FOM: Friedman, second order, CH, arithmetic and large cardinals ...

John Case case at
Tue Dec 2 14:35:42 EST 1997

1. Yes, I missed Harvey Friedman.  I've been out of touch with him and other
members of the fom community for too long.  

2. Professor Feferman says

   Incidentally, I don't agree with Case that statements of second-order
   arithmetic in general have a definite meaning. 

Whereas, I had parenthetically referred to the standard model for second-order
arithmetic.  Let see ... I'll indicate without details my basis for thinking 
`the standard model for second-order arithmetic' has denotation.  Responses to
this may indirectly shed light on Professor Tennant's question about others'
beliefs that

       the *component expressions* in CH had indefinite *senses*

Then, again, responses may merely end up being good for my education. (-8 
Either outcome (as well as some others) would be nice.

Consider the standard tree of characteristic functions of the members of
scriptP(N), the power set of the set of natural numbers depicted (in part)
below.  (I hope no ones mail server garbles this picture.)

                                         /   \
                                        /     \
                                       0       1
                                      / \     / \
                                     /   \   /   \
                                    00   01 10   11

As one follows any infinite branch downward from the root, the values on the 
nodes represent the successive values of the corresponding characteristic 
function on successively larger arguments.  For example, the infinite branch
downwards from the root which first goes right, then left, then right ... and 
so on alternating forever between right and left ... is the characteristic
function of the set of even natural numbers {0,2,3, ...}.  

My picture is finite. However, my finitely many words above it and the ellipses 
are to convey the (I believe) obvious infinite tree I have in mind.  I
(personally) am confident that this infinite tree is clear, unique, and
definite.  Furthermore, for me it presents a perfectly clear and definite
(even pictorial) standard model for scriptP(N):  scriptP(N) is just the 
collection of sets of natural numbers whose characteristic functions are the
infinite branches of the tree.  At least I conclude that `the standard model of
scriptP(N)' has denotation --- and a pretty clear and (infinite) pictorial one
at that.  In my thinking about denotation for `the standard model for 
second-order arithmetic' I had mostly worried about having denotation for 
`the standard model of scriptP(N)' since I imagined any problems were with the 
powerset operation, not with natural numbers, but I haven't worked out any 
more details.  )-8  

Now for some connection to CH:

When I try to get a clear and definite unique standard model of
scriptP(scriptP(N)) or of scriptP(the set of reals) I have problems getting 
a picture as good as the _infinite_ one _described_ above and which makes me 
think I actually know a unique thing that I'm talking about.  CH may for 
analysis of its component expressions require knowing for sure what one is 
talking about when one speaks of _more than one_ application of the powerset 
operation applied to N.  It would be neat if there were some other analysis of 
it that avoided problems of unique standard models of too many iterations of 
powerset applied, e.g., to N.  

3.  Again, I've been too removed from this fom community.  I'd be very grateful
if Professor Tennant or Harvey would slip me bib ref(s) or copies of Harvey's
results on statements of arithmetic equivalent to consistency of existence of
large cardinals.  My various coordinates are below.

(-8 John

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