FOM: Infallible and eligible cardinals

Joe Shipman shipman at
Fri Jan 29 13:29:12 EST 1999

John Bell writes:
>A notion very closely related to Joe Shipman's "infallible" cardinals
>appeared (in essence at least) in "Natural Models of Set Theories," by
>Montague and Vaught, Fundamenta Mathematicae, 1959. Call an ordinal k
>impeccable if (V-sub-k, epsilon) is an elementary substructure of
>(V,epsilon). (Clearly every impeccable cardinal is infallible.) Then
>Montague and Vaught show that, for example, if an impeccable ordinal
>exists, then the least one must be cofinal with omega, and so fails to
be a
">large" cardinal in any of the usual intrinsic senses.

This is clearly true for the first "infallible" cardinal as well since
it is the limit of n-infallible cardinals (cardinals k such that V_k
satisfies all true n-quantifier sentences), unless you require
infallibles as I originally did to be inaccessible as well.  I like my
later definition which did not require this better, because I am
interested in how high up these cardinals must be rather than their
internal properties.  What did Montague and Vaught name what you call
"impeccable" cardinals?

It seems that an infallible cardinal is larger than any cardinal
definable in ZFC, but is proof-theoretically weaker than an
inaccessible, so there is not much more that can be said.  However
1-infallible, 2-infallible, and 3-infallible cardinals are interesting
because they may fall in between known large cardinals (they can be
shown to exist in ZFC, I am interested in a question like "if measurable
cardinals exist can anything be said about the relative sizes of the
first measurable cardinal and the first 2-infallible cardinal?").

My other concept, an "eligible" cardinal (a k such that V_k satisfies
all sentences which are "eventually true"), is definable within ZFC but
appears to be larger than any cardinal that is *definable without
referring to sets of arbitrary rank*.  Where else is this distinction
important in set theory?

-- Joe Shipman

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