FOM: Large but weak

[Joe Shipman]

The new large cardinals I defined last weak are very large indeed in
set-theoretic terms, but they are proof-theoretically weak.  That is,
you can show they are larger than all the ordinarily defined large
cardinals IF those exist, but that doesn't help you show what actually
exists or is consistent.

Recall that a cardinal k is "eligible" if V_k satisfies all "eventually
true" statements.  This is clearly definable in ZF.  Since any
eventually true statement is actually true (by the reflection principle,
any true statement is true for some V_j containing any given set; if an
eventually true statement were actually false, it would be false in some
V_j above the place it was supposed to stop being false), V itself is
"eligible", and applying the reflection principle again we find that an
eligible cardinal exists.  Formalizing this argument fully seems to
require Morse-Kelley Set Theory or some other theory slightly beyond
ZFC, but in any case proof-theoretically weaker than the existence of an
inaccessible cardinal.  In fact, any "ordinary" large cardinal definable
in ZFC, whose definition does not refer to sets of arbitrary rank, must
*if it exists* be less than the first "eligible cardinal", because the
statement that such a cardinal exists would be eventually true.

Say that a cardinal k is "n-infallible" if V_k satisfies all true
n-quantifier sentences.  Because n-quantifier truth is definable,
arbitrarily large n-infallible cardinals provably exist for each n (but
I am arguing in a metatheory here, in ZFC you can independently define
n-quantifier truth for each n but not all at once).  A cardinal is
simply "infallible" if it is n-infallible for all n (again, this
definition requires a metatheory).  Taking the intersection of the
countably many classes of n-infallible cardinals, we find that the
smallest infallible cardinal exists and is larger than any cardinal that
can be defined in ZFC.  But again, this argument requires only a slight
extension of ZFC, so infallible cardinals are also proof-theoretically
weak.  Still, the n-infallible cardinals form an interesting "scale";
can anyone say where in the size hierarchy of large cardinals the
smallest 1-infallible, 2-infallible, and 3-infallible fall?  (I'm asking
here for results of the form "if a measurable cardinal exists, the first
2-infallible cardinal is less than it".)

-- Joe Shipman