FOM: Do We Need New Axoms? Some Quotes
friedman at math.ohio-state.edu
Sun Feb 13 20:16:28 EST 2000
The purpose of this posting is to give source material in the form of
direct quotes that lie behind my impressions of the views of Sol Feferman
and Penelope Maddy that I gave in my posting of 12.13PM 2/11/00 Do We Need
New Axioms? Upcoming Panel Discussion.
NOTE: This fulfils an offline request I received .
All references from Maddy quoted here are from her book:
Naturalism in Mathematics, 1997, Clarendon Press, Oxford.
All reference from Feferman quoted here are from his article:
Does Mathematics Need New Axioms?, American Mathematical Monthly, vol. 106,
no. 2, pp. 99-111, February 1999.
>1. Maddy. My impression of her view is this. There is an obviously
>fundamental problem in set theory of great interest in and out of
>mathematics, namely the continuum hypothesis. It is known that we cannot
>settle the continuum hypothesis using the usual axioms for mathematics as
>represented by ZFC - unless those axioms are inconsistent. Since ZFC is
>consistent, we therefore need new axioms to settle the continuum
Maddy, page 62:
"All the objects and theorems of classical mathematics, and much more, can
be derived in this system [ZFC]. The next order of business is to see why
these very powerful axioms are not powerful enough."
Maddy, page 63:
"The first and most famous of the independent questions is Cantor's
Continuum Hypothesis (often abbreviated CH), which was first formulated in
a weak form in Cantor (1878)."
>Maddy is aware that many mathematicians are not interested in the continuum
>hypothesis. However, Maddy would emphasize the overriding fundamental
>nature of the continuum hypothesis, and thereby give a strong affirmative
>answer to "Do We Need New Axioms?"
Maddy, page 69:
"This study of simple sets of reals -- initiated by the analysts Borel,
Baire, and Lebesgue -- is now called "descriptive set theory". Though some
of its fundamental questions have been shown to be independent of ZFC, like
CH, it is wsorth noting that the questions themselves (especially the
question of Lebesgue measurability) are significantly more down-to-earth.
The notions involved -- Borel set, Lebesgue measure, projective set --
arose in the ordinary pursuit of natural questions in analysis; by
comparison, Cantor's CH involves one of his greatest conceptual
innovations, the notion of infinite cardinality. Furthermore, the sets of
reals involved are simply definable; by contrast, CH essentially involves
the notion of all arbitrary, combinatorially-determined subsets of an
infinite set. So it is striking that simpler questions like these also turn
out to be independent of ZFC."
>As a "naturalist", Maddy would be expected to be interested in taking into
>*identifiable reasons why mathematicians are not interested in the
>in her view on these matters. Working primarily in the philosophy
>community, Maddy is unlikely to have had detailed interactions with
>mathematicians who are not interested in the continuum hypothesis, and so
>probably has not worked the disinterest of the mathematicians in the
>continuum hypothesis into her views yet. However, Maddy might be interested
>in doing this in connection with this upcoming panel discussion.
Maddy starts Chapter 4, Mathematical Naturalism, with:
"The version of naturalism to be outlined here begins, just as Quine's
does, within antural science; it takes the actual methods of natural
science as its own."
Maddy, page 184:
"...my naturalist takes mathematics to be independent of both first
philosophy and natural science (including the naturalized philosophy that
is continuous with science) -- in short, from any external standard."
Maddy, page 185:
"As mathematical naturalists, then, we approach mathematics on its own
terms. ... we now ask what mathematical practice can tell us about the
ontology of mathematics."
Maddy, page 191:
"At this point, I propose the following simple-minded historical analysis.
Many of the methodological debates in question have been settled: ... On
the other hand, the philosophical questions remain open: ... My point is
simply that the methodological debates have been settled, but the
philosophical debates have not, from which it follows that the
methodological debates have not been settled on the basis of the
"In fact, I think the historical record gives a fairly clear indication of
what did finally resolve the methodological debates: impredicative
definitions are needed for a classical theory of real numbers (among other
things)... the Axiom of Choice is so fruitful in so many branches of
mathematics that mathematicians refused to give it up ... In other words,
these debates were decided on straightforwardly mathematical grounds. With
the philosophical questions floating free, neither deciding nor decided by
the methodological conclusions of the practice, our mathematical naturalist
relinquishes her last hope of ontological guidance from the practice of
mathematics. Furthermore, our pursuit of mathematical naturalism has led us
to the same suggestion that tempted us at the end of ... if you want to
answer a question of mathematical methodology, look not to traditionally
philosophical matters about the nature of mathematical entities, but to the
needs and goals of mathematics itself."
"A passing mea culpa. Set theoretic realism was intended to be a
'naturalistic' theory: e.g., it steadfastly refuses to recommend reform of
mathematics on philosophical grounds; it scrupulously adheres to
epistemology naturalized. We've already seen that it failed in this goal by
relying on an indispensability argument that is at odds with scientific and
mathematical practice. But now another failing comes into view: though it
recommends no reforms, it does attempt to defend mathematical practice on
the basis of a philosophical realism about sets. It took me a very long
time to realize that if philoosophy cannot criticize, it cannot defend,
>The mainline set theorists argue that large cardinals are good axioms and
>the axiom of constructibility is bad. Maddy has tried to formalize this
>view, in terms of a principle she calls MAXIMIZE.
Maddy, page 210:
"The pay-off of these observations for set-theoretic methodology is simple.
If mathematics is allowed to expand freely in this way, and if set thoery
is to play the hoped-for foundational role, then set theory should not
impose any limitations of its own: the set theoretic arena in which
mathematics is to be modelled should be as generous as possible; the set
theoretic axioms from which mathematical theorems are to be proved should
be as powerful and fruitful as possible. Thus, the goal of founding
mathematics without encumbering it generates the methodological admonition
>It is my impression that Feferman disagrees with virtually
>every view of Maddy and Steel on these matters. My impression is that
>Feferman is not only aware of the disinterest of the mathematicians in the
>continuum hypothesis and the projective hierarchy, but feels that their
>attitude is fully justified in that the continuum hypothesis in particular
>is not clearly meaningful. Feferman has, in fact, come at least close to
>stating that it is meaningless. Specifically, he regards it as of such a
>fundamentally different character than any currently celebrated
>mathematical conjecture as to cause its nonprovability and nonrefutability
>from ZFC to be irrelevant to the appropriate meaning of the question "Do We
>Need New Axioms?"
Feferman, page 99:
"Logicians have learned a great deal in recent years
that is relevant to G¨odel's program, but there is considerable disagreement
about what conclusions to draw from their results. I'm far from unbiased
in this respect, and you'll see how I come out on these matters by the end
of this essay, but I will try to give you a fair presentation of other
along the way so you can decide for yourself which you favor."
Feferman, page 107:
"But the striking thing, despite all this progress, is that contrary to
G¨odel's hopes, the Continuum Hypothesis is still undecided by these further
axioms, since it has been shown to be independent of all remotely plausible
axioms of infinity, including MC, that have been considered so far (assum-
ing their consistency). That may lead one to raise doubts not only about
G¨odel's program but also about its very presumptions. Is CH a definite
problem as G¨odel and many current set-theorists believe? Is the continuum
itself a definite mathematical entity? If it has only Platonic existence, how
can we access its properties? Alternatively, one might argue that the con-
tinuum has physical existence in space and/or time. But then one must ask
whether the mathematical structure of the real number system can be iden-
tified with the physical structure, or whether it is instead simply an
mathematical model of the latter, much as the laws of physics formulated in
mathematical terms are highly idealized models of aspects of physical real-
ity. (Hermann Weyl raised just such questions in his 1918 monograph Das
Kontinuum, .) But even if we grant some kind of independent existence,
abstract or physical, to the continuum, in order to formulate CH we need to
refer to arbitrary subsets of the continuum and possible mappings between
them, and then we are dealing with objects of a higher level of abstraction,
the nature of whose existence is even more problematic than that of the
Feferman, page 107:
"While G¨odel's program to find new axioms to settle CH has not been re-
alized, what about the origins of his program in the incompleteness results
for number theory? As we saw, throughout his life G¨odel said we would
need new, ever-stronger set-theoretical axioms to settle open arithmetical
problems of even the simplest, purely universal, form---problems he called of
Goldbach type. Indeed, the Goldbach conjecture can be written in that form.
But the incompleteness theorem by itself gives no evidence that any open
arithmetical problems---or, equivalently, finite combinatorial problems---of
mathematical interest will require new such axioms. I emphasize the `math-
ematical interest', because G¨odel's own examples of undecidable statements
for each consistent S extending PA were of two kinds: the first, ` S ,
by a diagonal construction in order to establish incompleteness and evidently
true by the very theorem that it is used to prove, and the second, Con(S),
of definite metamathematical interest, but not of mathematical interest in
the ordinary sense of the word. "
"Beginning in the mid-1970s, logicians began
trying to rectify this situation by producing finite combinatorial statements
of prima-facie mathematical interest that are independent of such S. The
first example was provided by Jeff Paris and Leo Harrington who showed
in  that a modified form (PH) of the finite Ramsey theorem concerning
existence of homogeneous sets for certain kinds of partitions is not provable
in PA. PH is recognized to be true as a simple consequence of the infi-
nite Ramsey theorem; its independence rests on showing that PH implies
Con(PA); in fact PH is equivalent to 1-Con(PA)... While in each case, the
statement phi shown independent of S is equivalent to its 1-consistency,
the argument for the truth of phi is by ordinary mathematical
"Friedman later found a finite version of Kruskal's theorem KT which is
ATR0 . The infinitary theorem KT, a staple of graph-theoretic
combinatorics, asserts the
well-quasi-ordering of the embeddability relation between finite trees.
Friedman's work in
this respect is reported in ."
"Friedman found an extended version EKT of KT which is independent of the
impredicative Pi-1-1 comprehension principle in analysis (cf. ). EKT
later turned out to have close mathematical and metamathematical
relationships with the graph minor theorem of Robertson and Seymour, as
shown in ."
Feferman, page 109:
"To conclude, I hope I have given you some food for thought that will
help you come to your own conclusions about whether questions like the
Continuum Hypothesis are determinate, and, if so, what is going to settle
them, given that present axioms are insufficient. At the beginning of this
piece I promised to tell you my own views of these matters. By now, you
have probably guessed what these are, but let me say them out loud: I am
convinced that the Continuum Hypothesis is an inherently vague problem
that no new axiom will settle in a convincingly definite way. Moreover, I
think the Platonistic philosophy of mathematics that is currently claimed
to justify set theory and mathematics more generally is thoroughly unsatis-
factory and that some other philosophy grounded in inter-subjective human
conceptions will have to be sought to explain the apparent objectivity of
mathematics. Finally, I see no evidence for the practical need for new ax-
ioms to settle open arithmetical and finite combinatorial problems. The
example of the solution of the Fermat problem shows that we just have to
work harder with the basic axioms at hand."
>I am not sure what Feferman's view is with regard to the meaningfullness or
>meaninglessness of standard statements about the projective hierarchy such
>as the Lebesgue measurability of projective sets and the like. Such
>statements have been shown to follow from large cardinals, whereas it is
>already clear from Godel that such statements are refutable from the axiom
>of constructibility. I don't know if Feferman puts such statements in the
>same category as the continuum hypothesis.
Feferman, page 109:
"CH is just the most prominent example of many set-theoretical statements
consider to be inherently vague. Of course, one may reason confidently
within set theory
(e. g. , in ZFC) about such statements as if they had a definite meaning."
>Nevertheless, it is my impression that Feferman also regards these as so
>fundamentally different in character than currently celebrated mathematical
>conjectures, that their nonprovability and nonrefutability from ZFC does
>not bear on the appropriate meaning of the question "Do We Need New Axioms?"
Feferman, page 109:
"Moving beyond the domains of arithmetic and finite combinatorics, what
is the evidence that we might need new axioms for everyday mathematics?
Here it is certainly the case that various parts of descriptive set theory
have been shown to require higher axioms of infinity, in some cases well
beyond the range of ``small'' large cardinals. But again we are in a question-
begging situation, since our belief in the truth of these new results depends
essentially on our belief in the consistency or correctness to some extent or
other of these ``higher'' statements. Also, I think it is fair to say that
kinds of results are at the margin of ordinary mathematics, that is of what
mathematicians deal with in daily practice. 5 What is not at the margin can
be readily formalized within ZFC, and in fact in much weaker systems, as
has been demonstrated by many case studies in recent years."
>It is also my impression that in Feferman's view, philosophical coherence,
>explainability, and direct connections with matters of current interest to
>the mathematics community must be crucial issues for mainline set theory
>research. It is my impression that for Feferman, they must play a role in
>the direction and evaluation of that research. And it is my impression that
>on these grounds, Feferman believes that current research in mainline set
>theory is deficient.
>It is my impression that Feferman does not recognize any canonical nature
>of the large cardinal hierarchy as the unique extension and extrapolation
>of the ideas behind the axioms of ZFC. Nor does Feferman find any reaons
>put forth by the set theory community that large cardinals exist or are
Feferman page 106:
"Higher axioms of infinity, or so-called ``large cardinals'' in set theory have
been the subject of intensive investigation since the 1960s and many new
kinds of cardinals with special set-theoretical properties have emerged in
these studies. A complicated web of relationships has been established, as
witnessed by charts to be found in the recent book by Aki Kanamori, The
Higher Infinite [16, p. 471], and the earlier expository article by Kanamori
and Menachem Magidor . A rough distinction is made between ``small''
large cardinals, and ``large'' large cardinals, according to whether they are
weaker or stronger, in some logical measure or other, than measurable car-
dinals. Attempts to justify acceptance of both kinds of cardinals have been
made by set theorists involved in this development. The philosopher, Pene-
lope Maddy, in two interesting articles called ``Believing the axioms'', an-
alyzed the various kinds of arguments for these and other kinds of strong
axioms and summarized the evidence for them ."
"The elaboration of this subject has almost outrun the names that have been
duced for various large cardinal notions, witness (in roughly increasing
order of strength):
`inaccessible', `Mahlo', `weakly compact', `indescribable', `subtle',
`measurable', `strong', `Woodin', `superstrong', `strongly compact',
most huge', `huge', and `superhuge'."
>In fact, it is my impression that Feferman goes much further. It is my
>impression that Feferman dismisses the ongoing efforts to uncover natural
>questions in discrete mathematics that are neither provable nor refutable
>in ZFC on two grounds.
>Firstly, the questions in discrete mathematics that are dealt with in this
>work are, as I understand Feferman, not sufficiently natural in order to
>bear on the question "Do We Need New Axioms?"
>Secondly, the questions in discrete mathematics are proved using certain
>large cardinal axioms. But Feferman is not convinced that these large
>cardinal axioms are true, or even consistent. In fact, Feferman raises the
>possibility that these questions in discrete mathematics are outright
>refutable with no additional axioms (which would be the case if these large
>cardinal axioms are inconsistent). Therefore, as I understand it, Feferman
>concludes that these examples, even if they were sufficiently natural -
>which in his view they are not - would not clearly answer the question "Do
>We Need New Axioms?" in the affirmative.
Feferman, page 108:
"For some years, Friedman has been trying to go much farther, by pro-
ducing mathematically perspicuous finite combinatorial statements phi whose
proof requires the existence of many Mahlo axioms or even stronger axioms
of infinity and has come up with various candidates for that ( contains
the latest work in this direction). From the point of view of metamathe-
matics, this kind of result is of the same character as the earlier work just
mentioned; that is, for certain very strong systems S of set theory, the phi
produced is equivalent to the 1-consistency of S. But the conclusion to be
drawn is not nearly as clear as for the earlier work, since the truth of phi is
now not a result of ordinary mathematical reasoning, but depends essen-
tially on acceptance of 1-Con(S). It is begging the question to claim this
shows we need axioms of large cardinals in order to settle the truth of such
phi, since our only reason for accepting that truth lies in our belief in the
1-consistency of those axioms. However plausible we might find that, per-
haps by some sort of picture we can form of the models of such axioms, it
doesn't follow that we should accept those axioms themselves as first-class
mathematical principles. Finally, we must take note of the fact that up to
now, no previously formulated open problem from number theory or finite
combinatorics, such as the Goldbach conjecture or the Riemann Hypothesis
or the twin prime conjecture or the P=NP problem, is known to be inde-
pendent of the kinds of formal systems we have been talking about, not
even of PA. If such were established in the same way as the examples (PH,
FGP, etc.) mentioned above, then their truth would at the same time be
verified. I think it is more likely, as has been demonstrated in the case
Fermat ``last theorem'', that the truth of these will eventually be
at all---by ordinary mathematical reasoning without any passage through
metamathematics, and that only afterward might we see just which basic
axiomatic principles are required for their proofs."
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