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From: Rupert on 20 Jun 2010 01:18
In

http://math.stanford.edu/~feferman/papers/unfolding.pdf

Feferman gives a sense in which Mahlo cardinals can be seen as part of
an "unfolding" of ZF in a similar sense to that in which predicative
analysis is the "unfolding" of PA.

However he only permits reflection formulas in which the class
variables are universally quantified, and so draws the line at weakly
compact cardinals.

I find this quite a nice justification of the small part of the large-
cardinal spectrum. I would be interested if anyone could offer me any
reasons why I should accept weakly compact cardinals.
From: William Elliot on 20 Jun 2010 04:43
nOn Sat, 19 Jun 2010, Rupert wrote:

> Feferman gives a sense in which Mahlo cardinals can be seen as part of
> an "unfolding" of ZF in a similar sense to that in which predicative
> analysis is the "unfolding" of PA.
>
> However he only permits reflection formulas in which the class
> variables are universally quantified, and so draws the line at weakly
> compact cardinals.
>
> I find this quite a nice justification of the small part of the large-
> cardinal spectrum. I would be interested if anyone could offer me any
> reasons why I should accept weakly compact cardinals.
>
For what reason would anyone accept an inaccessible cardinal?
Their denial does not cause set theory to be inconsistent.
It can't be proven assuming their existence doesn't make set theory inconsistent.
In addition they violate Occam's rule not to wontedly multiply ententies.







From: Rupert on 20 Jun 2010 09:10
On Jun 20, 6:43 pm, William Elliot <ma...(a)rdrop.remove.com> wrote:
> nOn Sat, 19 Jun 2010, Rupert wrote:
>
> > Feferman gives a sense in which Mahlo cardinals can be seen as part of
> > an "unfolding" of ZF in a similar sense to that in which predicative
> > analysis is the "unfolding" of PA.
>
> > However he only permits reflection formulas in which the class
> > variables are universally quantified, and so draws the line at weakly
> > compact cardinals.
>
> > I find this quite a nice justification of the small part of the large-
> > cardinal spectrum. I would be interested if anyone could offer me any
> > reasons why I should accept weakly compact cardinals.
>
> For what reason would anyone accept an inaccessible cardinal?
> Their denial does not cause set theory to be inconsistent.
> It can't be proven assuming their existence doesn't make set theory inconsistent.
> In addition they violate Occam's rule not to wontedly multiply ententies.

Well, I linked to Feferman's article. It's a bit hard to summarise. Do
you want me to try?
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