[FOM] Set Theory semantics

joeshipman@aol.com joeshipman at aol.com
Wed Oct 2 23:03:49 EDT 2002

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).

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".

-- Joe Shipman

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