FOM: Re: V=L vs. PD

[Harvey Friedman]

Reply to Steel 12:37PM May 5/11/2000.

>    I agree that V=L is a restriction (more below). How "in tune" it
>is with the math community seems to me a sociological question entirely
>beside the point. It is not a rational restriction (more below). Is
>Harvey arguing that this is a rational restriction? Why or why not?

Let's put aside recent developments that promise to entirely change the
picture. There are two rational positions for the normal mathematician with
regard to new axioms for mathematics.

1. Add no new axioms for mathematics. This is very natural since there are
no statements in normal mathematics that cannot be settled within ZFC.

2. However, some normal mathematicians may not take such a strict attitude,
and regard the set theoretic problems as worrisome. In this case, V = L is
the obovious axiom to add since it is an axiom of restriction which serves
their purposes very well.

3. In particular, is very much in the spirit of the normal mathematician
who consistently backs off from abstraction and generality when it leads to
pathology. The normal mathematician is not interested in studying
abstraction and generality for its own sake, and wishes to completely rid
themselves of the incompleteness phenomenon once and for all. If this is to
be done by restriction, then all the better. Especially, if that
restriction rids oneself only of pathological objects. From the point of
view of a normal mathematician, there is not a single natural example of a
real number or set of integers or set of ordinals that is not known to be
constructible. So what normal mathematician wants to consider
nonconstructible sets if they cause problems?

4. An objection has been raised that V = L forever separates the normal
mathematician from certain constructions that may prove useful and
interesting to them later. But this is not true at the crucial practical
level.

Specifically, if events later prove that such things as measurable
cardinals turn out to be relevant for normal mathematics, then one can
later add an additional axiom to ZFC + V = L like this:

1) ZFC + V = L + there are arbitrarily long countable transitive models of
ZFC + there exists a measurable cardinal

and get the full power of having

2) ZFC + an actual measurable cardinal

for the purposes of normal mathematics.

This is because any Pi-1-3 consequence of 2) is also a Pi-1-3 consequence
of 1), and normal mathematics lives well within Pi-1-3.

>1. The descriptive set theory of projective sets one gets from V=L
>trivializes at precisely the point where it diverges from that
>given by PD. The DST given by PD is the natural extension of the
>DST of lower level projective sets which is provable in ZFC. Indeed,
>the development of the theory based on PD has led to a better
>understanding of the fragment of this theory which can be proved in ZFC.
>(Much of it can be seen as based on open determinacy, which is provable
>in ZFC.)

This paragraph is completely out of touch with the normal mathematician.
For a normal mathematician, there is no difference between, say, Borel sets
of reals and projective sets of reals. This is because there aren't any
natural examples - by the standards of a normal mathematician - of
projective sets that are not Borel sets.

One can come close to refuting what I just said by the result that there is
a Borel set A of reals such that the sum set A + A is analytic but not
Borel. However, which Borel set A? As far as I know, there are no natural
examples known of Borel A for which A + A is analytic and not Borel.

Since for a normal mathematician, the difference between projective sets
and Borel sets is one of pathology, the difficulties associated with the
statement

all projective sets are measurable

is associated with abstraction and generality and pathology, not genuine
mathematical issues.

So the fact that V = L makes most questions about the higher projective
hierarchy of a standard descriptive set theoretic nature trivial (for a
logician) is a positive development for the normal mathematician. It is
nice to see that pathology no longer causes difficulties.

>    On the other hand, the whole idea behind descriptive set theory(that
>definable sets of reals are well-behaved, free of the pathologies one
>can get from a wellorder of the reals) goes down the tubes under V=L.
>So in a sense, there is NO DST of higher level projective sets under V=L.

And from the point of view of the normal mathematician, this is good, since
the higher levels of the projective hierarchy only contain pathology as far
as he is concerned.

In fact, the overwhelming majority of normal mathematicians regard an
arbitrary Borel set or Borel function as a very general object that he/she
would dispense with in a second, the moment it becomes more of a problem
than it is worth. But, fortunately for Borel sets/functions, so much can be
done without messing with the axioms for mathematics, and with nice
understandable arguments, that they are tolerated. But the overwhelming
majority of normal mathematicians really only care about, say, piecewise
analytic functions (in the sense of power series), and perhaps the greatest
concentration strongly prefer to deal only with sets of solutions of
differential and partial differential equations, which are generally a lot
better behaved than merely analytic (power series) functions. Certainly
piecewise continuous functions are at the outer limits of normalcy for most
normal mathematicians.

>2. Even if there were a wonderful, useful DST of projective sets based
>on V=L (as there is in fact a wonderful infinitary combinatorics),
>adopting PD in no way eliminates it. The language of set theory as
>used by the believer of V=L can be interpreted in the language of
>set theory as used by the PD believer: one translates phi as (phi)^L.
>The V=L believer will agree that this preserves the meaning of the
>language as he is using it. In this way, all his mathematics becomes a
>chapter in the PD-believer's mathematics. There is, however, no
>meaning-preserving translation in the other direction.

It makes no sense for the normal mathematician to consider adopting PD or
large cardinals on the basis of the situation before some upcoming expected
developments take hold. Where do the winning strategies of the games
appear? Besides, infinite games are not normal mathematics anyways. In
addition, why get involved in axioms that raise serious issues about
consistency when one has V = L? In addition, PD and large cardinals do not
even solve the continuum hypothesis and related problems. So they are
weaker from the point of view of the normal mathematician than V = L. Of
course, upcoming developments promise to radically change the picture
forever. The ultimate forms that axioms should take for the normal
mathematician are up in the air, but they certainly involve some
formulation of large cardinals.

V = L can obviously be objected to for the normal mathematician on the
grounds that its formulation is technical. However, it can be appropriately
and convincingly blackboxed for them in this way:

*) It is the canonical strongest axiom of restriction compatible with the
usual axioms (ZFC).

>   In this light, we can see that adopting V=L in fact lets us settle NO
>questions which weren't already settled by ZFC!

What you propose above is not only awkward and roundabout, but as I said
above, raises concerns about consistency and also meaning - where do the
game strategies come from, and why infinite games in the first place?
Clearly *) is straightforward and convincing. In fact, *) is very much like
moves to look at, say, the real or complex algebraic numbers as the least
subfield of the real or complex numbers. A very familiar kind of move,
indeed! Of course, that move omits nonpathological objects like e and pi,
and cannot be used as a universal restriction on mathematics. However, L
can be so viewed since all of ZFC relativizes.

>Settling phi on the basis
>of ZFC + V=L is precisely the same as settling (phi)^L on the basis of
>ZFC. Adopting V=L just prevents us from asking as many questions!

And preventing us from asking questions - such as the ones you are
referring to - is exactly the kind of thing that the normal mathematician
would like to do - so that he/she can get back to
arithmetic/algebra/geometry or science/engineering applications or
computational issues.

>   Of course, it might have been reasonable to avoid asking about the
>world outside L ( compare the restriction V=WF given by the foundation
>axiom). But in fact, we have found all sorts of deep mathematical
>structure outside L. Adopting V=L amounts to prohibiting the investigation
>of that structure. Why would we want to do that?

Because this "deep mathematical structure" as you call it is simply "deep
set theoretic structure" - something the normal mathematician is deeply
disinterested in because of its lack of connection to
arithmetic/algebra/geometry or science/engineering applications or
computational issues.

I wrote:

>>However, V = L is powerless in connection with Boolean relation theory.
>>In
>>contrast to set theoretic problems, Boolean relation theory cannot be
>>finessed away by becoming more concrete or restricted.

Steel responded:

>   Boolean relation theory is not a very good example here, because it
>needs only Mahlos, which are consistent with V=L. But the basic point is
>one I agree with: in order to convince the die-hard V=L advocate that
>he needs more, you should prove theorems in the first order theory of
>L using large cards inconsistent with V=L. ( Or more dramatically, to
>convince the advocate of V= heredetarily finite sets, ...) A lot of
>Harvey's work goes in this direction.

>   ( By the way, I want to correct my last post, which suggested
>Boolean relation theory might be the first systematic, interesting,
>non-metamathematical, Pi^1_2 theory needing large cardinals. Harvey
>has done earlier work of this kind.)

Thank you.

>   (Aside: Martin first proved Borel determinacy is true in L using
>measurable cardinals. So at one point this was an example of the kind
>above, although later Martin got the hypothesis down to ZFC.)

History: Martin first proved BD is true in L using measurable cardinals and
actually kappa arrows alpha, alpha countable. I then proved that BD could
not be proved without uncountably many cardinals. Martin then proved that
Bd could be proved with uncountably many cardinals. I then gave a closely
related descriptive set theoretic statement with the same metamathematical
properties that is far more friendly to the normal mathematician:

Let E be a symmetric Borel subset of the unit square in the Euclidean
plane. Then E or its complement contains the graph of a Borel measurable
function.

Using all of the above work and some additional ideas, I proved that it was
necessary and sufficient to use uncountably many cardinals in order to
prove this statement, even if you assume V = L.

AN ADDITIONAL COMMENT OR TWO.

Normal mathematics is horizontal- not vertical. By this I mean that in
normal mathematics, the focus of attention is not on making theorems more
general - especially when no natural examples are covered by that added
generality. So passing from the Borel to the projective is a move that is
totally foreign to normal mathematics.

There certainly is a tradition in set theory and descriptive set theory
where it is rational to prefer large cardinals there to V = L.

Recent developments promise to radically change the picture with regard to
the normal mathematician. In partciular, if large cardinals are necessary
(in some form, e.g., reflection on them) for dealing with normal
mathematics that is accepted as normal by normal mathematicians, then the
entire picture changes.