FOM: Does Mathematics Need New Axioms?

[Harvey Friedman]

Does Mathematics Need New Axioms?
ASL Meeting, Urbana
Panel discussion: June 5, 2000

Here is a third version of remarks that I intend to make at the upcoming
panel, and the second to appear as a posting on the FOM. Fifteen minutes is
all the time allotted to us for our opening remarks. That limitation has
caused me to refocus what I will say there, although this still needs to be
shortened.

Recall my posting of 10:59AM 5/1/00 (previous version) and also my review
of Feferman's article 6:05AM 4/26/00.

I intend to circulate some detailed abstracts of recent work at the
beginning of and during the Urbana meeting to interested participants in
hard copy form. I am still preparing these documents. These documents will
NOT contain any remarks concerning the panel discussion question, but
rather be self contained abstracts concerning results and conjectures that,
in my view, are relevant to the panel discussion and will be referred to by
me during the panel discussion. I will keep you fully informed of the
content of these documents, mostly surrounding Boolean relation theory.

***********************

DOES MATHEMATICS NEED NEW AXIOMS?

My 15 minute remarks will follow this outline.

1. A summary of my views.
2. Circumstances surrounding actual adoption of new axioms.
3. Two open questions in set theory.
4. Comments on some alternative views.

1. A summary of my position.

It is well known that mathematics needs new axioms in order to settle
various mathematically natural questions, some of which are well known open
problems - e.g., the continuum hypothesis (CH).

However, the mathematics community does not currently recognize the need
for new axioms. How do we reconcile these two facts?

Firstly, these questions differ so radically from those occurring in normal
mathematics in such easily recognizable and fundamental ways that they are
considered to be anomalies that can be systematically avoided without
restricting mathematics in any substantial way. This view is rarely
articulated, but is understood intuitively and instinctively without
explanation or examination.

Secondly, there is no specific axiom presented to the mathematics community
for the purpose of settling questions like CH. Thus the math community is
not being pressed to focus on a specific axiom candidate in connection with
problems such as CH.

Thus the issue of new axioms is not properly joined by CH and related
matters. As significant steps in the direction of joining the issue, let me
mention three favorite examples low down in the projective hierarchy.

a) every PCA set is Lebesgue measurable.
b) every uncountable coanalytic set has a perfect subset.
c) any two analytic sets of reals that are not Borel are Borel isomorphic.

By the early 70's, these were known to follow from the existence of a
measurable cardinal, yet independent of ZFC (under various consistency
assumptions).

These questions are significantly closer to normal mathematics than CH and
related questions. Furthermore, a specific axiom candidate has been
presented in this connection - the existence of a measurable cardinal.

However, it is still true that  these questions differ so radically from
those occurring in normal mathematics in such easily recognizable and
fundamental ways that they are considered to be anomalies that can be
systematically avoided without restricting mathematics in any substantial
way.

In particular, the level of interaction of such regularity properties of
projective sets with normal mathematics is nearly zero.

CH and related questions lie in intergalactic space, whereas the projective
hierarchy lies in the Milky Way and the lower projective hierarchy lies in
our Solar System. However, we need to get on Earth and into the math
buildings and into the math offices and perhaps onto the math desks - in
order to join the issue of new axioms with the math community.

Boolean relation theory and anticipated related developments will get into
the math buildings, and even into the math offices. This will prove
sufficient to join the issue of axiom candidates for many well known
mathematicians working in the normal arithmetic/algebraic/geometric
tradition. The next major breakthough along these lines (when?) will place
it on the mathematicians' desks, thereby joining the issue for virtually
all mathematicians.

Because of the thematic nature of these developments, and the interaction
with nearly all areas of mathematics, large cardinal axioms will be begin
to be accepted as new axioms for mathematics - with controversy. Use of
them will still be noted, at least in passing, for quite some time, before
full acceptance. This was the case with the axiom of choice. This will be
accompanied by various improved intrinsic justifications of large cardinals
along various lines.

2. Circumstances surrounding actual adoption of new axioms.

The cirumstances that I envision are a coherent body of consequences of
large cardinals of a new kind.

a. They should be entirely mathematically natural. This criterion is so
important because it is always met by even minor mathematicians, who
invariably publish either entirely natural - but perhaps not deep or
important - theorems, or partial results on entirely natural conjectures.
CH is entirely natural, but the subject matter is unsuitable for present
purposes.

b. They should be concrete. E.g., functions of several integer variables,
or continuous functions of several real variables. The more concrete the
better - e.g., piecewise linear functions of several integer variables, or
analytic functions of several real variables. Even better are linear
functions on halfplanes in several integer variables, or semialgebraic
functions of several real variables. Better still are finite functions on
finite sets of integers.

c. They should be thematic. If they are isolated, they will surely be
stamped as a curiosity, and the math community will find a way to attack
them through an ad hoc raising of the standards for being entirely natural.
However, if they are truly thematic, then the theme itself must be
attacked, which may be difficult to do. For instance, the same theme may
already be inherent in well known basic, familiar, and useful facts.

d. They should have points of contact with a great variety of mathematics.
This is important, because "out of sight, out of mind." The points of
contact spawn an endless stream of publications, opening up connections
with more and more fields, creating one reminder after another with more
and more subcommunities. These subcommunities talk to one another.

e. They should be open ended. This way, the math community has no idea
where the incompleteness will strike next. It will never be over. I.e., the
pain will never end until the adoption of large cardinals.

f. They should be elementary. E.g., at the level of early undergraduate or
gifted high school. That way, even scientists and engineers can relate to
it, so it is harder for the math community to simply bury it, keeping it
hidden from view.

g. Their derivations should be accessible, with identifiable general
techniques. E.g., the proofs from large cardinals should be readily
understandable by any mathematician without any experience in logic. The
proofs should be easy to understand and entirely natural, with a familiar
underlying technique - except for the use of large cardinals.  The relevant
large cardinals should be blackboxed in simple combinatorial terms. This
way, the math community can readily immerse itself in hands on crystal
clear uses of large cardinals that beg to be removed - but cannot.

We have omitted an additional curcumstance:

h. They should be used in normal mathematics as pursued before such
thematic results.

For some mathematicians, h will be required before they consider the issue
really joined. I already know that for some well known core mathematicians,
h is definitely not required - that the issue is already sufficiently
joined for them by Boolean relation theory.

Implicit in criteria a-g is that the body of examples and the theme launch
a new field, with an eventual AMS classification number, etcetera. This new
field will be accepted as part of the general unremovable furniture of
contemporary mathematics whose intrinsic interest is comparable to other
fields in mathematics which are generally accepted as important. In this
way, the issue of large cardinal axioms will be joined for a critical
number of important core mathematicians.

3. Two open questions in set theory.

The following are relevant to the panel discussion.

a. Prove that large cardinals provides a complete theory of the projective
hierarchy.

b. Prove that there are no "simple" axioms that settle the continuum
hypothesis.

4. Comments on some alternative views.

View A. For the purposes of the main question, we should ignore
distinctions between different kinds of mathematics for various reasons.

Firstly, it doesn't make sense to me to disucss the main question without
considering the views and attitudes of the mathematics community. After
all, who is going to adopt new axioms? If one chooses to ignore
mathematicians, then the question could have been put very differently:

Does set theory need new axioms?

Secondly, the distinctions are not only reasonably well defined, but also
are intrinsically important. There are apparently fixed fundamental views
and attitudes towards mathematical abstraction and generality that have
began in earnest in the 60's and have been strengthened since then.

Specifically, mathematicians do like to state theorems in great generality,
but only when convenient and elegant. But once generality creates its own
difficulties which are atypical of the important cases, the mathematicians
back away from the generality.

View B. 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 all
specialists in set theory worldwide that I know.

But V = L is properly viewed only 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, V = L is
much more in tune with them than large cardinal axioms are.

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.

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.

View C. Statements such as those which appear in Boolean relation theory,
which are provably equivalent to the 1-consistency of large cardinals, do
not suggest the adoption of the large cardinal axioms themselves, but
rather their 1-consistency.

When put into proper perspective, this is a criticism of form over
substance. Adopting large cardinals amounts to asserting

"every consequence of large cardinals is true."

Adopting the 1-consistency of large cardinals amounts to asserting

"every Pi-0-2 consequence of large cardinals is true."

The obviously more natural choice is to accept large cardinals, since the
latter is syntactic and not an attractive axiom candidate.

When I publish that "phi needs large cardinals to prove" I explicitly
formalized this as "any reasonable formal system that proves phi must
interpret large cardinals in the sense of Tarski." This gives a precise
sense to "needs."

There is an interesting point of some relevance here. Statements in Boolean
relation theory are also consequences of the existence of a real valued
measurable cardinal - another kind of large cardinal axiom.

Let me put it somewhat differently. There is a substantial and coherent
list of non syntactic axiom candidates, including large cardinal axioms and
other axioms. In this list, only certain axiom candidates settle questions
in Boolean relation theory. The most appropriate ones from various points
of view are in fact the small large cardinal axioms.  That is the obvious
move to make from the point of view of a working scientist. If they later
prove to be inconsistent, then we can undergo theory revision. The key
advance is that the issue of new axioms finally promises to get joined in a
serious way for the mathematics community.

View D. We should give up trying to obtain intrinsic justifications for
large cardinals.

I am optimistic about the program of finding convincing intrinsic
justifications of large cardinals via some non set theoretic notion such as
predication. There are some suggestive developments that make me optimistic
about this. Work on transfer principles shows how large cardinals can be
obtained by transferring statements about functions on the natural numbers.
Work on multiple worlds provides new kinds of axiomatizations of large
cardinals. In both cases, the large cardinals involve are big enough to
refute the axiom of constructibility.

View E. Naturalism strongly suggests that mathematics needs large cardinal
axioms.

I acknowledge that naturalism strongly suggests that set theorists need
large cardinal axioms. But at the moment, naturalism does not suggest that
mathematics - in any broader sense - needs large cardinal axioms. However,
I anticipate that to change radically through the development of Boolean
relation theory and its extensions. The naturalist point of view needs to
expand to incorporate normal mathematics. This requires confronting and
making sense of the proud anti-foundational anti-philosophical perspective
of the normal mathematician.