Unformatted text preview: 1 SEARCH FOR CONSEQUENCES
by
Harvey M. Friedman
Ohio State University
LC ‘06
Nijmegen, Netherlands
Delivered August 2, 2006
Edited August 9, 2006
NOTE: This is an edited version of my lecture at LC ‘06. It differs
from my earlier lecture at the Gödel Centenary in Vienna, April 29,
2006 most notably in section 5, where “Finite Graph Theory” is replaced
by “Order Calculus”. 1.
2.
3.
4.
5. General Remarks.
Wqo theory.
Borel selection.
Boolean relation theory.
Order Calculus. 1. GENERAL REMARKS.
I would like to open with the same general remarks that I
delivered at the Gödel Centenary meeting in Vienna a few
months ago.
Gödel's legacy is still very much in evidence. It must be
noted that a careful analysis reveals that his great
insights raise more issues than they resolve. The Gödel
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 2
Gödel's legacy, incompleteness, will diminish over time if
such informal conjectures are not addressed in a
substantial way. I have devoted a major part of my efforts
over forty years to this endeavor.
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 for the
foreseeable future. EXOTIC CONJECTURE
Every interesting mathematical theorem can be
recast as one among a natural finite set of
statements, all of which can be decided using well
studied extensions of ZFC, but not within ZFC
itself.
The recasting of mathematical theorems as elements of
natural finite sets of statements represents an inevitable
general expansion of mathematical activity. This applies to
any standard mathematical context. This program has been
carried out, to a very limited extent, by BRT – and
hopefully soon by Order Calculus. Some details are below. 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. 3
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 published 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.
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. 4
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 the trees 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.
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. 5
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.
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.
Andreas Weiermann and his colleagues have been vigorously
pursuing striking “threshold” or “phase transition”
phenomena surrounding finite KT (Theorem 2.10 above) and
many other combinatorial statements. In finite KT, for
fixed k, i+k is a linear function. I.e., the functions f(i)
= i+k form a class of functions used for finite KT. For a
given class of functions, appropriately presented, we can
ask about the metamathematical status of finite KT based on 6
these functions. The results tell us that if the class of
functions represent a growth rate below (above) a certain
threshold growth rate, then finite KT (or the combinatorial
statement being treated) is provable (unprovable) in a
relevant formal system. 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.
We can say more.
THEOREM 3.6. The existence of the cumulative hierarchy up
through every countable ordinal is sufficient to prove 7
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.
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. 8
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. In fact, it cannot be proved in MAH +
V = L.
Here MAH+ = ZFC + “for all n there exists a strongly nMahlo cardinal”. MAH = ZFC + {there exists a strongly nMahlo cardinal}n.
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. 9
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.
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 (or even MAH + V = L). An instance of the Template
holds if and only if in that instance, “infinite” is
replaced by “arbitrarily large finite”.
From the Introduction, recall EXOTIC CONJECTURE
Every interesting mathematical theorem can be
recast as one among a natural finite set of
statements, all of which can be decided using well
studied extensions of ZFC, but not within ZFC
itself. 10
Theorem 4.1 as it stands is not exactly a case of this
Exotic Conjecture, as the Exotic Case (Proposition A) is
not a mathematical theorem. It can also be criticized as
being too ad hoc to be interesting.
However, consider the considerably more natural statements:
THEOREM I. For all f,g Œ ELG there exist A,B Œ INF such
that
A ». fA Õ B ». gB.
THEOREM II. For all f,g Œ ELG there exist A,B,C Œ INF such
that
A ». fA Õ B ». gB
A ». fB Õ C ». gC.
These are both Theorems of RCA0.
We could start from any one or both of these mathematical
theorems and then embed them into our class of 6561
statements in order to provide an instance of this Exotic
Conjecture.
We freely admit that this is not a very satisfactory
instance of this Exotic Conjecture, and is certainly poor
evidence for it. Nevertheless we believe in this Exotic
Conjecture. 5. ORDER CALCULUS.
There is a well known theorem in graph theory, with many
essentially equivalent formulations. It is really our
complementation theorem in a graph theory setting.
THEOREM 5.1. In any finite dag G there exists A Õ V(G) such
that GA = A’. Furthermore, A is unique.
Here GA is the set of all heads of edges in G whose tail
lies in A, and A’ = V(G)\A.
According to Steve Hedetniemi [email protected],
“A is what is called a solution in digraph theory, and if
you reverse all arcs, you get the wellknown kernel. In our
book Fundamentals of Domination in Graphs, by T.W. Haynes, 11
S.T. Hedetniemi and P.J. Slater, Marcel Dekker, 1998, we
have a 1,224 entry bibliography in the back. This
bibliography must contain about 100 papers on kernels in
graphs.”
We will instead use ordinary relation notation, and take
this in a purely order theoretic direction.
Let R Õ [1,n]k ¥ [1,n]k. We think of R as the digraph G
whose vertex set is [1,n]k, and whose edges are the elements
of R.
For A Õ [1,n]k, we write RA = {y: ($x Œ A)(R(x,y))}. Note
that RA is the same as the GA used above.
We say that R is strictly dominating iff R(x,y) Æ max(x) <
max(y).
Note that if R is strictly dominating then the
corresponding digraph G is obviously a dag.
THEOREM 5.2. For all n,k ≥ 1 and strictly dominating R Õ
[1,n]k ¥ [1,n]k, there exists A Õ [1,n]k such that RA = A’.
Furthermore, A is unique.
We say that B Õ [1,n]k is order invariant iff for all
order equivalent x,y Œ [1,n]k, x Œ B iff y Œ B.
We state the weakened order invariant form of Theorem 5.2.
THEOREM 5.3. For all n,k ≥ 1 and strictly dominating order
invariant R Õ [1,n]k ¥ [1,n]k, there exists A Õ [1,n]k such
that RA = A’. Furthermore, A is unique.
We now make our fundamental informal move.
INFORMAL. For all n,k ≥ 1 and strictly dominating order
invariant R Õ [1,n]k ¥ [1,n]k, there exists A Õ [1,n]k such
that RA and A’ are related.
Let B,C Õ [1,n]k and a,b be finite sequences from [1,n]. We
say that B,C are lower triple equivalent from a to b iff
("x,y,z Œ B » C)($u,v,w Œ B « C)
((x,y,z,a),(u,v,w,b) are order equivalent and
min(x,y,z,a,b) ≥ min(u,v,w,a,b)). 12
PROPOSTION A. For all n >> k and strictly dominating order
invariant R Õ [1,4n]k ¥ [1,4n]k, there exists A Õ [1,3n]k
such that RA,A’ are lower triple equivalent from n,2n to
2n,3n, and from 2n,3n to n,2n.
PROPOSTION B. For all n >> k and strictly dominating order
invariant R Õ [1,5n]k ¥ [1,5n]k, there exists A Õ [1,3n]k
such that RA,A’ are lower triple equivalent from n,2n,3n to
2n,3n,4n, and from 2n,3n,4n to n,2n,3n.
PROPOSITION C. For all n >> k,p and strictly dominating
order invariant R Õ [1,pn]k ¥ [1,pn]k, there exists A Õ
[1,pn]k such that RA,A’ are lower triple equivalent from
n,2n,...,pn2n to 2n,3n,...,pnn, and from 2n,3n,...,pnn
to n,2n,...,pn2n.
THEOREM 5.4. The following is provable in EFA. Proposition
A implies Con(ZFC + "there exists a totally indescribable
cardinal") and is implied by Con(ZFC + "there exists a
subtle cardinal"). Proposition B implies Con(ZFC + “there
exists a subtle cardinal”) and is implied by Con(ZFC
+”there exists a 2 subtle cardinal”). Proposition C is
equivalent to Con(SUB).
Here SUB = ZFC + {there exists an n subtle cardinal}n.
Note that Propositions AC are only explicitly P03. They can
be put into an equivalent explicitly P01 form by eliminating
n in favor of an expression in k,m (and k,m,p) as follows.
PROPOSITION A’. For all n ≥ (8k)! and strictly dominating
order invariant R Õ [1,4n]k ¥ [1,4n]k, $ A Õ [1,3n]k such
that RA,A’ are lower triple equivalent from n,2n to 2n,3n,
and from 2n,3n to n,2n.
PROPOSITION B’. For all n ≥
order invariant R Õ [1,5n]k
that RA,A’ are lower triple
2n,3n,4n, and from 2n,3n,4n (8k)! and strictly dominating
¥ [1,5n]k, $ A Õ [1,4n]k such
equivalent from n,2n,3n to
to n,2n,3n. PROPOSITION C’. For all n ≥
order invariant R Õ [1,pn]k
that RA,A’ are lower triple
to 2n,3n,...,pnn, and from
2n. (8kp)! and strictly dominating
¥ [1,pn]k, $ A Õ [1,pn]k such
equivalent from n,2n,...,pn2n
2n,3n,...,pnn to n,2n,...,pn 13
We now come to plans for Templating these Propositions.
Recall the successful Templating of Proposition A by the
6561 statement Template above in section 4.
We haven’t yet worked on any of these Templating plans.
Nevertheless, we think that it is informative to present
them.
To begin with, we can use an arbitrary list of pairs of
tuples of multiples of n.
TEMPLATE I. For all n >> k,p and strictly dominating order
invariant R Õ [1,pn]k ¥ [1,pn]k, there exists A Õ [1,p]k
such that RA,A’ are lower triple equivalent from a1 to b1,
from a2 to b2, ..., from aq to bq. Here p,q are specific
positive integers, and the a’s and b’s are finite tuples
from {n,2n,...,pn}.
We at least know that Template I holds if and only if for
all i, ai and bi have the same order type, and the entry pn
occurs in the same positions. What remains to be seen is
just what the logical strength is of each statement.
We can also use various Boolean combinations of A and RA.
TEMPLATE II. For all n >> k,p and strictly dominating order
invariant R Õ [1,pn]k ¥ [1,pn]k, there exists A Õ [1,pn]k
such that B1(A,RA),B1’(A,RA) are lower triple equivalent
from a1 to b1, B1(A,RA),B1’(A,RA) are lower triple
equivalent from a2 to b2, ..., Bq(A,RA),Bq’(A,RA) are lower
triple equivalent from aq to bq,. Here p,q are specific
positive integers, the a’s and b’s are finite tuples from
{n,2n,...,pn}, and the B1(A,RA),B2(A,RA)’ are specific
Boolean combinations of A,RA.
This should be fully analyzable. Considerably more
ambitious would be to allow the following wider class of
compound Boolean expressions in A,R. These are defined
inductively by
i. A is an expression.
ii. Boolean combinations of expressions are expressions.
iii. R(X) is an expression if X is an expression.
Thus we arrive at 14
TEMPLATE III. For all n >> k,p and strictly dominating
order invariant R Õ [1,pn]k ¥ [1,pn]k, there exists A Õ
[1,pn]k such that B1(A,RA),B1’(A,RA) are lower triple
equivalent from a1 to b1, B1(A,RA),B1’(A,RA) are lower triple
equivalent from a2 to b2, ..., Bq(A,RA),Bq’(A,RA) are lower
triple equivalent from aq to bq,. Here p,q are specific
positive integers, the a’s and b’s are finite tuples from
{n,2n,...,pn}, and the B1(A,RA),B2(A,RA)’ are specific
compound Boolean combinations of A,RA.
Much more ambitious Templates arise by asking for more than
one set A. We can ask for sets A1,...,At such that some set
of lower triple equivalences holds between various pairs of
Boolean combinations of A1,...,At, R(A1),...,R(At) on various
sets of multiples of n. We can even use compound Boolean
combinations of A1,...,Ak,R.
We can even go much much further by considering more than
one strictly dominating order invariant R.
A whole new dimension of difficulty arises when we wish to
template the definition of
B,C are lower triple equivalent from a to b.
Recall that this takes the "$ form
("x,y,z Œ B » C)($u,v,w Œ B « C)
((x,y,z,a),(u,v,w,b) are order equivalent and
min(x,y,z,a,b) ≥ min(u,v,w,a,b)).
We can replace B » C, and B « C, by various Boolean
combinations of B,C. We can replace ≥ by >,<,=, and use
various patterns of the letters x,y,z,u,v,w,a,b and allow
multiple clauses inside the quantifier free part. We can
also allow multiple "$ sentences of this form, taken
conjunctively. We can, of course, use two sets of
variables, each of finite cardinalities other than 3 each,
and also more Greek variables.
Furthermore, we can combine the previous paragraph with the
earlier Templating ideas.
As above, these Templates are, technically speaking, P03
Templates. However, we can appropriately replace n >> k,t
with n ≥ (8kt)! and obtain P01 Templates. I.e., each 15
instance is a P01 sentence. (Here t represents the size of
the conclusion statement). ...
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