[FOM] Justifying SRP?
Harvey Friedman
hmflogic at gmail.com
Sat Sep 13 19:00:58 EDT 2014
http://www.cs.nyu.edu/pipermail/fom/2014-September/018168.html Rupert
McCallum wrote:
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.
http://onlinelibrary.wiley.com/doi/10.1002/malq.201200015/abstract
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I'm sure that the FOM readers would be most interested if you could
give a simple brief account of the ideas behind some of the reflection
principles that work - at least in the sense that they can be
obtained from standard large cardinal hypotheses. Of course, subtle
cardinals themselves are based on a very simple idea - but that idea
would not normally be characterized as reflection.
For just subtle, we have
kappa is essentially subtle if and only if kappa is a cardinal such
that for all binary relations R on kappa, there exists infinite alpha
< beta < kappa such that the sections of R at alpha,beta agree below
alpha.
Note that essentially subtle is closed upward, so it is not quite the
same as being subtle. HOWEVER, the first subtle cardinal is exactly
the first essentially subtle cardinal. ALSO "there exists a subtle
cardinal" is equivalent to "there exists an essentially subtle
cardinal".
If FOM readers relate to your simple brief account, they can of course
delve into publications. FOM readers can also get a chance to interact
online starting from what you write.
Harvey
PS: Maybe I see how to do this using some arguable reflection using
multiple universes. Let's consider two universes V and V', where V' is
longer than V. Let's not worry about the most philosophically honest
way to formalize this just yet.
Let R be a binary relation on V and let phi be a sentence that holds
in (V',V,R). "Reflection" says that there exists kappa in V such that
phi holds in (V',V(kappa),R|V(kappa)). This seems to prove Con(ZFC +
"there exists a subtle cardinal"). I think that if you use
V,V',V'',V''',... then you will get Con(SRP).
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