[FOM] On the Strengths of Inaccessible and Mahlo Cardinals
Dmytro Taranovsky
dmytro at MIT.EDU
Sat Mar 5 11:21:15 EST 2005
By using weak set theories, one can clarify the strengths of certain cardinals.
Often, a weak set theory + large cardinal is inter-interpretable (that is there
is a correspondence of models) with a stronger set theory + slightly weaker
cardinals. That large cardinal corresponds to the class
of ordinals of the stronger theory. More precisely,
Theorem:
1. ZFC minus infinity is equiconsistent with rudimentary set theory plus the
axiom of infinity.
2. ZFC minus power set is equiconsistent with rudimentary set theory + there is
an uncountable ordinal.
3. ZFC is equiconsistent with rudimentary set theory + there is an inaccessible
cardinal
4. ZFC + {there is Sigma_n correct inaccessible cardinal}_n is equiconsistent
with rudimentary set theory + there is a Mahlo cardinal.
In general, ZFC + {there is Sigma_n correct large cardinal}_n is
inter-interpretable with rudimentary set theory + there is a regular limit of
stationary many of these cardinals.
Rudimentary set theory is meant to be the weakest set theory in which some basic
things can be done. Levels of Jensen hierarchy for L satisfy it. I am not
sure how strong rudimentary set theory should be--suggestions to that effect
are welcome--but the following version/axiomatization works for the above
theorem: extensionality, foundation, empty set, pairing, union, existence of
transitive closure, existence of the set of all sets with transitive closure
less numerous than a given set, and bounded quantifier separation.
In the theorem, "rudimentary set theory" can be strengthened by extending ZFC
with a stronger "logic", and modifying Sigma_n correct and the replacement
schema to the full expressive power of the logic. Second order logic
corresponds to ZFC minus power set as the weak theory.
More about expressive logics and other topics can be found in my paper:
http://web.mit.edu/dmytro/www/NewSetTheory.htm
Dmytro Taranovsky
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