[FOM] 729: Consistency of Mathematics/1

[martdowd at aol.com]

Dmytro Taranovsky writes:

 


 think that GCH is true, and that once we sufficiently understand set 
theory, it will become obvious. However, it is possible that no 
consensus on CH will be reached until we have a true reasonably complete 
axiomatization of third order arithmetic.  I wrote about CH in previous 
FOM postings ("On the Continuum Hypothesis, part 1" and "On the 
Continuum Hypothesis, part 2"):
http://cs.nyu.edu/pipermail/fom/2006-May/010554.html
http://cs.nyu.edu/pipermail/fom/2006-May/010555.html
In simple terms, GCH makes for a more natural and coherent theory, and 
arguments against CH usually depend on applying intuitions beyond their 
domain of applicability.


 

 
I also have argued that CH is true, in
   http://onlinelibrary.wiley.com/doi/10.1002/malq.19930390111/pdf.
The argument is more or less a refinement of points 4 and 6 of the first posting cited above.  AC is a "non-constructive existence principle", stating the existence of a set which cannot be "constructed" using the other axioms of set theory.  Likewise, CH is a 
nonconstructive existence principle, stating the existence of an enumeration of P(omega) by aleph_1.

Unlike AC, this principle cannot be claimed to be clearly true.  However, if it were false there would be an injection of aleph_2 into P(omega).  To quote my paper, "One would surely expect that if such objects existed they would be apparent".  Also, if one temporarily forgets the construction of L and tries to construct such an embedding, one realizes that there is no embedding via well-orders of aleph_1, in contrast to the embedding of aleph-1 via well-orders of aleph_0.

A choice must be made between a well-ordering by aleph_1 and an injection of aleph_2.  In response to Harvey Friedman's posting initiating this thread, this would seem to qualify as one of "the most monumental issues in the foundations of mathematics".  Even at this point in the history of set theory, the foregoing argument seems to be a reasonable one.

Another fundamental question is whether there are uncountably many reals in L_aleph_1.  Again, it seems reasonable to argue that there must be.

Martin Dowd



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