[FOM] Large cardinals and small sets

[joeshipman@aol.com]

Many large cardinal axioms come in "Strong" and "Weak" versions, where 
the condition defining the cardinal includes, or does not include, 
being a strong limit cardinal.

For example:

a regular limit cardinal is weakly inaccessible, a regular strong limit 
cardinal is (strongly) inaccessible.

a regular limit cardinal in which regular limit cardinals form a 
stationary subset is weakly Mahlo; if a weakly Mahlo cardinal is a 
strong limit then strong inaccessibles form a stationary subset and the 
cardinal is said to be strongly Mahlo.

an uncountable cardinal kappa with a kappa-additive real-valued measure 
is called measurable if kappa is a strong limit. otherwise it is called 
a real-valued measurable cardinal and can be shown to be no larger than 
the continuum.

In all these cases, the logical strength and many of the other 
consequences of the "strong" version of the axiom follow from the weak 
version of the axiom; and the weak version has the advantage that it 
can be stated in terms of "small sets". Thus, if there is a countably 
added measure defined on all subsets of the continuum, then there is a 
real-valued measurable cardinal <=c, which furthermore is weakly 
inaccessible, weakly Mahlo, etc.

There are other cardinals whose logical strength is "large" but which 
do not themselves have to be inaccessibly large sets, for example 
Jonsson cardinals.

On there other hand, there are cardinals whose definition refers in an 
essential way to sets of arbitrary rank; for example supercompact 
cardinals.

My question is, how much logical strength (beyond measurable cardinals) 
can we get with natural axioms which refer only to "small sets"? I will 
leave "natural" and "small" vague here, except to say that I don't want 
to allow axioms of the form "Large Cardinal X is consistent" as 
"natural", and DO want to regard "there exists a countably additive 
measure on subsets of the continuum" as referring only to "small" sets.

Friedman has many examples of statements about "small sets" (even just 
about integers) which have "large strength"; the question is how 
"natural" they can be made compared with the definitions of, say, 
weakly inaccessible or weakly Mahlo or real-valued measurable cardinals 
(all of which are at least as easy to define as their "strong" 
counterparts).

-- JS
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