[FOM] Axiomatization through Reflection Principles
Dmytro Taranovsky
dmytro at MIT.EDU
Wed Mar 2 18:56:05 EST 2005
A theory of high consistency strength may still be weak at high expression
level. For example, ZFC + "there is a proper class of supercompact cardinals"
(if it is consistent) does not prove existence of Sigma-ZF-4 correct model of
ZFC. Reflection principles can fill in the gap.
A reflection principle states that a certain true theory has a sufficiently
well-behaved model. Well-behaved can mean existing (consistent), Sigma-0-n
correct, well-founded, ... or Sigma-ZF-n correct. The strength of a large
cardinal notion phi can be extended to higher expression levels by writing for
every statement psi, (for every ordinal k (phi(k) --> V(k) satisfies psi))
implies psi. The extension can be weakened by replacing (for every ordinal k
(phi(k) --> V(k) satisfies psi)) with (ZFC proves that for every ordinal k
(phi(k) --> V(k) satisfies psi)). The weaker extension requires lesser
ontological commitment since one does not need to believe phi to accept the
extension.
The kth level of the cumulative hierarchy, V(k), serves as an image of the
universe, so we can formalize vague intuitions about V as requirements (in the
above-mentioned reflection principles) on k, using higher order statements
about V(k) as necessary.
This posting explains some of the topics in my paper
http://web.mit.edu/dmytro/www/NewSetTheory.htm
Dmytro Taranovsky
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