[FOM] role of large cardinals

Colin McLarty colin.mclarty at case.edu
Thu Sep 22 04:29:17 EDT 2016

This is true when you consider consistency strengths above that of ZF,
and if you think of elementary embedding principles as a form of large
cardinal axioms.  At the level of Second order Arithmetic and below,
comprehension and induction axioms are the usual measuring stick.  I
guess that is true below the level of Simple Type Theory. Between that
level and ZF I like a lot of results i have seen but I do not have a
strong idea of how those results are or should be seen by experts.

On Wed, Sep 21, 2016 at 10:08 PM,  <meskew at math.uci.edu> wrote:
> 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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