# FOM: Reformulation of errors

JoeShipman@aol.com JoeShipman at aol.com
Tue Sep 5 06:19:04 EDT 2000

```In a message dated 9/5/00 1:42:27 AM Eastern Daylight Time,
solovay at math.berkeley.edu 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
to
> 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
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
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

```