens
when you want to be closed under the operation of finding the next
larger inaccessible?