[FOM] Set Theory semantics

Rupert McCallum rupertmccallum at yahoo.com
Thu Oct 3 02:22:55 EDT 2002

--- joeshipman at aol.com wrote:
> I wrote:
> > 3) Suppose the Universe is Mahlo, and there is a stationary class
> of 
> > inaccessibles such every sentence is either eventually true or
> > eventually 
> > false in the corresponding set of V_j. 
> Rupert McCallum wrote:
> >As follows from a discussion in my previous post, that's not
> consistent
> >- the liar paradox applies.
> I see I was imprecise.
> "Phi eventually true" is too strong a condition.  A better notion of
> "election" is as follows:
> Suppose there is a class of cardinals with a countably complete
> ultrafilter.  Then every sentence Phi partitions the class into
> cardinals k where V_k satisfies Phi and the rest where V_k satisfies
> (not Phi).  The "dogmatically approved" sentences are the Phi where
> the class of V_k satisfying Phi is in the ultrafilter.  There are
> countably many such sentences, and the intersection of the
> corresponding subclasses is also in the ultrafilter, and any member
> of this intersection is eligible for election as a Pope (a cardinal
> to whom all questions of dogma may be referred).

How about a countably complete ultrafilter on a class of cardinals
cofinal in kappa with kappa a measurable? And to avoid the liar paradox
the ultrafilter must not be definable in V_kappa. 

> But there may be other ultrafilters lying about which give rise to a
> different set theory (theology), and there may be a notion of
> election which applies to classes of cardinals which do not
> necessarily have a countably complete ultrafilter.
> If "The Universe is Measurable", is it possible that there exist two
> ultrafilters giving rise to distinct theologies? In such a case
> "Popes are not necessarily infallible".

Well, how about asking "On V_kappa with kappa a measurable, can there
exist two ultrafilters yielding two distinct theologies?" 

When the ultrafilter is kappa-complete, easily prove from basic facts
about measurable cardinals that the "theology" has to be the actual
theory of V_kappa, so that with this restriction the answer is no.

Countably complete... unsure at this point.

> -- Joe Shipman
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