FOM: The Martin-Steel Theorem

[Joseph Shoenfield]

    This posting is an answer to a challenge to explain the 
significance of the Martin-Steel Theorem.
     In many years of teaching and writing about axiomatic set
theory, I have developed a picture of the subject.   The picture
has three chapters (up to the present).   The first consists of 
the formation of the axiom system ZFC and the development in it of
set theory through, say, elementary ordinal and cardinal theory.
A conclusion of this chapter is that virtually all of accepted
mathematics can be formulated and derived in ZFC.   This conclusion
has many applications, but it does not, in my opinion, shed much
light upon the nature of set theory.
     Chapter 2 is devoted to showing that certain statements are
unprovable in ZFC.   One does this by producing a model of ZFC in
which the statement is false; so the main problem is to find ways
of constructing models of ZFC with special properties.   The first
important step was taken by Godel, who in 1934 introduced a model
L.   Godel was able to prove that many results, such as GCH, were
true in L; this implied that their negations were not provable from
the axioms.   This work was continued by Jensen and his followers,
who made an intense study of the structure of L.
     The big step in Chapter 2 took place in 1963, when Cohen dis-
covered the technique of forcing.   This was a very versatile
method of constructing models of set theory in which a particular
statement is true or false.   Cohen applied it to CH, thus showing
the unprovability of CH from the axioms.   The method was developed
and extended by many researchers; they showed the independence of
virtually all the known unsolved problems as well as many new
problems.   This continues today due to the efforts of Shelah and
his followers. It is perhaps now drying up, not from a lack of
ingenuity by the researchers but from a lack of new unsolved problems.
     A natural problem suggested by Chapter 2 is to find new axioms 
which solve some of these problems; this is the subject of Chapter 3.
The problem was to find the new axioms.   There were not too many
candidates; and the known ones did not help much.   For example,
many were willing to at least consider large cardinal axioms; but
the large cardinals known at the time did not solve many problems.
Things changed in 1960, when Hanf showed that the measurable cardinals,
which had been studied earlier by Ulam and Tarski, were much larger
that previously cardinals.   It turned out that the existence of a
measurable cardinal has important consequences for set theory.   The
first such consequence, due to Scott, was that there is a set not in L;
many more followed.
    At this point, the theory of projective sets enters the scene.   The
basic theory was developed by researchers in the twenties; they showed
that sets at low positions in the projective hierarchy had certain
regularity properties, such as being measurable.   Godel had shown,
using L, that these results could not be extended in ZFC.   Solovay
proved that if there is a measurable cardinal, these regularity
properties extended to one higher level.   This suggested finding a
large cardinal which would extend them throughout the hierarchy.
With this in mind, Solovay defined supercompact cardinals; but he did
not have much success with them at this time.
    The next progress came from quite a different direction.   Some
Polish logicians had discovered the regularity property of deter-
minacy, and shown that it implied most of the known regularity pro-
perties.   This suggested the new axiom PD: all projective sets
are determinate.   It had many interesting consequences and appeared
to be consistent; but there was no other reason to accept it.
     The next task was clearly to find the relation between large
cardinal axioms and determinacy axioms.   The first result was due to
Martin: if there is a measurable cardinal, then every pi-1-1 set is
determinate.   Martin tried to use his methods to find a large car-
dinal which implied the determinacy of other projective sets.   He
obtained such a cardinal for pi-1-2 sets; but it was extremely large
(much larger than supercompacts), and set-theorist were reluctant to
use it.
     The big break came several years later.   In a paper by Foreman,
Maggidore, and Shelah, it was shown that some problems which had been
thought to require such very large cardinals to solve could be solved
by cardinals smaller than a supercompact.   This was taken up by Shelah
and Woodin, who showed that some of these not-so-large cardinals
implied regularity properties for the projective sets.
     The next step was clear: show that some of these cardinals imply
PD.   This was accomplished by Martin and Steel, who proved: If there
are n Woodin cardinals and a measurable cardinal larger than all of
them, then every pi-1-n+1 set is determinate.   Here a Woodin cardinal
is one of the not-so-large cardinals of the previous paragraph; it's
definition is a bit complicated.   Martin and Steel did not use the
techniues of the FMS paper; the new techniques which they developed
were suggested by the definition of a Woodin cardinal.
    To see that Martin-Steel really gives the connection between large
cardinals and determinacy, it was necessary to show that the large
cardinal hypotheses in Martin-Steel are minimal.   This cannot be done
by proving the converse of Martin-Steel, since no  determinacy hypothe-
sis implies the existence of large cardinals.   The right idea is to
extend Godel's method for showing regularity properties could not be
proved in ZFC.   For this, one needed a model like L which contains
Woodin cardinals. (By Scott's result, L does not contain a measurable.)
Models which extend L but have the propoerties needed to show non-
provability of regularity conditions are called core models (since the
essential property of the model is being the smalelst model of a
certain kind).   The efforts of Mitchell, Martin and Steel constructed
the required core models.   More recent work of Steel has extended our
knowledge of core models; but there is still much work to do here.
    None of the chapters of set theory will ever be completely
finished, since significant new theorems always suggest new problems.
I think, however, that the basic ideas of Chapter 3 are now in place
for the theory of sets of reals.   About sets of higher types, such as
sets of sets of reals, we know almost nothing (although there are a
couple of recent results on pi-2-1 sets).   We have no analogue of
determinacy, and no idea of what large cardinals (if any) will be
useful.   It would not surprise me if the solution to these problems
became the subject of Chapter 4.
     What impresses me in the whole story is how the solution of
problems leads to new concepts, which are then developed and, after
a time, integrated with the old concepts.   An example: Solovay's
original proof of the extension of regularity properties from a
measurable cardinal made use of forcing.   To me, this shows the
folly of trying to decide in advance of doing the mathematics what
the fundamental concepts of set theory are.