FOM: Reformulation of errors JoeShipman at
Tue Sep 5 06:19:04 EDT 2000

In a message dated 9/5/00 1:42:27 AM Eastern Daylight Time, 
solovay at writes:

>> Joe,
    There are a number of technical errors in your comment on my last
 posting which I am going to point out. As per my usual custom, I shall
 abstain from "discussing the philosophy".
Thanks for your prompt response.  This will teach me not to compose postings 
at 1AM!
>>  Any assertion about what happens in an V(alpha) for alpha weakly
 characterizable can be phrased as saying that a closely related phi is
 second-order valid. In particular, Borel determinacy can be expressed as
 the fact that a certain sentence holds in V(omega+1).<<

Of course I should have seen (and in fact at other times HAVE seen) that only 
a sentence speaking about very large sets could possibly fail to be 
equivalent to the validity of a sentence of SOL.  But this is not strictly an 
error since I formulated it as a question, it is merely a stupid oversight.
>>  1) It is completely trivial to "reduce SOL to set-theory". That
 is, for each sentence phi of SOL, there is a closely related sentence phi'
 such that phi is valid in SOL iff phi' holds in V.
    Of course, it is certainly *not* true that all second-order
 validities are provable in ZFC. This is an immediate consequence of the 
 unsolvability of the halting problem for TM's and Godel's work on the
 expressibility of arithmetic. [Godel beta function, etc.]<<

Here of course I did not mean "reduce" in this trivial sense.  I meant that 
there would be some large enough set the first-order theory of which would 
suffice to determine second-order validity.  The point is that any such set 
would have to be extremely large.
 >> 2) It is standard to say that two structures are elemntarily
 equivalent iff they have the same sentences obtaining in them. Thus a
 trivial upper bound on the elementary equivalence classes of V(alpha)'s is
 c. In L, this bound is obtained. In the model where we start with L and
 generically collapse aleph_1 to aleph_0, the cardinality of this set is
 aleph_0. It is also trivial that there are always at least aleph_0 such
 elementary equivalence classes.<<

I got confused with the definition of elementary submodel, which is stronger 
than simply being a submodel satisfying the same sentences, but in any case I 
expected the answer to be at most c, I was just curious about what the 
possible values were between aleph-0 and c.

 > 2) On the other hand, there ARE theorems of ZFC which could not be derived 
 > even with an oracle for second-order validity, so the ZFC approach is 
 > stronger in a sense (my 5th question was to clarify this, though I'd like 
 > see a simpler example).
    Let's put it this way. If I had an oracle for the truth of
 second-order validities, all my practical curiosities about mathematical
 questions would be settled. There is much more information contained in
 the set of second order validiteies than in the theorems of ZFC. In
 particular, any question about V(omega + omega) would be settled. All the
 questions that concern Harvey's "ordinary mathematician" live there.<<

This may be true nowadays, but ordinary mathematicians used to be concerned 
about GCH, a statement about arbitrarily high ranks, before set theory had 
acquired its taint of nonabsoluteness.  Now GCH *is* equivalent to the 
validity of a sentence of SOL, but aren't there any other questions about 
sets of uncountable or arbitrary rank which have ever been considered 
important by an "ordinary mathematician"?

Maybe not -- V=L is too technical, so maybe only statements about cardinal 
arithmetic would be good enough, and they might be expressible in SOL the 
same way GCH is.

-- Joe Shipman

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