[FOM] CH and mathematics

Charles Silver silver_1 at mindspring.com
Sat Jan 19 19:20:46 EST 2008

	A question was raised about one of Solomon Feferman's questions in  
his online paper entitled: "Philosophy of mathematics: 5  
questions."   This was the Feferman question that was itself questioned:

                                        (i) Is the Continuum  
Hypothesis a definite mathematical problem?

What seemed puzzling to me was why Feferman's own succinct  
explanation was not taken into consideration as part of the ensuing  


"Concerning (i), I came to the conclusion some years ago that CH is  
an inherently vague
problem (see, e.g., the article (2000) cited above).  This was based  
partly on the results
from the metatheory of set theory showing that CH is independent of  
all remotely
plausible axioms extending ZFC, including all large cardinal axioms  
that have been
proposed so far.  In fact it is consistent with all such axioms (if  
they are consistent at all)
that the cardinal number of the continuum can be “anything it ought  
to be”, i.e. anything
which is not excluded by König’s theorem.  The other basis for my  
view is philosophical:
I believe there is no independent platonic reality that gives  
determinate meaning to the
language of set theory in general, and to the supposed totality of  
arbitrary subsets of the
natural numbers in particular, and hence not to its cardinal number.   
Incidentally, the
mathematical community seems implicitly to have come to the same  
conclusion: it is not
among the seven Millennium Prize Problems established in the year  
2000 by the Clay
Mathematics Institute, for which the awards are $1,000,000 each; and  
this despite the fact
that it was the lead challenge in the famous list of unsolved  
mathematical problems
proposed by Hilbert in the year 1900, and one of the few that still  
remains open.
I have been asked to explain what I mean by the statement of a  
problem being inherently
vague.  The idea is that, not only is it vague, but there is no  
reasonable way to sharpen the
notion or notions which are essential to its formulation without  
violating what the notion
is supposed to be about.  For example, the notion of feasibly  
computable number is
inherently vague in that sense.  And, for the statement of CH, the  
notion of arbitrary
subset of N can’t be sharpened to arbitrary constructible subset of  
N, or any specific
relativization thereof, without violating the idea of arbitrary  
subset of a set, independent
of any means of definition.  I think progress can be made on  
elaborating the idea of
inherently vague notions; whether that can be used to strengthen the  
case that CH is an
inherently vague problem remains to be seen. "

	(The rest of the article contains other relevant points.)

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