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MY FORTY YEARS ON HIS SHOULDERS
by
Harvey M. Friedman
Ohio State University
Gödel Centenary
April 29, 2006
1.
2.
3.
4.
5. General Remarks.
Wqo theory.
Borel selection.
Boolean relation theory.
Finite graph theory. 1. GENERAL REMARKS.
Godel's legacy is still very much in evidence. It must be
noted that a careful analysis reveals that this great
insights raise more issues than they resolve. The Godel
legacy practically begs for renewal and expansion at
a fundamental level.
When I entered the field some forty years ago, I seized on
one glaring opportunity for renewal and expansion. The
independence results from ZFC and significant fragments
lied in a very narrow range, and had systemic features that
are glaringly unrepresentative of mathematics and
mathematical subjects generally.
This state of affairs suggests obvious informal conjectures
to the effect that there are severe systemic limitations to
the incompleteness phenomena, culminating in informal
conjectures to the effect that, in principle, there is no
relevance of set theoretic methods to "genuine"
mathematical activity.
Now, there is no question that this central aspect of
Godel's legacy, incompleteness, will diminish over time if
such informal conjectures are not addressed in a
substantial way. I have devoted the major part of my
efforts over forty years to this effort.
I view this effort as part of a perhaps slow but steady
evolutionary process. I have every confidence that this
process will steadily continue in a striking manner as long
as mathematics remains a vibrant activity. 2
2. WQO THEORY.
Wqo theory is a branch of combinatorics which has proved to
be a fertile source of deep metamathematical pheneomena.
A qo (quasi order) is a reflexive transitive relation (A,£).
A wqo (well quasi order) is a qo (A,£) such that for all
x1,x2,... from A, $ i < j such that xi £ xj.
Highlights of wqo theory: that certain qo’s are wqo’s.
J.B. Kruskal treats finite trees as finite posets, and
studies the qo
$ an inf preserving embedding from T1 into T2.
THEOREM 2.1. (J.B. Kruskal). The above qo of finite trees
as posets is a wqo.
We observed that the connection between wqo’s and well
orderings can be combined with known proof theory to
establish independence results.
The standard formalization of “predicative mathematics” is
due to Feferman/Schutte = FS. Poincare, Weyl, and others
railed against impredicative mathematics.
THEOREM 2.2. Kruskal’s tree theorem cannot be proved in FS.
KT goes considerably beyond FS, and an exact measure of KT
is known through work of Rathjen/Weiermann.
Kruskal actually considered finite trees whose vertices are
labeled from a wqo £*. The additional requirement on
embeddings is that
label(v) £* label(h(v)).
THEOREM 2.3. (J.B. Kruskal). The qo of finite trees as
posets, with vertices labeled from any given wqo, is a wqo.
Labeled KT is considerably stronger, proof theoretically,
than KT, even with only 2 labels, 0 £ 1. I have not seen a
metamathematical analysis of labeled KT.
Note that KT is a P11 sentence and labeled KT is a P12 in
the hyperarithmetic sets. 3 THEOREM 2.4. Labeled KT does not hold in the
hyperarithmetic sets. In fact, RCA0 + KT implies ATR0.
It is natural to impose a growth rate in KT in terms of the
number of vertices of Ti.
COROLALRY 2.5. (Linearly bounded KT). Let T1,T2,... be a
linearly bounded sequence of finite trees. $ i < j such that
Ti is inf preserving embeddable into Tj.
COROLLARY 2.6. (Computational KT). Let T1,T2,... be a
sequence of finite trees in a given complexity class. There
exists i < j such that Ti is inf preserving embeddable into
Tj.
Note Corollary 2.6 is P02.
THEOREM 2.7. Corollary 2.5 cannot be proved in FS. This
holds even for linear bounds with nonconstant coefficient
1.
THEOREM 2.8. Corollary 2.6 cannot be proved in FS, even for
linear time, logarithmic space.
By an obvious application of weak Konig’s lemma, Corollary
2.5 has very strong uniformities.
THEOREM 2.9. (Uniform linearly bounded KT). Let T1,T2,... be
a linearly bounded sequence of finite trees. There exists
i < j £ n such that Ti is inf preserving embeddable into Tj,
where n depends only on the given linear bound, and not on
T1,T2,...
With this kind of strong uniformity, we can obviously strip
the statement of infinite sequences of trees.
For nonconstant coefficient 1, we have:
THEOREM 2.10. (finite KT). Let n >> k. For all finite trees
T1,...,Tn with each Ti £ i+k, there exists i < j such that
Ti is inf preserving embeddable into Tj.
Since Theorem 2.10 Æ Theorem 2.9 Æ Corollary 2.5
(nonconstant coefficient 1), we see that Theorem 2.10 is
not provable in FS. 4
Other P02 forms of KT involving only the internal structure
of a single finite tree can be found in the Feferfest
volume.
I proved analogous results for EKT = extended Kruskal
theorem, which involves a finite label set and a gap
embedding condition. Only here the strength jumps up to
that of P11CA0.
I said that the gap condition was natural (i.e., EKT was
natural). Many people were unconvinced.
Soon later, EKT became a tool in the proof of the famous
graph minor theorem of Robertson/Seymour.
THEOREM 2.11. Let G1,G2,... be finite graphs. There exists i
< j such that Gi is minor included in Gj.
I then asked Robertson/ Seymour to prove a form of EKT that
I knew implied full EKT, just from GMT. They complied, and
we wrote a triple paper.
The upshot is that GMT is not provable in P11CA0. Just
where GMT is provable is unclear, and recent discussions
with Robertson have not stabilized. I disavow remarks in
the triple paper about where GMT can be proved.
An extremely interesting consequence of GMT is the subcubic
graph theorem. A subcubic graph is a graph where every
vertex has valence £ 3. (Loops and multiple edges are
allowed).
THEOREM 2.12. Let G1,G2,... be subcubic graphs. There exists
i < j such that Gi is embeddable into Gj as topological
spaces (with vertices going to vertices).
Robertson/Seymour also claims to be able to use the
subcubic graph theorem for linkage to EKT. Therefore the
subcubic graph theorem (even in the plane) is not provable
in P11CA0.
We have discovered lengths of proof phenomena in wqo
theory. We use S01 sentences.
*) Let T1,...,Tn be a sufficiently long sequence of trees
with vertices labeled from {1,2,3}, where each Ti £ i. 5
There exists i < j such that Ti is inf and label preserving
embeddable into Tj.
**) Let T1,...,Tn be a sufficiently long sequence of
subcubic graphs, where each Ti £ i+13. There exists i < j
such that Gi is homeomorphically embeddable into Gj.
THEOREM 2.13. Every proof of *) in FS uses at least 2[1000]
symbols. Every proof of **) in P11CA0 uses at least 2[1000]
symbols.
3. BOREL SELECTION.
Let S Õ ¬2 and E Õ ¬. A selection for A on E is a function
f:E Æ ¬ whose graph is contained in S.
A selection for S is a selection for S on ¬.
We say that S is symmetric if and only if S(x,y) ´ S(y,x).
THEOREM 3.1. Let S Õ ¬2 be a symmetric Borel set. Then S or
¬2\S has a Borel selection.
My proof of Theorem 3.1 relied heavily on Borel
determinacy, due to D.A. Martin.
THEOREM 3.2. Theorem 3.1 is provable in ZFC, but not
without the axiom scheme of replacement.
There is another kind of Borel selection theorem that is
implicit in work of Debs and Saint Raymond of Paris VII.
They take the general form: if there is a nice selection
for S on compact subsets of E, then there is a nice
selection for S on E.
THEOREM 3.3. Let S Õ ¬2 be Borel and E Õ ¬ be Borel with
empty interior. If there is a continuous selection for S on
every compact subset of E, then there is a continuous
selection for S on E.
THEOREM 3.4. Let S Õ ¬2 be Borel and E Õ ¬ be Borel. If
there is a Borel selection for S on every compact subset of
E, then there is a Borel selection for S on E.
THEROEM 3.5. Theorem 3.3 is provable in ZFC but not without
the axiom scheme of replacement. Theorem 3.4 is neither
provable nor refutable in ZFC. 6 We can say more.
THEOREM 3.6. The existence of the cumulative hierarchy up
through every countable ordinal is sufficient to prove
Theorems 3.1 and 3.3. However, the existence of the
cumulative hierarchy up through any suitably defined
countable ordinal is not sufficient to prove Theorem 3.1 or
3.3.
DOM: The f:N Æ N constructible in any given x Õ N are
eventually dominated by some g:N Æ N.
THEOREM 3.7. ZFC + Theorem 3.4 implies DOM (H. Friedman).
ZFC + DOM implies Theorem 3.4 (Debs/Saint Raymond).
4. BOOLEAN RELATION THEORY.
We begin with two examples of statements in BRT of special
importance for the theory.
THIN SET THEOREM. Let k ≥ 1 and f:Nk Æ N. There exists an
infinite set A Õ N such that f[Ak] ≠ N.
COMPLEMENTATION THEOREM. Let k ≥ 1 and f:Nk Æ N. Suppose
that for all x Œ Nk, f(x) > max(x). There exists an infinite
set A Õ N such that f[Ak] = N\A.
These two theorems are official statements in BRT. In the
complementation theorem, A is unique.
We now write them in BRT form.
THIN SET THEOREM. For all f Œ MF there exists A Œ INF such
that fA ≠ N.
COMPLEMENTATION THEOREM. For all f Œ SD there exists A Œ
INF such that fA = N\A.
The thin set theorem lives in IBRT in A,fA. There are only
22^2 = 16 statements in IBRT in A,fA. These are easily
handled.
The complementation theorem lives in EBRT in A,fA. There
are only 22^2 = 16 statements in IBRT in A,fA. These are
easily handled. 7
For EBRT/IBRT in A,B,C,fA,fB, fC,gA,gB,gC, we have 22^9 =
2512 statements. This is entirely unmanageable. It would
take several major new ideas to make this manageable.
DISCOVERY. There is a statement in EBRT in A,B,C,fA,fB,
fC,gA,gB,gC that is independent of ZFC. It can be proved in
MAH+ but not in MAH, even with the axiom of
constructibility.
The particular example is far nicer than any “typical”
statement in EBRT in A,B,C,fA,fB,fC,gA,gB,gC. However, it
is not nice enough to be regarded as suitably natural.
Showing that all such statements can be decided in MAH+
seems to be too hard.
What to do? Look for a natural fragment of full EBRT in
A,B,C,fA,fB,fC,gA,gB,gC that includes the example, where I
can decide all statements in the fragment within MAH+.
Also look for a bonus: a striking feature of the
classification that is itself independent of ZFC.
Then we have a single natural statement independent of ZFC.
In order to carry this off, we use somewhat different
functions.
We use ELG = expansive linear growth.
These are functions f:Nk Æ N such that there exist
constants c,d > 1 such that
cx £ f(x) £ dx
holds for all but finitely many x Œ Nk.
TEMPLATE. For all f,g Œ ELG there exist A,B,C Œ INF such
that
X ». fY Õ V ». gW
P ». fR Õ S ». gT.
Here X,Y,V,W,P,R,S,T are among the three letters A,B,C.
Note that there are 6561 such statements. We have shown
that all of these statements are provable or refutable in
RCA0, with exactly 12 exceptions. 8 These 12 exceptions are really exactly one exception up to
the obvious symmetry: permuting A,B,C, and switching the
two clauses.
The single exception is the exotic case
PROPOSITION A. For all f,g Œ ELG there exist A,B,C Œ INF
such that
A ». fA Õ C ». gB
A ». fB Õ C ». gC.
This statement is provably equivalent to the 1consistency
of MAH, over ACA’.
If we replace “infinite” by “arbitrarily large finite” then
we can carry out this second classification entirely within
RCA0.
Inspection shows that all of the nonexotic cases come out
with the same truth value in the two classifications, and
that is of course provable in RCA0.
Furthermore, the exotic case comes out true in the second
classification.
THEOREM 4.1. The following is provable in MAH+ but not in
MAH, even with the axiom of constructibility. An instance
of the Template holds if and only if in that instance,
“infinite” is replaced by “arbitrarily large finite”.
5. FINITE GRAPH THEORY.
THEOREM 1. For all strictly dominating order invariant R Õ
Nk x Nk there exists A Õ Nk such that RA = A¢.
THEOREM 2. For all strictly dominating order invariant R Õ
[1,n]k x [1,n]k there exists A Õ [1,n]k such that RA = A¢,
THEOREM 3. Every strictly dominating order invariant R Õ
[1,n]k x [1,n]k has an antichain A such that RA = A¢.
THEOREM 4. Every strictly dominating order invariant R Õ
[1,n]k x [1,n]k has an antichain A such that (RA)3,(A¢)3,
(A¢)3 \28k^21 have the same elements up to order equivalence
relative to the powers of 2. 9 This is a purely universal sentence provably equivalent to
the consistency of MAH. ...
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 Fall '08
 JOSHUA
 Math, Graph Theory

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