FOM: Review of Feferman
Harvey Friedman
friedman at math.ohio-state.edu
Wed Apr 26 06:05:35 EDT 2000
Here is my review of
"Does mathematics need new axioms?"
Solomon Feferman
American Mathematical Monthly
Volume 106, No. 2
February, 1999
pp. 99-111
I take some direct quotes from the article and comment on them.
>The question, ``Does mathematics need new axioms?'', is ambiguous in
>practically every respect.
>What do we mean by `mathematics'?
>What do we mean by `need'?
>What do we mean by `axioms'?
>You might even ask, What do we mean by `does'?
I agree with Feferman's opening, although he doesn't go nearly far enough
in clarifying the different senses of the question. For example, one
ambiguity in the title that he does not pay sufficient attention to is the
distinction between
i) mathematics needs new axioms;
ii) there is a new axiom that mathematics needs.
In particular, I believe that under many interpretations of the words, i)
is much more compelling than ii).
>The Oxford English Dictionary defines `axiom' as used in Logic and
>Mathematics, by: ``A self-evident proposition requiring no formal demon-
>stration to prove its truth, but received and assented to as soon as men-
>tioned.'' I think it's fair to say that something like this definition is
>the first
>thing we have in mind when we speak of axioms for mathematics: I'll call
>this the ideal sense of the word. It's surprising how far the meaning of ax-
>iom has been stretched from that in practice, both by mathematicians and
>logicians. Some even take it to mean an arbitrary assumption, and so refuse
>to take seriously what purpose axioms are supposed to serve.
Note how important the distinction between self-evident axioms and other
kinds of axioms (compelling axioms, productive axioms, plausible axioms,
etcetera) is to the question at hand. E.g., contrast these four:
i) mathematics needs new (self-evident) axioms;
ii) there is a new (self-evident) axiom that mathematics needs.
where these are read with and without the phrase in parentheses.
Feferman then states that he will focus on Peano Arithmetic and the ZFC
axioms. He considers PA as a prime example of what is meant by self-evident
axioms. This is a very reasonable position, and the most common position,
but not the only position. In particular, one can sensibly question even
the meaning of arbitrary combinations of quantifiers over all of the
natural numbers.
Feferman then contrasts PA with ZC = the Zermelo axioms with the axiom of
choice (a fragment of the ZFC axioms), in that ZC codifies principles that
were controversial at the time. As Feferman remarks, the full axioms of ZFC
came about because of the inability of ZC to make some additional
constructions of Cantor such as the limit of the first omega infinite
cardinals. Feferman's contrast between PA and ZFC (or even ZC) seems
appropriate, but it is not clear how to get at an instrinsic difference
that is not historical.
>What was left unsettled by this development is an explanation of what,
>exactly, the Zermelo-Fraenkel axioms are axioms for. If they are to be con-
>sidered to be axioms in the ideal, dictionary sense, they should be evident
>for some pre-axiomatic concept that we have in mind. The concept of ar-
>bitrary set, so to speak at large, which might first be offered as a candidate
>for this is unsatisfactory, because it seems to be an evident
>characteristic of
>this concept that for any property P (x) the set of all x satisfying P exists.
>But as we know, this results in contradictions, the simplest being that due
>to Russell...
Feferman goes on to describe the cumulative hierarchy. He continues:
>It is argued by set-theorists nowadays that the axioms of ZFC are evident
>for the universe V of sets consisting of all objects in some V ff . But the
>intuition for that is a far cry from what leads one to accept the Dedekind-
>Peano axioms. Among other things, what this takes for granted is that
>there is an objective notion of arbitrary subset of a given set. This is the
>Platonistic conception of mathematics applied to set theory, a conception
>which is philosophically controversial; we shall have more to say about that
>later on.
Feferman then goes on to give an account of the two incompleteness theorems
of Godel. He then presents an important quote from Godel:
>"...the true reason for the incompleteness inherent in all formal
>systems of mathematics is that the formation of ever higher types
>can be continued into the transfinite...[since] the undecidable
>propositions constructed here become decidable whenever ap-
>propriate higher types are added."
Feferman then discusses the axiom of constructibility, originally supported
by Godel.
>But within a decade he was clearly rejecting
>it as an axiom, on the basis of a strongly Platonistic point of view of
>what set
>theory is supposed to be about.
Feferman later discusses strong axioms of infinity (large cardinals).
>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 [17]. 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
>1 The elaboration of this subject has almost outrun the names that have
>been intro-
>duced for various large cardinal notions, witness (in roughly increasing
>order of strength):
>`inaccessible', `Mahlo', `weakly compact', `indescribable', `subtle',
>`ineffable', `Ramsey',
>`measurable', `strong', `Woodin', `superstrong', `strongly compact',
>`supercompact', `al-
>most huge', `huge', and `superhuge'.
The above paragraph does not present a balanced view of the existing large
cardinal hierarchy in that it does not comment on its striking internal
coherence and apparent robustness. To me, the hierarchy has the feeling of
being forced on us from a higher reality. I have little doubt that these
same cardinals - perhaps with additional intermediate ones - have been
investigated by all civilizations in the universe that are at least as
technologically advanced as we are.
However, the precise nature of this coherence and robustness is very much
up in the air, and I can't help but have the feeling that they are a shadow
of a much more fundamental theory - that somehow there is a missing
parameter, perhaps purely numerical, which, when varied, produces the large
cardinal hierarchy in a clear and convincing way.
Incidentally, it is commonplace in science for "the elaboration of [a]
subject [to] almost outrun the names that have been introduced" - e.g., in
the classification programs in biology. So this complexity should not
reflect negatively on the large cardinal hierarchy or its investigation.
"Broadly speaking, the
>arguments [for large cardinal axioms] are classified as being based on
>intrinsic or extrinsic reasons. The
>above-mentioned reflection principles are examples of intrinsic reasons, but
>these do not take us beyond the ``small'' large cardinals. Among the extrinsic
>reasons for going higher are that the assumption of ``large'' large cardinals
>has been fruitful---through the dazzling work of Solovay, Martin, Foreman,
>Magidor, Shelah, Steel, Woodin and others---in extending ``standard'' prop-
>erties of Borel and projective subsets of the continuum, such as Lebesgue
>measurability, the Baire property, the perfect subset property, determinate-
>ness of associated infinite games, etc. to substantially larger classes.
There is a small technical confusion here worth mentioning. Feferman writes
"extending "standard" properties of Borel and projective subsets of the
continuum ... to substantially larger classes." What he should have said
was "extending "standard" properties of Borel subsets of the continuum ...
to substantially larger classes such as the projective sets and beyond."
Whereas the Borel sets have plenty of "standard" properties to extend, no
"standard" property of projective sets of the kind referred to can be
proved without additional axioms such as what Feferman refers to as large
large cardinals.
At this point in the article, it would make sense for Feferman to take
stock in how the work he has just referred to impinges on the original
question. For example, let us take the question
*) are all projective sets Lebesgue measurable?
This is already known to be independent of ZFC for PCA sets, which are low
down in the projective hierarchy. I.e., the special case
**) are all PCA sets Lebesgue measurable?
is independent of ZFC.
So don't we need new axioms to settle **)? Obviously we do, assuming that
"new" means "not available in ZFC."
However, there are still at least a couple of ways around the use of **) to
answer the question - at least in one of its myriad forms.
A. It can be argued that **) is not part of mathematics. Or is at least so
atypical of mathematics that it cannot be the basis for the adoption of new
axioms for mathematics.
B. It can be argued that there is no specific new axiom that we need to
settle *) or **). This is because *) and **) can be settled one way with
large large cardinals, and the opposite way with V = L (the axiom of
constructibility).
So we now have eight relevant formulations of the original question:
i) (normal) mathematics needs new (self-evident) axioms;
ii) there is a new (self-evident) axiom that (normal) mathematics needs.
V = L is not considered to be a candidate for a new axiom for mathematics
by the set theory community on the grounds that it limits the set theoretic
universe in the sense that it asserts that every set can be constructed by
a certain transfinite process. Of course, there is nothing evident about V
= L, but there is also nothing evident about large large cardinals. It is
well known that one cannot have both - large large cardinals are
incompatible with V = L.
I, personally, view V = L as an axiom of specificity. It "asserts" that we
are only going to be considering sets that can be constructed by a certain
transfinite process. I should add that this transifinite process - although
technical in all presently known descriptions - is rather robust. And I
believe that one can find elegant nontechnical descriptions of the process
in fundamental terms.
Furthermore, I will make the following sweeping conjecture:
CONJECTURE. Every natural mathematical statement in the current
mathematical literature - with the sole exception of various published and
unpublished manuscripts written by Harvey M. Friedman and one or two of his
close associates - that naturally lives within, say, the set theoretic
universe of rank at most omega + omega, as long as it is of a natural -
perhaps even highly set theoretic - nature, as opposed to of a logical or
metamathematical nature, including all of the statements from Cantor and
from classical descriptive set theory and from set theoretic combinatorics,
set theoretic measure theory, and abstract analysis, can be settled within
ZFC + V = L.
In this specific sense, ZFC + V = L is an incredibly powerful axiom system.
I have spent over 30 years intensively looking for appropriate statements
that cannot be settled within ZFC + V = L. In particular, ZFC + V = L
proves that *) and **) are false, and that the continuum hypothesis is
true.
>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).
Feferman could here take the position that this situation with CH means
that mathematics needs new axioms, but that there is no axiom that settles
it that mathematics needs.
Instead, Feferman takes a different tack:
>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
>idealized
>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, [28].) 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
>continuum. Here we are skirting deep philosophical waters; let us retreat
>from them for the moment.
Recalling our eight formulations of the question,
i) (normal) mathematics needs new (self-evident) axioms;
ii) there is a new (self-evident) axiom that (normal) mathematics needs.
what Feferman is setting up, at least in our terms, is that CH is not
normal mathematics. Therefore, the situation with regard to CH is seen to
not affect the formulations of i) and ii) with "normal."
Regardless of how one feels about the question of whether CH is a definite
or meaningful problem, I think that it is entirely appropriate to
distinguish CH - and other heavily set theoretic problems - from normal
matheamtics. So in this arena, I have sympathy with Feferman's position.
>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.
>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 [22] 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). Moving up to a stronger
>system, a few years later, ...
Feferman puts by far the most convincing results along these lines in
footnotes:
>Friedman later found a finite version of Kruskal's theorem KT which is
>independent of
>ATR0 . The infinitary theorem KT, a staple of graph-theoretic
>combinatorics, asserts the
>well-quasi-ordering of the embeddability relation between finite trees.
>Friedman found an extended version EKT
>of KT which is independent of the impredicative \Pi-1-1
>comprehension principle in analysis. EKT later turned out to have close
>mathematical and metamathematical relationships with the graph minor
>theorem of Robertson and Seymour.
>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
>reasoning.
I have found some considerably more elegant finite consequences of KT which
are also independent of systems like ATR_0, subsequent to the publication
of Feferman's article. Here is an example.
For any finite tree T and positive integer n, let T[<=n] be the subtree of
vertices with at most n predecessors, and let T[n] be the set of vertices
with n predecessors. Thus T[1] is the root. The height is the largest
number of predecessors any vertex has.
#) Let n >> k and T be a tree of valence (splitting) <=k and height n.
There exist 2 <= i,j <= n and an inf preserving embedding from T[<=i] into
T[<=j] which maps T[i] into T[j].
>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. 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.
Feferman makes what appears to be a logical error here. This last paragraph
suggests that
a) we need certain axioms of large cardinals in order to settle phi
implies
b) we should accept certain axioms of large cardinals as "first-class
mathematical principles."
We also see the introduction of a new phrase "first-class mathematical
principles" whose connection with the rest of his paper is left unclear.
Feferman also is blurring the distinction between i) and ii) in our
formulations:
i) (normal) mathematics needs new (self-evident) axioms;
ii) there is a new (self-evident) axiom that (normal) mathematics needs.
He seems to be addressing ii) and not i), but not drawing the distinction.
He seems to be questioning some specific axiom candidates as not
"first-class mathematical principles."
In particular, Feferman does not acknowledge the fact that this situation
with regard to phi and large cardinals can be quite convincing for i) in
the weak sense that "mathematics needs new axioms." Specifically, let us
look at
c) we need new axioms in order to settle phi.
Since the main result of my work here is that phi is provably equivalent to
the 1-consistency of certain large cardinals, c) reduces to
d) we need new axioms in order to settle the 1-consistency of certain large
cardinals.
Now using Godel's work, we see that
*) the statement d) is equivalent to the consistency of certain (slightly
larger) large cardinals.
Thus we can be morally certain that d) holds, at least at the level of
large cardinals we are talking about.
Let me put it differently. If we accept phi as part of mathematics for the
purpose of the discussion, or part of normal mathematics if we formulate
our question in terms of "normal", then either we need new axioms, or
something occurs of a totally unexpected dramatic nature that completely
changes the very nature of the discussion that we are engaging in (i.e.,
that small large cardinals are inconsistent!).
So for all practical purposes, we should simply conclude as working
scientists that mathematics needs new axioms - assuming that we embrace the
mathematical relevance of phi.
Furthermore, as I pursue this work, it is apparent that one can expect to
be able to adjust such phi so that they become equivalent to the
1-consistency of, say, ZFC, or, say, ZFC + an inaccessible cardinal. Then
the drawbacks in what Feferman is saying become even clearer.
>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.
This is certainly correct on the face of it, but it ignores the strategy
behind the work of mine under discussion. The plan is to apply certain
large cardinals in an essential way in order to derive new mathematical
results of a compelling nature. Moreover, to develop new penetrating
subject areas of mathematics of a thematic nature that are accepted by the
mathematical community as fundamentally interesting contributions of
lasting significance with substantial points of contact with existing
normal mathematics.
>From this perspective, the fact that existing open problems may not be
amenable to this approach is not in the least bit troublesome. One has to
look and see what mathematical information of a normal kind can be obtained
through use of the large cardinals but not without.
If the quality and quantity are sufficiently striking, then the
mathematical community can be expected to want to expand the available
axioms and rules to accomodate the new methods so as to further the
development of these new subject areas. They will be presented with the
relevant large cardinal axioms as the only known vehicle for obtaining
these results.
While it is true that the negation of these results can be obtained from
the inconsistency of the relevant large cardinals, that is likely to be
viewed as completely besides the point. Firstly, because no inconsistency
is going to be found, and secondly, because taking the existence of
inconsistencies in axiomatic systems of large cardinals as new axioms for
mathematics is far too bizarre to be contemplated.
But suppose a surprise inconsistency were found in some relevant large
cardinal axiom? I have no doubt that the results could be weakened so that
the relevant large cardinals are in a much safer region. Such is the
flexibility of the results being obtained. In this sense, the results are
immune to surprise inconsistencies - even within ZFC.
>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.
Again, I don't think that the question-begging issue is compelling. But
there is another aspect that is more telling.
The statement "various parts of descriptive set theory have been shown to
require higher axioms of infinity" is flawed. In fact, these parts of
descriptive set theory have also been done with V = L and no higher axioms
of infinity - except that the results go the other way. In fact, as I have
said before, all problems in descriptive set theory have been solved
without higher axioms of infinity, but with V = L instead.
This is in contrast to the results of mine discusssed above. Here if you
want to settle them without using large cardinals, you have to use what
amounts to the inconsistency of large cardinals. This is not remotely in
the realm of what we mean by an axiom, or even what we mean by a
mathematical principle.
>Also, I think it is fair to say that these
>kinds of results are at the margin of ordinary mathematics, that is of what
>mathematicians deal with in daily practice.
>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.
I agree with this criticism of the situation in set theory and descriptive
set theory. Meeting this criticism is the primary goal of the work of mine
discussed earlier by Feferman, and also which is being hotly pursued right
now.
>I am
>convinced that the Continuum Hypothesis is an inherently vague problem
>that no new axiom will settle in a convincingly definite way.
I am far more convinced that CH is not "normal" mathematics than I am that
no new axiom will settle it in a convincingly definite way. I think that
there may be striking results to the effect that certain general approaches
to CH must fail.
>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.
I think that the Platonistic philosophy of mathematics is approximately as
good as any other philosophy that we have. They all have their strengths
and weaknesses, and seminal results can always be obtained that profoundly
affect our assessment of these strengths and weaknesses.
>Finally, I see no evidence for the practical need for new ax-
>ioms to settle open arithmetical and finite combinatorial problems.
As indicated above, this may well be true for currently open questions, but
I am convinced that current work will show the great value in necessarily
using large cardinals to obtain new results of this type.
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