[FOM] 454: Three Milestones in Incompleteness
Paul Budnik
paul at mtnmath.com
Mon Feb 21 11:45:52 EST 2011
This is an observation on the post of 02/06/2011 03:33 AM by Harvey
Friedman and related posts.
It is worth keeping in mind that any claim that a hyperarithmetic
statement requires a large cardinal axiom to decide is relative to the
existing development of mathematics. Such results are very interesting,
but they can change.
Since Kleene's O is a \Pi_1^1 complete set, any hyperarithmetic
statement can be decided in second order arithmetic augmented with a
finite number of axioms that say specific integers are or are not
notations for recursive ordinals under Kleene's definition, because a TM
with oracle that makes a decision, must do so after a finite number of
queries.
Cardinal hierarchies seem to be powerful ways to implicitly define
combinatorial iteration of a sort that is very difficult to tackle head
on. Of course it may be too complex to develop such structures
explicitly even with the aid of computers. However, because this
iteration can be specified implicitly in relatively brief axiom systems,
I suspect we can eventually uncover the explicit iteration schemes that
are implicitly defined by using the computer to help manage the
combinatorial complexity. Of course this is speculation. No one knows,
but I think it is a possibility that deserves consideration.
Paul Budnik
www.mtnmath.com
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