[FOM] role of large cardinals

[John Kadvany]

  Small to large cardinals do indeed provide such a measure, e.g. Friedman's
proof that aleph_1 power set iterations are required for Borel determinancy,
or Woodin cardinals for projective determinacy, with determinacy in turn
implying many other statements.  

  But there are many other such measures of relative strength, including ones
involving only countable sets. 

  I'm thinking of: the arithmetical and analytic hierarchies, in which a
just-higher quantifier depth is tied to representing all sets of lower
complexity; the related Kleene jump operation in recursion theory; the myriad
patterns of incomparable relative recursiveness possible through priority
method constructions; complexity hierarchies of Ritchie, Grzegorzyk, etc.; the
use of countable ordinals from epislon_0 onward vis a vis arithmetic
provability; the restricted use of second-order quantification in analysis and
reverse mathematics to sharply isolate relative consistency strength. Where
set definitions are involved, they will typically translate to some
provability condition. 
  
  On 'possibly-consistent', the continuum hypothesis is that; but CH cannot be
determined by large cardinal assumptions alone. I'm unsure that's is what's
meant here by 'strength'. 

   John Kadvany 
  

-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of
meskew at math.uci.edu
Sent: Wednesday, September 21, 2016 7:08 PM
To: fom at cs.nyu.edu
Subject: [FOM] role of large cardinals

I recently wrote the following paragraph-fragment.  I would appreciate any
critiques of the assertions, especially if you disagree with the last thing
starting with "the fact that..."

In contemporary logic, there is a wide-ranging consensus that the traditional
large cardinal axioms are the appropriate measuring-stick for gauging the
logical strength and showing the consistency of any mathematical statement.
The main reasons for this are their mutual compatibility, their success in the
role so far, and the fact that there is no known example of a
possibly-consistent hypothesis whose strength can be shown to transcend the
large cardinal notions.

Thanks!
Monroe

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