FOM: Does Mathematics Need New Axioms?

[Harvey Friedman]

Here is a version of remarks that I intend to make at the upcoming panel.

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

DOES MATHEMATICS NEED NEW AXIOMS?

This question has many versions. Some recent work is proving to be highly
relevant to some of these versions. There is not enough time to discuss
details of this relevant work. I have copies here of a detailed abstract
describing the recent work.

1. Is there a mathematically natural question that cannot be settled using
the current axioms?

Yes, as is well known. In particular, CH = continuum hypothesis. It is
known that if ZFC is consistent then CH cannot be settled in ZFC. One could
actually take this to be the definitive answer to the panel question, and
hence no need for any panel at all. But let us explore further.

2. Do we need new axioms in order to settle the continuum hypothesis?

Yes, as we have just said. However, let me mention a quibble. If ZFC is
inconsistent, then we don't need new axioms to settle the continuum
hypothesis, since already the present axioms suffice! But this only means
that we cannot prove within the usual axioms that 2 has an affirmative
answer. This should not stop us as scientists from concluding that we do.
If we get shocked by an inconsistency, we can then undergo theory revision.

3. Is there an axiom that we need to adopt in order to settle the continuum
hypothesis?

The most well studied hypothesis that settles the continuum hypothesis
(positively) is the "axiom" of constructibility, V = L. However, there are
other lesser known hypotheses which also settle the continuum hypothesis,
some positively and some negatively, and which are incompatible with V = L.
And there is some controversy as to which, if any, of these hypotheses
should be adopted as axioms. So in this sense, 3 either has a negative
answer, or at least does not have an affirmative answer at this time.

The hypotheses that settle the continuum hypothesis mostly fall into two
categories: inner model axioms, and forcing axioms. The former, of which V
= L is the most basic, prove CH, whereas the latter refute CH.

There is no assertion that settles the continuum hypothesis that is
currently being strongly put forth by the set theory community as a new
axiom.

4. Does mathematics need to settle the continuum hypothesis? Will the math
community adopt a new axiom in order to settle the continuum hypothesis?

This impertinent question needs to be asked if we are to conclude from 2
that mathematics needs new axioms. Here there really is an issue. The
independence of CH from ZFC has been known since 1962, yet there is now
virtually no attention paid to CH and related questions by the math
community. The most obvious explanation is that CH and related problems are
sharply disconnected from virtually all current mathematical activity. It
is definitely true that CH and related problems were once more connected
with an appreciable amount of mathematical activity than now. There has
been a general trend in mathematics, starting with the 60's, towards more
concrete questions, and this trend was underway before 1962 and is
accelerating.

In fact, the tacit attitudes of the math community towards generality have
now become clear. They do like to state theorems in great generality, but
only when convenient and elegant. However, once generality creates its own
difficulties which are atypical of important cases, the community loses
interest in the generality.

Thus settling problems like CH - as noble an enterprise for set theorists
as it is - is not going to form the basis for the math community accepting
a new axiom.

5. Do we need new axioms in order to settle basic questions about the
projective sets of real numbers?

Yes, as is well known. 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.

It is known that if ZFC is consistent then a) cannot be settled within ZFC;
if ZFC + "there exists a strongly inaccessible cardinal" is consistent then
b) cannot be settled within ZFC; if ZFC + "for all sets of integers x, x#
exists" is consistent then c) cannot be settled within ZFC.

Once again, we can quibble that we cannot prove in ZFC that we need new
axioms in order to settle these basic questions about PCA, coanalytic, and
analytic sets. But that should not stop us from concluding, as scientists,
that we do. If we get shocked by an inconsistency, we can then undergo
theory revision.

6. Is there an axiom that we need to adopt in order to settle basic
questions about the projective sets of reals?

Here we enter some controversy. As is the case with the continuum
hypothesis, there are hypotheses that settle these questions with different
answers. However, there is a difference, in that some of these hypotheses
(large cardinals) are being actively put forth by set theorists as new
axioms for mathematics. There is a minimum large cardinal axiom that serves
this purpose - infinitely many Woodin cardinals. The other kind of
hypotheses that settle these questions are inner model axioms, the simplest
of which is the axiom of constructibility; these settle the questions in
opposite ways, and even alternative ways, and even different ways from each
other.

In particular, the three favorites above in the low projective hierarchy
are settled positively with large cardinals and negatively with V = L and
related inner model axioms.

But the axiom of constructibility and related inner model axioms are
categorically rejected by set theorists as axioms for mathematics - they
strongly advocate rejection. The reason is that V = L is not really an
axiom, but a restriction - to replace the general concept of set with the
restricted concept of constructible set.

This kind of restriction is totally against the grain of the mainstream
thinking of set theorists, but, in my opinion, in agreement with mainstream
thinking of the math community. The math community will do just about
anything within the law to defend against the intrusion of foreign objects
into their domain, especially if they are not grounded in arithmetic,
algebraic, or geometric thinking. Thus if forced to adopt new axioms, V = L
would be the most attractive to them, settling the continuum hypothesis and
all related questions to boot. But we must ask an impretinent question:

7. Does mathematics need to settle basic questions about projective sets of
reals? Will the math community adopt a new axiom in order to settle basic
questions about projective sets of reals?

This impertinent question needs to be asked if we are to conclude from 7
that mathematics needs new axioms. Here there really is an issue. The
independence of some of these questions has been known since the early
70's, yet there is now virtually no attention paid to such questions and
related questions by the math community. The most obvious explanation is
that such questions - even in the low projective hierarchy - are sharply
disconnected from virtually all current mathematical activity. It is
definitely true that such questions were once more connected with an
appreciable amount of mathematical activity than now. There has been a
general trend in mathematics, starting with the 60's, towards more concrete
questions, and that movement is accelerating.

In fact, the tacit attitudes of the math community towards generality have
now become clear. They do like to state theorems in great generality, but
only when convenient and elegant. However, once generality creates its own
difficulties which are atypical of important cases, the community backs
away from the generality.

Thus settling such problems in descriptive set theory - as noble an
enterprise for set theorists as it is - is not going to form the basis for
the math community accepting a new axiom.

8. As far as the math community adopting new axioms is concerned, how does
CH and related questions compare with basic questions about projective sets?

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 - perhaps even into the math offices - in order to cause even the
consideration of the adoption of new axioms by the math community.

9. What kind of incompleteness of ZFC do we need in order that the math
community will be receptive to the adoption of new axioms?

The following features are preferable, if not outright necessary.

a. It 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.

b. It 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. It should be thematic. If it is isolated, it will surely be stamped as a
curiosity, and the math community will find a way to attack it through an
ad hoc raising of the standards for being entirely natural. However, if it
is truly thematic, then the theme must be attacked, which may be difficult
to do. For instance, it may be that it is an obvious thematic extension of
known basic and familiar facts.

d. It 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. It 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. It 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. Derivations should be accessible, with identifiable general techniques.
E.g., the proofs from large cardinals can be understood by any
mathematician without any experience in logic. The proofs should be easy to
understand and entirely natural, even to the point of being almost
familiar, with a basically 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.

10. Is there any prospects for such a magic bullet?

Boolean relation theory, and its expected extensions, seem to be developing
all of these features. These expected extensions are expected to give
Pi-0-1 statements about finite functions equivalent to the consistency of
any large cardinal axioms yet considered, including the highest ones
incompatible with the axiom of choice.

11. But are such results subject to some of the same objections? Can we now
conclude that large cardinals are needed to settle these new statements?

Apparently yes. The new statements will be proved from large cardinals, but
their negations do not seem to be provable from alternative hypotheses that
are axiom candidates. The obvious alternative hypothesis that proves their
negations is the inconsistency of large cardinals. But that is not an axiom
candidate if only because of its syntactic form, which is completely
unsuitable. Note that V = L is not a syntactic statement, and is an axiom
candidate - even if rejected by set theorists. V = L is an axiom of
restriction, whereas "large cardinals are inconsistent" is not an axiom of
restriction.

Also, whereas it is true that we cannot prove within ZFC that large
cardinals are needed to settle these new statements, this should not stop
us from acting as scientists and concluding that large cardinals are indeed
needed. If we get shocked by an inconsistency, then we can undergo theory
revision.

One can object that we should only conclude that the consistency or
1-consistency of large cardinals is needed to settle these new statements,
and not the large cardinals themselves. But those are syntactic statements,
utterly unsuitable as axiom candidates. So as far as the available list of
axiom candidates are concerned, it is clear that we are compelled to adopt
the large cardinals for this purpose.

We can be more formal about this in the following way. We say that we need
large cardinals in the sense that any formal system which proves the
statements, must interpret large cardinals.

We can also make the following practical scientific point. That settling
these statements is outright equivalent to settling the consistency (or
1-consistency) of large cardinals. In this way, the status of large
cardinals promises to becaome an essential topic for the math community.