[FOM] Justifying SRP?
rupertmccallum at yahoo.com
Sat Sep 13 03:18:25 EDT 2014
William Tait wrote an essay that appeared in "The Provenance of Pure Reason" called "Constructing Cardinals from Below" which discussed a set of reflection principles that justify SRP. Unfortunately Peter Koellner later observed that some of the reflection principles he considered were inconsistent. I wrote down my own thoughts in a recent Mathematical Logic Quarterly article about how one might find principled grounds for distinguishing the consistent ones from the inconsistent ones.
On Saturday, 13 September 2014, 2:43, Harvey Friedman <hmflogic at gmail.com> wrote:
One of the more well known senior Fields Medalists (senior even by my
standards) has gotten interested in the recent Pi01 independence.
These are provable using the subtle cardinal hierarchy but not with
less. In fact, provably equivalent to the consistency thereof.
The subtle cardinal hierarchy is the same as the ineffable cardinal
hierarchy, and this is also the same as the SRP hierarchy - each
cofinal in each other. Recall that SRP stands for "stationary Ramsey
property", which asserts that every partition of the k-tuples into two
pieces has a stationary homogenous set.
He writes me "why should I believe that there is a subtle cardinal (or
k-subtle cardinal), or even that it is consistent?"
I already answered that it is my business to get the issue joined by
proving equivalence with Con(SRP). I said I would get back to him
about the so called "arguments" for SRP and Con(SRP).
Now how should I respond for a senior Fields Medalist in core
mathematics who clearly is a combination of dubious/not knowledgeable
about higher set theory?
If the cardinals were somewhat smaller, than I would just give the
second order reflection story as my best shot - with a class
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