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DOES MATHEMATICS NEED NEW AXIOMS?
ASL Meeting, Urbana
Panel discussion: June 5, 2000
Harvey M. Friedman
The point of view of the set theory community is well
represented here. I want to concentrate on the perspective of
mathematicians outside set theory.
1. MATHEMATICIANS’ VIEWPOINT.
New axioms are needed in order to settle various mathematically natural questions. Yet no well known mathematician
outside set theory is even considering adopting any new
axioms for mathematics, even though they are aware of at
least the existence of the independence results.
The difference in perspective of set theorists versus mathematicians who are not set theorists, is enormous. Recall that
mathematics goes back, say, 2,500 years  whereas set theory
in the relevant sense dates back to the turn of the 20th
century.
For 2,500 years, mathematicians have been concerned with
matters of counting and geometry and physical notions.
These main themes gave rise to arithmetic, algebra, geometry,
and analysis.
The interest in and value of mathematics is judged by
mathematicians in terms of its relevance and impact on the
main themes of mathematics.
It is generally recognized by most mathematicians that set
theory is the most convenient vehicle for achieving rigor in
mathematics.
For this purpose, there has evolved a more or less standard
set theoretic interpretation of mathematics, with ZFC
generally accepted as the current gold standard for rigor.
It is simply false that a number theorist is interested in
and respects set theory just as they are interested in and
respect group theory, topology, differential geometry, real
and complex analysis, operators on Hilbert space, etcetera. 2
The reason for this attitude is quite fundamental and extremely important. A number theorist is of course interested
in complex analysis because he uses it so much. But not so
with operators on Hilbert space. Yet there is still a distant
respect for this because of a web of substantive and varied
interconnections that chain back to number theory. Set theory
does not have comparable interconnections.
For the skeptical, the degree of extreme isolation can be
subjected to various tests including citation references broken down even into their nature and quality. Using a
critical notion from statistics, set theory is an extreme
outlier.
Nor is set theory regarded as intrinsically interesting to
mathematicians, independent of its lack of impressive interconnections. Why?
For the mathematician, set theory is regarded as a convenient way to provide an interpretation of mathematics that
supports rigor. A natural number is obviously not a set, an
ordered pair is obviously not a set, a function is obviously
not a set of ordered pairs, and a real number is obviously
not a set of rationals.
For the mathematician, mathematics is emphatically not a
branch of set theory. The clean interpretation of mathematics into set theory does not commit the mathematician
to viewing problems in set theory as problems in mathematics.
The mathematician therefore evaluates set theory in terms of
how well it serves its purpose  providing a clean, simple,
coherent, workable way to formalize mathematics.
This point of view hardened as many mathematicians experimented for several decades with what has come to be known as set
theoretic problems which turned out to be independent of ZFC.
There was a growing realization that the cause of these
difficulties was excessive generality in the formulations of
the problems which allowed for pathological cases which were
radically different in character than normal mathematical
examples. That if the problems were formulated in more
concrete ways that still covered all known interesting cases,
then the difficulties completely disappeared. 3
Furthermore, distinctions between these set theoretic
problems causing difficulties and the most celebrated
theorems and open problems in mathematics can be given FORMALLY. This is in terms of quantifier complexity and the
closely related matter of absoluteness. Thus set theory comes
out as an extreme outlier which can be documented FORMALLY.
2. THE MALIGNED AXIOM OF CONSTRUCTIBILITY  MORE IS LESS AND
LESS IS MORE.
The set theorist is looking for deep set theoretic phenomena, and so V = L is anathema since it restricts the set
theoretic universe so drastically that all sorts of phenomena are demonstrably not present. Furthermore, for the set
theorist, any advantage that V = L has in terms of power can
be obtained with more powerful axioms of the same rough type
that accomodate measurable cardinals and the like  e.g., V
= L(m), or the universe is an inner model of a large
cardinal.
However, for the normal mathematician, since set theory is
merely a vehicle for interpreting mathematics so as to
establish rigor, and not mathematically interesting in its
own right,
the less set theoretic difficulties and phenomena the better.
I.e., less is more and more is less. So if the mathematician
were concerned with the set theoretic independence results and they generally are not  then V = L is by far the most
attractive solution for them.
This is because it appears to solve all set theoretic
problems (except for those asserting the existence of sets of
unrestricted cardinality), and is also demonstrably
relatively consistent.
Set theorists also say that V = L has implausible consequences  e.g., there is a Sigma12 well ordering of the reals,
or there are nonmeasurable PCA sets.
The set theorists claim to have a direct intuition which
allows them to view these as so implausible that this provides “evidence” against V = L. 4
However, mathematicians disclaim such direct intuition about
complicated sets of reals. Many say they have no direct
intuition about all multivariate functions from N into N!
3. QUESTION ANSWERED BY CLASSICAL DESCRIPTIVE SET THEORY?
The classical descriptive set theory coming from large cardinals is most often cited by set theorists as the reason why
mathematics needs large cardinal axioms. I have several
objections to this claim.
a. Part of the argument is that large cardinals are needed to
establish these results. But large cardinals are not needed
to establish an alternative series of such results. E.g., V =
L provides another, entirely different, set of answers to
these questions. The set theorists answer saying V = L gives
the wrong answers and large cardinals give the right answers,
citing their direct intuition about projective sets of reals.
I am very dubious about this direct intuition. I don’t have
it, and mathematicians in general disclaim it.
b. Another part of the argument is that, in light of a,
set theory needs large cardinals
and therefore
mathematics needs large cardinals.
But this inference depends on a reading of our question that
makes this tautological.
Reading the question this way simply avoids the really interesting questions, replacing them by much less interesting
questions. For instance, it avoids questions of how and under
what circumstances the general mathematical community or
individual mathematicians will adopt new axioms, should adopt
new axioms, and if so, how this will be manifested.
Here is the closest I can come to the set theorists’ point of
view on our question.
There is an interesting notion of “general set theory in its
maximal conceivable form” and that V = L has no basis in this
context. However, the notion is at present virtually
completely unexplained, and no work that I have seen provides 5
any serious insight into what this really means. We simply do
not know how to explicate any relevant notion of maximality.
I agree that
“general set theory in its maximal conceivable form” needs
large cardinals axioms
is very likely to be true. But I can’t conclude even that
set theory needs large cardinal axioms
let alone
mathematics needs large cardinal axioms.
4. GENERAL PREDICTIONS.
The picture is going to change radically with the new Boolean
relation theory and related developments, joining the issue
of new axioms and the relevance of large cardinals in a
totally new and unexpectedly convincing way.
Because of the thematic nature of these developments, and the
interaction with nearly all areas of mathematics, large
cardinal axioms will 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.
5. 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
standard is very high for a logician trying to uncover such
consequences, yet is routinely met in mathematics (set theory
included) by professionals at all levels of achievment.
b. They should be concrete. At least within infinitary discrete mathematics. Most ideally, involving polynomials with
integer coefficients, or even finite functions on finite sets
of integers. 6
c. They should be thematic. If they are isolated, they will
surely be stamped as curiosities, 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.
e. They should be open ended. 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 them.
g. Their derivations should be accessible, with identifiable
general techniques. 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 ag is that the body of examples and the
theme launch a new area, with an eventual AMS classification
number. This new area will be accepted as part of the general
unremovable furniture of contemporary mathematics whose
intrinsic interest is comparable to other established areas
in mathematics. In this way, the issue of large cardinal
axioms will be joined for a critical number of important core
mathematicians.
6. LARGE CARDINALS OR THEIR 1CONSISTENCY? 7 The statements coming out of Boolean relation theory are
provably equivalent to the 1consistency of large cardinals.
So instead of adopting the large cardinal axioms themselves,
one can instead adopt their 1consistency.
When put into proper perspective, this is more of a criticism of form than over substance. Adopting large cardinals
amounts to asserting
"every consequence of large cardinals is true."
Adopting the 1consistency of large cardinals amounts to
asserting
"every ’02 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.
However, for the purposes of proving ’02 sentences, these
two choices are essentially equivalent.
Another consideration is more practical. When the working
mathematician wants to develop Boolean relation theory, the
proofs are incomparably more direct and mathematically
elegant when done under the assumption of the large cardinal
axioms themselves than under the 1consistency.
When I publish that "j needs large cardinals to prove" I
explicitly formalized this as "any reasonable formal system
that proves j 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  a related
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. 8
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.
APPENDIX
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.
Here a major challenge is to come up with an appropriate
definition of “complete.”
b. Prove that there are no "simple" axioms that settle the
continuum hypothesis.
Here I mean “simple” in the same sense that the axioms of ZFC
are “simple.” For example, very short in primitive notation. ...
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