FOM: Further comments on CH and "inherent vagueness"

[John Case]

On Dec 4, 11:05, Torkel Franzen wrote:
                                  .
                                  .
                                  .
	Subject: FOM: Further comments on CH and "inherent vagueness"
	Date: Thu, 04 Dec 1997 11:05:23 +0100
                                  .
                                  .
                                  .
    ... explicit or implicit in our understanding of the world of sets.
	                                             ^^^^^^^^^^^^^^^^^
                                  .                          |
                                  .                          |
                                  .                          |
	  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
 
                                           .
                                         /   \
                                        /     \
                                       0       1
                                      / \     / \
                                     /   \   /   \
                                    00   01 10   11
                                           .
                                           .
                                           .

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 well-ordering of N to get the tree.  If one has
some handy well-ordering 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 well-orderings 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 well-ordering?  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

________________
* AND know what one is talking about, e.g., wrt unique standard models.