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UNPROVABLE THEOREMS
by
Harvey M. Friedman
[email protected]
http://www.math.ohiostate.edu/~friedman/ April 15, 2004 INTRODUCTION.
We discuss the growing list of examples of simply stated
Theorems where it is known that there are no concrete
proofs.
Here is our agenda.
A. In sufficiently long finite sequences from a finite set,
certain blocks are subsequences of certain later blocks.
B. In sufficiently tall finite trees, certain truncations
are embedded in certain taller truncations.
C. Multivariate functions on the integers have
nonsurjective infinite restrictions.
D. Any two countable sets of reals are pointwise
continously comparable.
E. Any permutation invariant Borel function from infinite
sequences of reals into infinite sequences of reals maps
some sequence into a subsequence.
F. For every symmetric Borel set in the plane, either it or
its complement has a Borel selection.
G. Any Borel set in the plane that has a Borel selection on
every compact set has a Borel selection.
H. For any two multivariate functions on the natural
numbers of expansive linear growth there are three infinite
sets which bear a certain Boolean relation with their
images under the two functions. (Boolean relation theory).
1. BLOCK SUBSEQUENCE THEOREM.
The block subsequence theorem involves a single finite
string in k letters. The following binary case is
elementary, and a good challenge for gifted high school
students.
THEOREM 1.1. There is a longest finite sequence x1,...,xn in
two letters such that no consecutive block xi,...,x2i is a
subsequence of a later consecutive block xj,...,x2j. 2 The longest length is 11, with 12221111111 and 21112222222
as the only examples.
THEOREM 1.2. There is a longest finite sequence x1,...,xn in
3 letters where no consecutive block xi,...,x2i is a
subsequence of a later consecutive block xj,...,x2j.
Theorem 1.2 merely states the existence of a natural number
with a specific testable property.
But the simplest known way to prove this involves not only
infinite sequences but also defining infinite sequences
using all infinite sequences (impredicativity). This is a
weak use of the uncountable.
A logically more down to earth but more involved proof can
be given for any number of letters. The most logically
economical proof uses induction with an induction
hypothesis that has three alternating quantifiers over the
natural numbers. Two quantifiers does not suffice.
Coming back to the case of three letters, the exotic nature
of all proofs is illustrated by the following lower bound
on the length of the longest such finite string. In the
case of two letters it is 11. In the case of three letters,
it is greater than the 7198th Ackermann function at
158,386.
The 3rd Ackermann function at 158,386 is already an
exponential tower of 2’s of height 158,386.
See my paper: Long finite sequences, Journal of
Combinatorial Theory, Series A 95, 102144 (2001).
2. EMBEDDINGS OF FINITE TREES.
A tree is a finite poset with a least element (root), where
the predecessors of any vertex are linearly ordered.
Note the obvious inf operation on the vertices of any
finite tree.
J.B. Kruskal works with inf preserving embeddings between
finite trees. I.e., h is a oneone map from vertices into
vertices such that h(x inf y) = inf(h(x),h(y)). These are
homeomorphic embeddings as topological spaces. 3 THEOREM 2.1. In any infinite sequence of finite trees, one
tree is inf preserving embeddable into a later one.
J.B. Kruskal also considers finite trees whose vertices are
labeled from a finite set (and more generally).
THEOREM 2.2. In any infinite sequence of finite trees with
vertices labeled from a finite set, one tree is inf and
label preserving embeddable into a later one.
Kruskal’s, and the simpler NashWilliams proof, are rather
exotic. NW uses the “minimal bad sequence” argument, which
represents a weak use of the uncountable:
Suppose there is an infinite sequence that forms a
counterexample. Let T0 be a tree of minimal size which
starts such a counterexample. Let T1 be a tree of minimal
size such that T0,T1 starts such a counterexample. Continue
in this way. This is the “minimal bad sequence”.
NashWilliams goes on to derive a contradiction from this
minimal bad sequence.
We showed that, in an appropriate sense, all proofs must
have this exotic nature. We also gave finite forms, and
showed that all proofs of even these finite forms must have
this exotic nature.
Here is one of our state of the art finite forms.
THEOREM 2.3. Let T be the full k splitting tree with labels
1,...,r which is sufficiently tall relative to k,r. There
is an inf preserving label preserving terminal preserving
embedding from some truncation of T into a taller
truncation of T.
By a truncation of T, we mean the subtree of vertices at or
below a certain height.
The growth rate associated with this finite form
corresponds exactly to the necessarily exotic nature of its
proof.
See my paper 4
Internal finite tree embeddings, in: Reflections on the
Foundations of Mathematics: Essays in honor of Solomon
Feferman, ed. Wilfried Sieg, Richard Sommer, Carolyn
Talcott, Lecture Notes in Logic, Association for Symbolic
Logic, pp. 6293, AK Peters, 2002.
We extended this work to the graph minor theorem
(Robertson/Seymour). We show that all proofs are yet more
exotic, involving arbitrary finite iterations of the
minimal bad sequence argument. There are also some finite
forms.
3. CONTINUOUS COMPARISON OF COUNTABLE SETS OF REALS.
The following is in the classical folklore.
THEOREM 3.1. For any two closed sets of real numbers, one
is continuously embeddable into the other.
The proof necessarily uses the CantorBendixson countably
transfinite decomposition of closed sets.
Theorem 3.1 follows from the following main case:
THEOREM 3.2. For any two countable closed sets of real
numbers, one is continuously embeddable into the other.
This also uses the
transfinite decomposition of countable closed sets.
The following is also from the classical folklore.
THEOREM 3.3. For any two countable compact metric spaces,
one is continuously embeddable into the other.
By a more careful argument, we have shown the following.
THEOREM 3.4. For any two countable sets of real numbers,
one is continuously embeddable into the other. For any two
countable metric spaces, one is continuously embeddable
into the other.
Even for countable sets of rationals, we know that, in
various appropriate senses, we must use transfinite
induction of arbitrary countable well ordered lengths.
See my paper 5 Metamathematics of comparability, to appear in Reverse
Mathematics, ed. Simpson, ASL. Available at http://www.math.ohiostate.edu/~friedman/ 4. NONSURJECTIVE RESTRICTIONS.
The following result of ours is deeply connected with the
infinite Ramsey theorem.
THIN SET THEOREM. Let f:Nk Æ N. There exists infinite A Õ N
such that f[Ak] ≠ N.
TST can be derived from the infinite Ramsey theorem, but it
is not known if the infinite Ramsey theorem can be derived
from it. However, we do know that there is no constructive
proof of TST even for k = 2, and any proof must be about as
exotic as he proof of the infinite Ramsey theorem.
This is the simplest example of what we call inequational
Boolean relation theory. BRT will be discussed later in the
talk.
See the paper
Free Sets and Reverse Mathematics, by Cholak, Guisto,
Hirst, and Jockusch, to appear in Reverse Mathematics, ed.
Simpson, ASL. Available at
http://www.nd.edu/~cholak/papers/vitae.html
5. PERMUTATION INVARIANT BOREL FUNCTIONS.
Here is one form of Cantor’s theorem.
THEOREM 5.1. For any infinite sequence of real numbers,
some real number is not a coordinate of the sequence.
There is a reasonable way of getting a real number that is
off the given sequence, from the point of view of
descriptive set theory.
THEOREM 5.2. There is a Borel measurable function F:¬• Æ ¬
such that for all x Œ ¬•, F(x) is not a coordinate of x. 6
The construction of F is by diagonalization, and we expect
that the value of F depends on the order in which the
arguments are given.
THEOREM 5.3. Every permutation invariant Borel function
from ¬• into ¬ maps some infinite sequence to a coordinate.
Permutation invariance makes sense for F:¬• Æ ¬•. One
notion is that if x,y are permutations of each other then
F(x),F(y) are permutations of each other. Another is that
F(p(x)) = p(F(x)) for all permutations p. The results hold
under a variety of related notions.
THEOREM 5.4. Every permutation invariant Borel function
from ¬• into ¬• maps some infinite sequence into an
infinite subsequence.
The proofs use a Baire category argument on the highly
nonseparable and exotic space ¬•, where ¬ is given the
DISCRETE topology.
This is a highly nonseparable argument for a separable
result. We know that there is no separable argument. The
necessary use of the uncountable here is substantially
stronger than what we have encountered up till now.
The necessarily exotic nature of the proof is much more
dramatic when we consider Borel equivalence relations on ¬.
I.e., equivalence relations E Õ ¬ ¥ ¬ which are Borel
measurable.
THEOREM 5.5. Let F:¬• Æ ¬• be a Borel function that
respects the Borel equivalence relation E. Then F maps some
sequence to a subsequence up to E.
In order to prove this,
theoretic way, but also
even more than this. We
iterations of the power we must not only use ¬ in a set
℘(¬), ℘℘(¬), ℘℘℘(¬), etc., and
must use all countably transfinite
set operation. See my paper
On the Necessary Use of Abstract Set Theory, Advances in Math.,
Vol. 41, No. 3, September 1981, pp. 209280.
6. BOREL SELECTION AND SYMMETRIC BOREL SETS. 7
Let E Õ ¬ ¥ ¬. We say that E is symmetric iff (x,y) Œ E ´
(y,x) Œ E.
We say that f is a selection for E on ¬ iff for all x Œ ¬,
(x,f(x)) Œ E.
Here is some background regarding selection.
THEOREM 6.1. Let E be a Borel set in the plane such that
every vertical cross section is nonempty. There is a
Lebesgue measurable selection for E on ¬, but maybe not be
a Borel selection for E on ¬.
The proof of 6.1 is not exotic. However consider the
following.
THEOREM 6.2. Let E be a symmetric Borel set in the plane.
Then E or ¬\E has a Borel selection on ¬.
The proof of 6.2. uses all countable transfinite iterations
of the power operation in a demonstrably essential way. The
number of iterations of the power set operation needed
corresponds to the level of E in the Borel hierarchy.
We proved Theorem 6.2 using a theorem of infinite game
theory due to Donald Martin, called Borel determinacy. This
theorem was first proved by Martin in the mid 1960’s using
large cardinals going way beyond the ZFC axioms. In 1968 we
proved that any proof of Borel determinacy must use all
countably transfinite iterations of the power set
operation.
In 1974, Martin proved Borel determinacy using exactly all
countably transfinite iteration of the power set operation.
This is exactly what is necessary and sufficient here.
See my paper
On the Necessary Use of Abstract Set Theory, Advances in Math.,
Vol. 41, No. 3, September 1981, pp. 209280.
7. BOREL SELECTION IN BOREL SETS.
There is a series of joint papers by Debs and Saint Raymond
concerning selection theorems (they use different
terminology). 8
THEOREM 7.1. Let S be a Borel set in the plane 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.
A proof of Theorem 7.1 using Borel determinacy is implicit
in Debs/Saint Raymond. We have shown that if we use only a
transfinite iteration of the power set operation up to a
single countable ordinal, then we cannot prove Theorem 7.1.
The following is also implicit in Debs/Saint Raymond.
PROPOSITION 7.2. Let S be a Borel set in the plane. If
there is a Borel selection for S on every compact subset of
E, then there is a Borel selection for S on E.
“Proposition” indicates that Debs/Saint Raymond use an
axiom that goes beyond ZFC. We have shown that Proposition
7.2 is independent of ZFC.
See my paper
Selection for Borel relations, Proceedings of Logic
Colloquium ‘01, to appear, 2004. Available at
http://www.math.ohiostate.edu/~friedman/ 8. 6561 CASES OF BOOLEAN RELATION THEORY.
We have discovered a general class of mathematical problems
which makes good sense in a great variety of contexts, but
which presents severe logical difficulties even in concrete
contexts.
Boolean Relation Theory (BRT) concerns the Boolean
relations between sets and their images under multivariate
functions.
More specifically, let f be a multivariate function and A
be a set. We define
fA = {f(x1,...,xk): k is the arity of f and x1,...,xk Œ A}.
I.e., if the arity of f is k then
fA = f[Ak].
It is very convenient to suppress the arity of f and use
the notation fA. 9 Let f:Nk Æ N. We say that f is strictly dominating iff for
all x Œ dom(f), T(x) > max(x).
Here are two simple examples of equational Boolean relation
theory.
1. For all strictly dominating f there exists infinite A Õ
N such that N = A ». fA. I.e., A = N\fA. Or A,fA partitions
N.
2. For all strictly dominating f,g there exists infinite
A,B,C Õ N such that C « fA = C « gB = fA « gB = ∅.
1 is called the Complementation Theorem and plays a special
role in BRT.
We leave the proof of both statements to the audience.
2 involves two functions and three sets. Here we know of
interesting concrete contexts where BRT with two functions
and three sets leads to severe logical difficulties.
Let f be a multivariate function from N into N. We say that
f is of expansive linear growth iff there exist c,d > 1
such that for all but finitely many x Œ dom(f),
c max(x) £ f(x) £ d max(x).
We use X ». Y for X » Y if X,Y are disjoint; undefined
otherwise.
PROPOSITION 8.1. For all f,g
there exist infinite A,B,C Õ
A ». fA
A ». fB of expansive linear growth,
N such that
Õ C ». gB
Õ C ». gC. We have given a proof of Proposition 8.1 using certain
large cardinals that go well beyond the usual axioms of
ZFC. We have also shown that ZFC alone does not suffice. In
fact, we know exactly what large cardinals are required.
It is clear that Proposition 8.1 has a particularly simple
structure compared to a typical statement in BRT. In fact,
the two clauses in Proposition 8.1 have the form
X ». fY Õ Z ». gW 10
S ». fT Õ U ». gV
where X,Y,Z,W,S,T,U,V are among the three letters A,B,C.
This amounts to a particular set of instances of Boolean
relation theory of cardinality 38 = 6561.
We have been able to show that all of these 6561 statements
are provable or refutable very explicitly, with ONE
exception (up to symmetry): Proposition 8.1.
Furthermore, there is a finite obstruction phenomena to the
effect that if we replace “infinite” by “arbitrarily large
finite” then we get the same classification.
Finite obstruction can be proved very explicitly for all
cases except 8.1 (up to symmetry). For 8.1, using
“arbitrarily large finite” makes 8.1 easily provable.
What if we require that f,g are concretely given – e.g.,
integral piecewise linear (finitely many pieces). Then 8.1
can be proved with the same large cardinals, and still
require them.
NOTE: At present, we have carried this out for these two
cases:
a. f,g are integral piecewise linear (finitely many
pieces), where g is of expansive linear growth.
b. f,g are defined by cases using addition, multiplication,
exponentiation, and round up subtraction and division, and
of expansive linear growth.
What if we also require that the sets A,B,C be explicitly
given? For a, we can require that A,B,C be finite unions of
integral piecewise linear images of infinite geometric
progressions. Then 8.1 again corresponds to the same large
cardinals.
The large cardinals in question are the Mahlo cardinals of
finite order – formulated in around 1905!
For a proof of 8.1 using these cardinals, see my paper
Equational Boolean Relation Theory, available at
http://www.math.ohiostate.edu/~friedman/ 11
9. HOW ARE LARGE CARDINALS USED?
We will give a brief explanation of how they are used to
prove Proposition 8.1. It is considerably easier to explain
this if we assume that f,g are integral piecewise linear
functions. In the brief sketch, we will only use that g is
of expansive linear growth.
Let Z* be the free Abelian group on a well ordered set of
generators of type k, where k is a large cardinal. We take 1
to be the first generator. Z* is naturally linearly
ordered.
Clearly the integral piecewise linear f,g have canonical
extensions f*,g* to N* = the nonnegative part of Z*.
N* is not well ordered. However, it easy to see that the
relation 2x < y on N* is well founded. This enables us to
use a transfinite form of the Complementation theorem. In
particular, we can construct a unique set W Õ N* such that
N* = W ». g*W.
With some care, we then build a tower A Õ B Õ C Õ W, such
that
A ». f*A Õ C ». g*B
A ». f*B Õ C ». g*C.
This lives in the transfinite, and so may not be embeddable
back into N Õ Z. However, using the combinatorics of large
cardinals, we can arrange not only that C is of order type
w, but the set of all generators used to represent elements
of C has order type w, and also that the representation of
elements of C have lengths bounded by a fixed integer. Then
we know that we can embed C back into N Õ Z, completing the
proof of 8.1. ...
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