[FOM] role of large cardinals

meskew at math.uci.edu meskew at math.uci.edu
Thu Sep 22 15:12:07 EDT 2016

Thank you, I neglected the gauges used for weaker propositions.

Regarding CH, look carefully at my claim.  There is no known example of a
statement whose strength provably *transcends* the large cardinals.  I
mean consistency strength; I should have been more clear.  LCs do not
decide CH of course, but Con(ZFC+CH) is not a strong assumption.

>   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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