FOM: V=L vs. PD
John Steel
steel at math.berkeley.edu
Thu May 11 15:37:06 EDT 2000
I would like to continue my reply to Harvey's panel comments. Harvey
wrote:
>View B. (Most closely associated with John Steel). Work on the projective
>hierarchy demonstrates the need for large cardinal axioms.
>There is an inherent weakness in this argument that surrounds the axiom
>candidate V = L (axiom of constructibility), categorically rejected by
>almost all self respecting set theorists worldwide (Shelah is an
> exception).
>But V = L is only properly viewed as a restriction - to replace the
>general
>concept of set with the restricted concept of constructible set. And when
>viewed that way, it is very much in tune with basic attitudes of the
>mathematics community. Namely, to refocus on more concrete aspects of
>things when generality leads to great difficulties. In this way, it is
>much
>more in tune with them than are large cardinal axioms.
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?
>Pragmatically, V = L is particularly attractive in connection with the
>set
>theoretic problems in that it appears to settle all set theoretic
>questions
>- including those in the lower projective hierarchy.
There are two replies here:
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.)
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.
(Kechris, Moschovakis, and others have emphasized these points. I find
them convincing, but I find the next point more so.)
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.
In this light, we can see that adopting V=L in fact lets us settle NO
questions which weren't already settled by ZFC! 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!
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?
>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.
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.)
(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.)
John Steel
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