FOM: Further comments on CH and "inherent vagueness"
John Case
case at eecis.udel.edu
Thu Dec 4 15:01:35 EST 1997
On Dec 4, 11:05, Torkel Franzen wrote:
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Subject: FOM: Further comments on CH and "inherent vagueness"
Date: Thu, 04 Dec 1997 11:05:23 +0100
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... explicit or implicit in our understanding of the world of sets.
^^^^^^^^^^^^^^^^^
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Torkel Franzen 
} End of excerpt from Torkel Franzen 






Again, the language (with its use of `the'  and `world' in the singular)
presupposes a unique standard model  which, _sadly_, we do not seem to
have.
In my just previous message to this group, I discussed power sets. Here, for
expository convenience, I'll discuss corresponding sets of characteristic
functions. I essentially indicated how to use the (implicitly) infinite
picture
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/ \
/ \
0 1
/ \ / \
/ \ / \
00 01 10 11
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to indicate we have a clear and definite denotation for `the standard model
of 2^N'. I also essentially indicated that I have trouble getting as good a
picture for 2^R, where R is the set of reals, etc. In the above model of 2^N,
I use the usual type omega wellordering of N to get the tree. If one has
some handy wellordering of R, one can use it to build such a tree
representation of 2^R (I think this is ok). I expect there are/could be
problems with this that I have not personally examined. There are problems
with getting wellorderings of R, problems, perhaps, with which order type to
use, ... . Maybe there are problems with sensitivity of the resultning models
of 2^R to which wellordering? Other problems? I also haven't reflected on
whether, to show elemetary facts re such models of 2^R, one has to resort to
knowing what one means by 2^(2^R) or some such stuff. Anyhow: has someone
looked at this bottom up approach to figuring out what might be pictorial
denotations for `the standard model of 2^R', `the standard model of
2^(2^R)', ... ? If so, are they problematic? Straightforward? Old hat and
useless? With standard coding of pairs as sets, one need only go a short ways
up this hierarchy to talk* about possible mappings between low levels. Hence,
IF one can succeed in getting THE standard models a little ways up this
hierarchy, one should be able to have a definite truth value for CH  even
if us mere humans subsequently have trouble finding out what that definite
truth value is. Set theory is not at all my field, so I have not tried the
above nor checked if others have. What is the situation about the above
approach?
Naively thinking,
(8 John
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* AND know what one is talking about, e.g., wrt unique standard models.
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