FOM: the Borel universe (a positive posting)

[Stephen G Simpson]

The recent flood of messages about the meaningfulness/vagueness of CH
does not seem very illuminating.  Let me attempt to steer this in a
more fruitful direction: the Borel universe (see below).

My springboard is Kanovei's reference to "Cinq lettres sur la th'eorie
des ensembles", an exchange of correspondence among Hadamard, Borel,
and Baire; published in Bulletin de la Societe Mathematique de France,
vol. 33, 1905, pages 261-273; reprinted in Borel's "Le,cons sur la
Th'eorie des Fonctions", 2nd edition, Gauthier-Villars, 1914.

The French analysts were deeply disturbed by set-theoretic pathology
flowing from the existence of a well-ordering of the reals.  They
considered various ways to remedy this pathology.  One of the remedies
that they considered was to modify the foundations by DISCARDING ALL
SETS OF REALS EXCEPT BOREL SETS.  Chapter IV of
Fraenkel/Bar-Hillel/Levy (written by van Dalen, I think) refers to the
French school of "semi-intuitionists".  Even if we reject the
semi-intuitionist proposal as overly dogmatic and radical, we can
still view it as interesting and fruitful.  This is because Borel sets
provide a rich context for classical analysis, yet are much better
behaved than arbitrary ZFC-style sets of reals.  For instance:

  1. There is no Borel well-ordering of the continuum.

  2. All Borel sets are Lebesgue measurable, have the property of
  Baire, etc etc.

  3. The continuum hypothesis holds for Borel sets:

     a. Every Borel set is either countable or contains a perfect set.
     Perfect sets are in canonical 1-1 correspondence to the Cantor
     set, hence are canonically of cardinality 2^{aleph_0}.  (This
     fails badly for arbitrary sets.)

     b. Silver: For every Borel equivalence relation E with
     uncountably many equivalence classes, there exists a perfect set
     of E-inequivalent points.  (This fails badly for arbitrary
     equivalence relations.)

  4. Martin: Borel games are determined.  (This fails badly for
  arbitrary games, the "axiom" of determinacy notwithstanding.)

  5. Harrington-Shelah: Every Borel partial ordering of the continuum
  is the union of countably many chains or has a perfect antichain.
  (This fails badly for arbitrary partial orderings of the continuum.)

  6. There are many other fascinating positive results about Borel
  sets along this line.  This has been an area of research for 90+
  years and has been especially active in the last 30 years right up
  to the present.  Some of the recent contributors are: Becker,
  Burgess, Friedman, Gao, Harrington, Hjorth, Kada, Kanovei, Kechris,
  Louveau, Marker, Martin, Sami, Shelah, Solovay, Stanley, Stern,
  Wadge, ....

  (Have I omitted anybody?)

By the Borel universe I mean a world where all sets of reals are
Borel.  The above results imply that the Borel universe is a much more
friendly and less pathological place than the ZFC universe.

I now propose an elegant axiomatization of a significant part of the
theory of the Borel universe.  Let's temporarily call it TBU_0.

The language of TBU_0 is that of third order arithmetic: natural
numbers, sets of natural numbers, sets of sets of natural numbers.
Within TBU_0, sets encode binary relations by means of a pairing
function.  The axioms of TBU_0 consist of a few trivial ones, plus the
following axiom: For every single-valued set X there exists a set Y
such domain(X)=Y.  (X is said to be single-valued if for all b there
exists at most one c such that (b,c) belongs to X.  domain(X)={a:
exists b such that (a,b) belongs to X}.)  This axiom expresses the
well-known fact that the domain of a single-valued Borel relation is a
Borel set.

In TBU_0, the intended range of the set variables is the Borel sets,
but analytic sets also have their place, as projections of Borel sets.
Many of the classical properties of Borel and analytic sets are
provable in TBU_0.  For instance, Souslin's theorem (every analytic
co-analytic set is Borel) can be expressed as a kind of third order
comprehension principle:

  forall X forall Y (forall a ( exists b X(a,b) <-> forall b Y(a,b) )

    -> exists Z forall a (a in Z <-> exists b X(a,b)) ) .

I can show that the second order part of TBU_0 is ATR_0.  ATR_0 is one
of a few mathematically natural subsystems of second order arithmetic
which arise in reverse mathematics.  It is the weakest natural system
in which various mathematical theorems can be proved.  See chapter V
of my forthcoming book on subsystems of second order arithmetic.

Unfortunately, some of the above-mentioned positive results about
Borel sets are non-classical, hence not provable in TBU_0, because
they require higher cardinals to prove.  The most famous result here
is that Borel determinacy requires aleph_1 cardinals.  More precisely,
the strength of determinacy for Borel sets of Borel rank gamma (a
countable ordinal number) is something like gamma applications of the
power set axiom, i.e. ZFC up to sets of set-theoretic rank gamma.
Friedman proved this in 1968, years before Martin's determinacy proof
in 1975.  More recently, Friedman and Stanley developed natural
statements about Borel sets whose strength is calibrated by large
cardinals (beyond ZFC).

[ Question for Harvey and Lee: What's the latest on this? ]

Kondo's theorem has an interesting status.  More on this later,
perhaps.

Much more recently, Hjorth (I think) showed that aleph_1 cardinals are
needed to prove Kanovei's theorem: the Borel constituents of a
non-Borel analytic set have unbounded rank.

[ Question for Vladimir and Greg: Have I got this right? ]

Research problem: Supplement TBU_0 with elegant axioms in the language
of TBU_0 to capture the bulk of the non-classical positive theorems on
Borel sets.

Does everybody agree that this is more fruitful than blathering on and
on about the meaningfulness of CH?

-- Steve