#### by Richard Lockhart

I was David Blackwell’s Ph.D. student from mid-1977 to early 1979, having asked him to supervise after hearing the clearest lectures I had, and have, ever heard. I was Ph.D. student number sixty-two of sixty-five, according to the Mathematics Genealogy Project, where it will be seen that David had four students finish in 1978, two in 1979, and two more in 1981. I remember the chair outside his office where I would wait for my weekly half hour. I remember that he suggested a problem to me by giving me a paper and saying he didn’t think he had done everything there that he could have. A few months later I gave up and asked to work on something he was doing currently — programmable sets.

David gave a talk at Stanford in the Berkeley–Stanford
colloquium series in which he described
these objects. I emerged from the talk amazed by
how clear and easy it all was. Trying to tell others
about it, I saw quickly that the lecture was a very
clear presentation of something not so easy at all
and that David had the capacity to organize arguments
far, far better than I. The idea is simple as he
described it. Imagine a computer represented as a
countably infinite sequence of lights. A computer
program is a function __\( f \)__ that turns off some lights.
Which lights are extinguished is a function of the
current pattern of on and off lights. You start with
some initial pattern of lights, __\( x_0 \)__ say, and let the
program run by computing __\( x_1 = f(x_0) \)__, __\( x_2 = f(x_1) \)__,
and so on inductively,
generating the sequence
__\( x_n; n = 0, 1, 2, \dots \)__. The sequence
__\( x_n \)__ decreases so
it has a limit, which we
call __\( x_{\omega} \)__. But for a general
function __\( f \)__ the program
might not be finished — __\( x_{\omega} \)__ need not be a fixed
point of __\( f \)__. So continue
applying __\( f \)__ inductively,
once for each countable
ordinal __\( \alpha \)__ arriving at __\( x_{\omega_1} \)__,
the limit at __\( \omega_1 \)__, the first
uncountable ordinal. The
limit must actually be
reached at some countable
ordinal, which may
depend on __\( x_0 \)__.

The iterates __\( x_{\alpha} \)__ are, of
course, iterates of the function __\( f \)__ applied to __\( x_0 \)__, so
that we may think of the functions __\( f_{\alpha} \)__ converging
(pointwise) to the limit function __\( f_{\omega_1} \)__, and we may
wonder what sorts of functions __\( g \)__ have the form of
such limits if the basic function __\( f \)__ is Borel? David
looked at the output program __\( f_{\omega_1} \)__ as a step in
defining general functions __\( g \)__ between two polish
spaces __\( U \)__ and __\( V \)__, which were defined by mapping __\( U \)__
into the computer’s state space by a Borel function,
running the program, and then mapping the result
into __\( V \)__ in a Borel way. Any function admitting such
a factorization is Borel programmable (BP), and
sets are BP if their indicator functions are. David’s
1978 paper, establishing the basic properties of
the sets and functions and showing that they had
potential as useful objects in probability theory,
is typical of his work — very clear, very concise,
questions not answered set forth, and not quite
four pages long.

In my thesis I solved one or two minor problems
connected with these ideas. I was trying to
clarify the relation between BP-sets, R-sets studied
by
Kolmogorov
and others, and C-sets. I wanted
to know, for instance, if letting the program, encoding,
and decoding functions be BP, rather than
just Borel, gave you a larger class of sets than the
BP class.
(John Burgess
did the problem properly
in *Fundamenta Mathematica* in 1981 using results
from logic concerning monotone operators, the
lightface and boldface set hierarchies and game
quantifiers.) I cannot remember how I came to realize
that I needed to think not about sigma fields
but about collections of sets like the analytic sets
which had fewer closure properties. I now think,
though, that David planted the seed of this idea
in my head one day — just by emphasizing the
importance of closure properties to me; I rather
suspect that my thesis would not have taken him
more than a few weeks to put together and that
the result would have been much shorter and far
clearer.

I wish I could say I knew him well, but my weekly meetings were all I knew — and they were all mathematics. I was young and shy and David was friendly but formal, I think. I wish we could all learn mathematics from people who see it as clearly as David did.