[FOM] Thoughts on CH 1

[martdowd at aol.com]

Sol Feferman has put forward the thesis that 


The Continuum Hypothesis is neither a definite mathematical problem nor a definite logical problem



which is in fact the title of one of his recent preprints. I am one of many scholars that Sol asked for comments. 



 

 In my opinion CH is a definite mathematical problem.  While there is ongoing
debate on the subject, most mathematicians would agree that aleph_1,
aleph_2, and P(omega) are definite mathematical objects.  Further, either
there is an injection from aleph_2 into P(omega), or thee is a bijection
of aleph_1 with P(omega).  This all follows in ZFC, so within the confines
of ZFC CH is a definite problem.

The choice between the injection and the bijection cannot be made based on
ZFC.  I argued in
``Remarks on Levy's reflection axiom'',
Mathematical Logic Quarterly 39 (1993), 79--95
that a choice in favor of nonexistence of the injection could be made on the
basis of mathematical intuition.  Namely, if there is an injection, where is
it?

The standard reply is that the same question can be asked regarding the
bijection.  But it can be argued that the symmetry can be broken in favor
of CH.  The bijection can't be constructed within ZFC, but it is there.
The injection would "stick out like a sore thumb" if it existed.

Further evidence can be found in the fact that a bijection can be
constructed from V=L.  Indeed, the suspicion that the bijection exists
is evidence that V=L.  Many set theorists reject V-L because it contradicts
0# (whence various large cardinal principles).  In my opinion this
position should be re-evaluated.

Friedmann states:
There is now a common idea put forth that "there is a
compelling case for the truth (weaker: the consistency) of a
large cardinal hypothesis if and only if there is an
appropriate inner model theory for it.

In a series of papers I have been arguing that large cardinals can and should
be accepted if they can be justified by "repeating" the argument that the
cumulative hierarchy should not be "arbitrarily truncated" at the first
inaccessible cardinal.  The size has continued to increase, but it is open
whether a weakly compact cardinal has been reached.

In sum, it is a reasonable position that V=L, and large cardinals should be
accepted only if they have been strongly justified, e.g., by arguments
involving extending the cumulative hierarchy.  Research on both
possibilities should continue.

- Martin Dowd


 



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