FOM: Re: infallibility and inexpressibility

[Joe Shipman]

Neil Tennant wrote:

> Joe Shipman's concept of an infallible cardinal is interesting,
> because it enables one to reach certain philosophical conclusions
> about the limitations of set theory without making use of the heavy
> armour of G"odel's second incompleteness theorem.
>
> Definition (Joe Shipman): alpha is an infallible cardinal just in case
> all sentences true in V are true in V_alpha.
>
> Theorem: If the concept "x is an infallible cardinal" is expressible
> in the language of set theory, then there is no infallible cardinal.
>

This follows most simply from the undefinability of truth.  In fact, not only can
you not define infallible cardinals as a class, you can't define any nonempty
subclass of them, because the smallest element alpha of this subclass would be
definable and
so would the set of sentences it satisfies.  Thus the first measurable cardinal
could not be infallible, etc.   However, one of the large cardinals which can
only be defined in higher theories might be infallible; my query is how the first
infallible compares to such "known" cardinals.

We are working simultaneously in set theory and a metatheory involving strong
classes or a truth predicate for the ordinary language of set theory LST (the
language with = and /epsilon but no class or truth predicates).

This allows me to answer my earlier question about cardinals which satisfy all
eventually true sentences of LST.  Assuming inaccessibles are unbounded in V, the
first such cardinal, if it exists, is definable and so cannot be infallible.

My idea here was to define a notion of "majority vote" -- if a class of
(inaccessible) cardinals says S is true and only a set says S is false [my
definition of "S is eventually true"], we can regard S as "dogmatically
established" by the college of cardinals.  A cardinal is eligible to be elected
as the official arbiter of questions of faith if it is "dogmatically sound", i.e.
satisfies all eventually true sentences.  So we have seen that the first
"eligible cardinal" is not infallible.

But we can argue according to informal reflection principles that any infallible
cardinal must be eligible, because if it were not there would be a true sentence
which would be false for all sufficiently large cardinals.  (This is not in
violation of any reflection principle I can formalize because "sufficiently
large" might be undefinably large, so that one cannot modify the sentence to get
a true sentence not reflected in ANY V_alpha.)  Thus if there are any infallible
cardinals (which one believes because of informal reflection principles--remember
that infallibility is limited to questions in LST and an infallible cardinal
doesn't need to reflect all true sentences in a higher-order set theory) then
they are eligible for election and it is possible for the Pope to be infallible.

Can anyone help me place the first eligible cardinal and the first infallible
cardinal more precisely in the hierarchy of large cardinals that have been
defined?

-- Joe Shipman