[FOM] Set Theory semantics

[Mitchell Spector]

On Wednesday, October 2, 2002, at 08:03  PM, joeshipman at aol.com wrote:
> ...
> "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
>

I think you need to impose additional requirements to do what
you want to do, Joe.  (In the following, all ultrafilters are
non-principal, as you clearly intended.)

For one thing, you haven't assumed that every member of your
countably complete ultrafilter is a proper class.  Assume
that there exist at least two measurable cardinals.  Let
rho be the least measurable cardinal, and let sigma > rho
be another measurable cardinal.  Let U be a rho-complete
ultrafilter U on rho, and let W be a sigma-complete
ultrafilter on sigma.  We can extend U and W to ultrafilters
on the class of all cardinals, of course (a class is in the
ultrafilter U or W if that class contains a set in U or W,
respectively).  Then U "says" that "there are no measurable
cardinals," while W "says" that "there exists at least one
measurable cardinal."  So U and W give rise to distinct
"theologies."

Or you can get by with a single measurable cardinal rho.
Let U be a normal ultrafilter on rho.  Then rho can be
placed in 1-1 correspondence with the set of all successor
cardinals less than rho; using this, you can define from U
a rho-complete ultrafilter W on rho that gives measure 1
to the set of successor cardinals less than rho.  Then W
"says" that "there exists a greatest cardinal"; in contrast,
U "says" that "there is no greatest cardinal."  So U and W
disagree.

Even if you add in the requirement that every member of
the countably complete ultrafilter is a proper class, it's
still not enough.  Any two proper classes of ordinals can
be placed into 1-1 correspondence, of course, so if there's
an ultrafilter on one such proper class, there's a
corresponding ultrafilter on any such class.  An ultrafilter
that concentrates on successor cardinals would "say" that
"there exists a greatest cardinal."  An ultrafilter that
concentrates on limit cardinals would "say" that "there
is no greatest cardinal."  Again, these two ultrafilters
disagree.

If, however, you require that the ultrafilter be normal,
then the sentences approved by your "pope" are precisely
the sentences true in V.  So any two "normal popes" agree.
(If you really want to carry this out, you need to be
careful with the formulation.  It clearly can't be fully
carried out in ZFC.)

Mitchell Spector