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WHAT YOU CANNOT PROVE 1: before 2000
by
Harvey M. Friedman
Department of Mathematics
Ohio State University
December 19, 2005
http://www.math.ohiostate.edu/%7Efriedman/
Most of my intellectual efforts have focused around a
single general question in the foundations of mathematics
(f.o.m.). I became keenly aware of this question as a
student at MIT around 40 years ago, and readily adopted it
as the principal driving force behind my research.
1. General question, general conjectures.
2. The most abstract: general set theoretic statements.
3. The second most abstract: limited set theoretic
statements.
4. The third most abstract: the projective hierarchy of
sets of reals.
5. The fourth most abstract: Borel sets of reals.
6. The fifth most abstract: infinite sets of integers.
7. The sixth most abstract: finite combinatorics, with
unbounded existence.
8. The most concrete: completely finite combinatorics.
1. GENERAL QUESTION, GENERAL CONJECTURES.
The currently standard formalization of mathematics came
into general acceptance before 1930, and is known as ZFC =
Zermelo Frankel set theory with the axiom of choice.
Enter the 1930s, the era of Kurt Gödel. Among his most
astounding results are his two incompleteness theorems.
They are very general, and assert the following when
applied to ZFC.
G1. There exist sentences in the language of set theory
which cannot be proved or refuted in ZFC (assuming that ZFC
does not prove that ZFC is inconsistent).
G2.
set
ZFC
and The consistency of ZFC is a sentence in the language of
theory, which cannot be proved in ZFC (assuming that
is consistent). In fact, ZFC does not prove Con(ZFC) if
only if ZFC is consistent. Note that G2 immediately implies G1. 2 Both of these results are proved finitistically, without
resorting to any kind of infinitary or set theoretic
methods.
Barkley Rosser improved G1 to
GR. There exist sentences in the language of set theory
which cannot be proved or refuted in ZFC (assuming that ZFC
is consistent). In fact, these exist if and only if ZFC is
consistent.
G1 is proved by the diagonalization method. GR is proved by
a trickier form of the diagonalization method. G2 is proved
by first applying the diagonalization method to obtain a
sentence A not provable in ZFC (assuming ZFC is
consistent). This is followed by a careful verification of
a miraculous fact  A is equivalent to Con(ZFC) – where
this equivalence is provable in ZFC. Hence Con(ZFC) is not
provable in ZFC (provided ZFC is consistent).
Note that Con(ZFC) is a statement concerning formal systems
of set theory, which is of the utmost importance for
f.o.m., but rather remote from normal mathematical
activity. So we are lead to the general question, which has
been an obsession of mine for over 40 years:
Q1. What kinds of mathematical questions cannot be settled
in ZFC? I.e., neither proved nor refuted in ZFC. I.e.,
independent of ZFC.
After G1 and G2, Gödel tackled the most well known question
in set theory prominently left open by Cantor – the
continuum hypothesis, CH. This asserts that every set of
real numbers is mappable onto the set of all real numbers,
or embeddable into the set of all integers.
G3. CH is not refutable in ZFC (assuming ZFC is
consistent).
The method used by Gödel to establish G3 is completely
different than the methods he used to establish G1,G2. For
G3, Gödel invented the method of inner models. He showed
that every model of ZFC can be cut back to a model of ZFC +
CH. This yields G3 via the completeness theorem for first
order predicate calculus with equality – also due to Gödel!
It is also possible to avoid the use of his completeness 3
theorem (as he does). The entire proof of G3 is given in a
tiny fragment of ZFC.
Cohen provided the other half.
GC. CH is independent of ZFC (assuming ZFC is consistent).
For Cohen’s half, he invented a third technique for showing
independence from ZFC, called forcing. In particular, Cohen
showed that every countable model of ZFC can be expanded to
a model of not CH. This yields GC via the completeness
theorem. The completeness theorem can be avoided (as Cohen
does). The entire proof of GC is given in a tiny fragment
of ZFC.
We now have three methods for proving independence from ZFC
– two due to Gödel, and one due to Cohen. The last two of
these methods was invented to show the independence of CH
from ZFC.
There is a fourth method that has proved fruitful for
showing independence from ZFC, which I use intensively.
This will be described in Lecture 2.
Thus independence from ZFC touches fundamental problems in
set theory. Subsequent work following Cohen showed that
among the large backlog of open set theoretic problems, a
considerable percentage are independent of ZFC.
Over the years, mathematicians have generally reacted by
steering clear of difficult set theoretic questions,
preferring to either ignore them or reduce their level of
generality if set theoretic difficulties arise.
Set theoretic difficulties are generally perceived to be so
remote from and sharply different than normal mathematical
difficulties (topological, geometric, analytic,
combinatorial, etcetera), that they should be separated.
In particular, the steering clear of set theoretic
difficulties is generally regarded as an effective
defensive move by the mathematical community with little or
no real cost to mathematics.
Mathematicians will generally embrace the singular
generality of set theoretically formulated results when the
set theory involved is readily manageable. E.g., every 4
field has an algebraic closure. But if there were set
theoretic difficulties in proving this for arbitrary
fields, mathematicians would rather quickly focus on, say,
countable fields. They would regard that extra generality
as definitely not worth the cost.
But what about questions that are not overtly set
theoretic? We make the following conjecture.
C1. Every mathematical question that is not heavily set
theoretic, which has been raised in a published paper
appearing in a refereed journal before today, the paper
having been assigned an AMS classification number not in
logic or set theory, can be settled in ZFC.
The closest exception that we know of to C1 lies in the
realm of Borel measurable functions on the real line. This
exception is almost explicit in the joint writings of the
Paris analysts Debs and Saint Raymond. We will discuss
these results in Lecture 2, as the proof of independence
came after 2000.
Some decades ago, Borel measurable functions – particularly
those at finite levels – were not considered to be set
theoretic at all, and lied at the heart of real analysis.
However, times have changed, and the focus of mathematical
activity has become more and more concrete. Normal
mathematical activity has become even more focused on the
finite, the discrete, and the well behaved continuous (or
almost continuous), in contexts where continuous objects
are readily approximable by finite objects.
The last part of Lecture 2 will focus on (nearly) the most
concrete kind of finite statements involving infinitely
many objects considered by mathematicians. This very recent
work suggests the following conjecture:
C2. There is no inherent limitation to the construction of
“natural” or “beautiful” mathematical statements that can
be shown to be independent of ZFC (assuming ZFC does not
prove its own inconsistency), except the obvious one: the
statement must involve infinitely many objects.
In fact, we make the following extremely strong conjecture. 5
C3. There is a core finite combinatorial structure that
emerges in a natural and uniform way in any nontrivial
mathematical context, which cannot be handled within ZFC.
Specifically, there is a natural, beautiful, and uniform
way to seek more refined information in any nontrivial
mathematical development involving infinitely many objects,
where the resulting statements can be proved only by adding
additional axioms to ZFC believed by set theorists to be
consistent. This is because this core finite combinatorial
structure is easily hidden in even the modest complexity
inherent in any nontrivial mathematical development
involving infinitely many objects.
In these two lectures, we will not only present statements
independent of ZFC, but also statements independent of
substantial fragments of ZFC. However, we will not discuss
statements independent of only finite set theory (or Peano
arithmetic), except in passing  finite set theory (or PA)
does not represent substantial set theoretic methods. This
will exclude any substantive discussion of historically
important work by Goodstein, Paris/Harrington, myself, and
others.
2. THE MOST ABSTRACT: GENERAL SET THEORETIC STATEMENTS.
We begin with general set theoretic statements. These
involve sets of unlimited cardinality. The oldest example
is implicit in Cantor. Here is a particularly attractive
form.
GST1. There exists an uncountable set A such that 1) A is
not the union of fewer than A sets that are smaller than A;
2) the power set of any set smaller than A is also smaller
than A.
Note that this property holds for sets of cardinality 0,w,
and for no finite cardinality ≥ 1. Such an uncountable set
must be awesomely enormous, as some simple investigations
reveal.
In modern set theoretic terminology, GST1 is equivalent to
the assertion “there exists a strongly inaccessible
cardinal”. GST1 asserts that the cardinality of A is a
strongly inaccessible cardinal.
It is known that “there exists a strongly inaccessible
cardinal” is not provable in ZFC, provided ZFC is 6
consistent. It is strongly believed that it is not
refutable in ZFC, although it is known that the
nonrefutability cannot be established by merely knowing
that ZFC is consistent, or that ZFC does not prove that ZFC
is inconsistent, etcetera.
There are much stronger well known statements of this
general kind – i.e., the existence of a set with a basic
set theoretic property that depends only on its
cardinality:
GST2. There exists a nonempty set A such that there is a
countably additive measure on all subsets of A that takes
on only the values 0 and 1, where points have measure 0,
and A has measure 1.
GST2 is one of many equivalent formulations of “there
exists a measurable cardinal”. This property obviously
fails for nonempty countable sets A. Basic work in the
1940s shows that the least cardinality k of such a set A is
a strongly inaccessible cardinal, and also that there are
lots of strongly inaccessible cardinals below k. Thus GST2
not only implies GST1, but is much stronger from many
points of view.
The set theory community expresses great confidence in the
consistency of ZFC, and equally great confidence in the
consistency of ZFC + “there exists a strongly inaccessible
cardinal”. However, with ZFC + “there exists a measurable
cardinal”, confidence abounds, but it is subdued.
A natural hierarchy of statements such as GST1 and GST2 has
emerged from the set theory community, and it is called the
hierarch of large cardinal hypotheses. The weakest natural
large cardinal hypotheses that have emerged are at the
level of GST1. GST2 is generally viewed as a medium large
cardinal hypothesis, with many interesting levels strictly
between GST1 and GST2. As we move up from GST2, we move
into the region of large large cardinal hypotheses, where
confidence in consistency is visibly restrained. Who cares? What good are such amorphous abstract monstrosities postulated by the likes of GST1 and GST2?
Stay tuned to Lecture 2, which concentrates on the
relevance of something like GST1.1 in surprisingly concrete
contexts. 7 3. THE SECOND MOST ABSTRACT: LIMITED SET THEORETIC
STATEMENTS.
In limited set theoretic statements, the cardinality of the
objects under consideration are bounded at the outset, or
can be demonstrated to be bounded.
The most well known of such highly set theoretic statements
is the continuum hypothesis (CH), which states:
CH. Every set of real numbers is either mappable onto the
set of all real numbers, or embeddable into the set of all
integers.
As mentioned before, Godel and Cohen proved the
independence of CH from ZFC (assuming ZFC is consistent).
The heavily set theoretic nature of CH is readily
identifiable by the use of arbitrary sets of real numbers,
and arbitrary functions on the real numbers. These sets and
functions are NOT required to be presentable in the forms
that are so common in ordinary mathematical activity.
Specifically, the sets and functions in CH are not required
to be presented sequentially.
Borel measurable functions are required to be presented
sequentially  for finite level Borel measurable functions,
the sequential presentations are particularly explicit.
The Borel sets in complete separable metric spaces form the
least family of subsets containing the open sets, and
closed under taking countable unions, countable
intersections, and complements.
The Borel functions between two complete separable metric
spaces can be defined in two equivalent ways. One is that
the inverse image of every open set is Borel. The other is
that they form the least family of functions containing the
continuous functions, and closed under pointwise limits of
sequences of functions.
These definitions naturally give rise to hierarchies of
length w1. But they can be cut off at w, resulting in the
finitely Borel sets and functions. Generally speaking, 8
phenomena appearing at some level already appear at levels
at most 2 or 3.
What happens if we Borelize CH?
CHB. Every Borel set of real numbers is either Borel
mappable onto the set of all real numbers, or Borel
embeddable into the set of all integers.
CHB is provable in a tiny fragment of ZFC.
There is a rather substantial literature on a variety of
set theoretic statements involving arbitrary sets and
functions on the real line (or in complete separable metric
spaces). A great many of these statements have been shown
to be independent of ZFC (generally, but not always,
assuming only that ZFC is consistent). These results use
the forcing method of Cohen in elaborate ways.
Here is a typical such result (although atypically
brilliant!), going back to the 1970s, due to Richard Laver.
A set of reals S is said to be strongly measure zero if and
only if for any sequence Œ1,Œ2,… of positive reals, there
exists a sequence I1,I2,… of intervals covering S, where
each In has length less than Œn.
Borel’s Conjecture (1919). Strongly measure zero is
equivalent to countable.
This is the same Emile Borel as in “Borel sets and Borel
functions”.
Borel’s Conjecture is independent of ZFC. It implies the
negation of CH, but is not implied by the negation of CH.
These results use only the consistency of ZFC.
Again, we can Borelize this statement.
“Every Borel set of real numbers that is strongly measure
zero is countable” is provable using a tiny fragment of
ZFC.
Not all important limited set theoretic statements live in
subsets of complete separable metric spaces. 9
Let ℘ be the power set operation. Form the set theoretic
point of view, the real line and complete separable metric
spaces amount to taking ℘w, where w is the set of all set
theoretic natural numbers (the first limit ordinal in the
sense of von Neumann).
Sets of real numbers live in ℘℘w, and we can form
℘℘℘…℘w, any finite number of times. This is incredibly
gargantuan from the point of view of ordinary mathematical
activity.
However, this is nothing compared to what is available in
ZFC, and thus we regard this as limited set theory.
In particular, in ZFC, we have the axiom of replacement,
and so we can form the entire sequence
w, ℘w, ℘℘w, ℘℘℘w, ℘℘℘℘w, …
and therefore the union of this sequence. Then we continue.
We can continue this way throughout all of the von Neumann
ordinals.
Of particular interest at the level of ℘℘℘w, or ℘℘¬, is
the existence of a probability measure on the unit
interval:
PMI. There is a countably additive measure on all subsets
of [0,1], where points have measure 0 and [0,1] has measure
1.
Even though PMI is a limited set theoretic statement, in
some ways it behaves like an unlimited set theoretic
statement, because of this result of Solovay from the
1960s:
ZFC + PMI is consistent if and only if ZFC + “there exists
a measurable cardinal” is consistent (Solovay). However,
these two hypotheses do not imply each other.
4. THE THIRD MOST ABSTRACT: THE PROJECTIVE HIERARCHY OF
SETS OF REALS.
The projective hierarchy of sets of reals starts off at the
bottom with the Borel subsets of ¬n, n ≥ 1. 10
At the next level of the projective hierarchy, we have the
so called A and CA sets. The A sets (analytic sets) are the
projections
{(x1,…,xn): ($y Œ ¬)((x1,…,xn,y) Œ S}
where S is a Borel subset of ¬n+1.
The CA sets (coanalytic sets) are the complements of
analytic sets.
It is classical that
Borel is properly included in A (or CA).
A « CA = Borel.
Analytic sets are much more abstract than Borel sets, in
the following sense. In order to determine
membership/nonmemberhip of the real number x in a Borel
set, only countably many questions have to be answered
regarding x, each question calling for a comparison of x
with a given rational number. This takes the form of a
countably branching decision tree.
On the other hand, in order to determine membership of x in
an analytic set, countably many questions have to be
answered regarding x to establish membership. However, no
countably many questions are generally sufficient to
establish nonmembership. Furthermore, there is no way to
know in advance the schedule of questions even to establish
membership (i.e., there is no appropriate countably
branching decision tree).
Then comes PCA and CPCA sets. These are projections of CA
and sets, and complements of PCA sets.
This process is normally continued through all finite
levels. The sets of reals that appear somewhere are called
the projective sets of reals. It is classical that the
hierarchy does not collapse.
A handful of basic questions naturally arise concerning
properties of projective sets. Generally speaking, such
properties are provable about Borel sets, but cannot be
proved at the PCA and CPCA levels and higher; sometimes
they can be proved at the A and CA levels. 11
i. Do uncountable sets have perfect subsets? This is
provable for Borel sets, and even analytic using a tiny
fragment of ZFC. But “every uncountable coanalytic set has
a perfect subset” is independent of ZFC. Also “every
uncountable projective set has a perfect subset” is
independent of ZFC.
ii. Are the sets Lebesgue measurable? Trivial for Borel
sets, and provable for analytic and coanalytic sets using a
tiny fragment of ZFC. But “every PCA set is Lebesgue
measurable” is independent of ZFC. Also “every projective
set is Lebesgue measurable” is independent of ZFC.
iii. Are the new projections at each level equally
complicated? E.g., “any two analytic sets that are not
Borel are in oneone correspondence via a Borel bijection
of the reals”. This is independent of ZFC. There is an
obvious analog higher up in the projective hierarchy.
There is an interesting aspect to this phenomena. There are
two rival extensions of ZFC that settle all such questions
about the projective hierarchy – but in opposite ways!
The first axiom is the so called Gödel axiom of
constructibility – which Gödel may have subscribed to at
one point, but sharply repudiated later in print. (For the
later Gödel, there is right and a wrong for set theoretic
sentences, no matter how unlimited).
This axiom, written V = L, settles not only all such
questions about the projective hierarchy (negatively), but
also CH (positively), PMI (negatively), and all related
questions about sets of limited size. In addition, we know
that V = L is independent just from the consistency of ZFC.
However, V = L does not settle the existence of a strongly
inaccessible cardinal and related questions, although it
does settle the existence of a measurable cardinal
(negatively).
The second axiom is a natural strengthening of the
existence of measurable cardinals. It is called “the
existence of Woodin cardinals”. Measurable cardinals will
suffice for questions about A, CA, and PCA (sometimes
CPCA), but not higher. The questions about the projective
hierarchy are settled positively. However CH, PMI, and
related questions about sets of limited size remain
independent. It does settle the existence of a strongly 12
inaccessible cardinal and, in fact, the existence of a
measurable cardinal (positively), but there is also a
hierarchy of yet stronger principles of a related kind, in
unlimited set theory, that remain independent.
In contrast to V = L, the consistency of the existence of
Woodin cardinals must be taken on faith – just as is the
case with the existence of measurable cardinals.
The set theorists usually are adamant about rejecting V = L
in favor of the existence of Woodin cardinals, and regard
the seemingly “complete” treatment of the projective
hierarchy given by V = L as FALSE – whereas the opposite
seemingly “complete” treatment of the projective hierarchy
given by Woodin cardinals as TRUE. It is the main
vindication put forth by contemporary set theorists of the
study of large cardinals.
My own view is that the projective hierarchy is so far
removed from normal mathematics, that even if we accept the
previous paragraph in the sense intended by the set
theorists, this development does not come close to
satisfactorily establishing the importance or relevance of
large cardinals to mathematics.
We want demonstrably necessary uses of large cardinals in
concrete mathematical contexts – the more concrete the
better.
5. THE FOURTH MOST ABSTRACT: BOREL SETS OF REALS.
We have already discussed Borel sets and functions on the
reals (and complete separable metric spaces), and
contrasted them with the higher levels of the projective
hierarchy.
The most striking independence result from ZFC in the realm
of Borel sets was obtained after 2000, and will be
discussed in Lecture 2. It grew out of work of the Paris
analysts Debs and Saint Raymond.
The independence results before 2000, in the realm of Borel
sets, were most convincing at important levels
significantly below ZFC. We discuss some of these now.
The lowest level that we consider in this section is
countable set theory. This amounts to simply eliminating 13
the power set axiom from ZFC. We can, optionally, also add
“every set is countable” to ZFC\P. We can also view ZFC\P
as “separable mathematics”.
In ZFC\P, we cannot construct ¬ or ℘w as a set. However,
we can speak of elements of ¬ and elements of ℘w. We can
develop a complete treatment of the Borel sets and
functions on the reals (and complete separable metric
spaces) well within ZFC\P.
We begin with one form of Cantor’s theorem that the reals
are uncountable.
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 (Borel theory).
THEOREM 5.2. There is a Borel function F:¬w Æ ¬ such that
for all x Œ ¬w, F(x) is not a coordinate of x.
The construction of F is by diagonalization. We would
expect that F(x) depends on the order of the arguments of
x.
THEOREM 5.3. Every permutation invariant Borel function
from ¬w into ¬ maps some infinite sequence to a coordinate.
Permutation invariance makes sense for F:¬w Æ ¬w. 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 of w. The following
result holds under a variety of related notions.
THEOREM 5.4. Every permutation invariant Borel function
from ¬w into ¬w maps some infinite sequence into an infinite
subsequence.
We proved Theorems 5.3 and 5.4 by going just beyond ZFC\P,
and showed that they were not provable in ZFC\P. Certainly,
the use of ℘℘w is more than sufficient to prove Theorems
5.3 and 5.4.
We now give an example at a much higher level. Recall that
in ZFC, we can iterate ℘ infinitely often, starting at w, 14
and then take the limit. We can continue transfinitely,
obtaining what is called the cumulative hierarchy of sets.
The level we are going to represent is by iterating the
power set through all countable ordinals, just shy of
putting them all together.
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 an old classical result. It is proved in a tiny
fragment of ZFC.
THEOREM 5.5. 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 ¬. However, there
may not be a Borel selection for E on ¬.
Now consider this.
THEOREM 5.6. Let E be a symmetric Borel set in the plane.
Then E or its complement has a Borel selection on ¬.
Theorem 5.6 can be proved using all countably transfinite
iterations of the power set operation, but not without.
The proof of Theorem 5.6 relies heavily on earlier work of
D.A. Martin in infinite game theory (Borel determinacy).
We will return to the Borel realm in Lecture 2 with some
post 2000 work concerning Borel selection that is
independent of ZFC.
6. THE FIFTH MOST ABSTRACT: INFINITE SETS OF INTEGERS.
The most striking independence results from ZFC in the
realm of infinite sets of integers appear in the
substantially developed post 2000 work on “Boolean Relation
Theory”, the title of my forthcoming research monograph.
Here we will discuss some pre 2000 highlights that
represent rather modest levels of independence – at least
by the standards of this talk. 15
It is an accepted part of ZFC to allow sets of natural
numbers to be formed based on quantifying over all sets of
natural numbers – including the one being formed.
This is well known to be akin to the least upper bound
principle for the real line, as opposed to the convergence
of Cauchy sequences, or the existence of convergent
subsequences of bounded infinite sequences.
Poincare and Weyl railed against the use of such a
principle on the general grounds of circularity. Recall
that in 1902, Russell had obtained his famous paradox via a
set formation principle that bears at least some
resemblance to this.
Current conventional wisdom is that the “circularity”
Poincare and Weyl disavowed is not vicious like the one
Russell identified, and it sits well within a tiny fragment
of ZFC.
We showed that a number of basic results require this
principle, including two that are celebrated existing
theorems in infinitary combinatorics.
The first is due to J.B. Kruskal. A tree is a finite poset
with a least element called the root, where the
predecessors of every vertex are linearly ordered.
There is an obvious inf operation on the vertices of any
finite tree.
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)).
Note that these h are homeomorphic embeddings as
topological spaces.
Kruskal’s Tree Theorem. In any infinite sequence of finite
trees, one tree is inf preserving embeddable into a later
tree.
The second example is a theorem of Robertson and Seymour
called the graph minor theorem.
Let G,H be graphs (undirected, with loops and multiple
edges allowed). We say that G is minor included in H if and 16
only if G can be obtained (up to isomorphism) from H by
applying the following operations zero or more times in H:
contracting edges to a point, removing edges, and removing
vertices.
Graph Minor Theorem. In any infinite sequence of finite
simple graphs, one graph is minor included in a later
graph.
We showed that in an appropriate sense, the use of the
principle in question must be applied arbitrarily many
finite times, in order to prove the graph minor theorem.
Just one application is sufficient for the Kruskal tree
theorem.
7. THE SIXTH MOST ABSTRACT: FINITE COMBINATORICS, WITH
UNBOUNDED EXISTENCE.
In finite combinatorics, all data is entirely finite. Most
commonly, we assert that any given finite object of a
certain kind has a specific property, where that property
may or may not be algorithmically checkable.
The most well known illustrative examples come from number
theory.
P01. A specific Diophantine equation, or family of
Diophantine equations, has no integral (rational) solution.
The Riemann hypothesis. Goldbach’s conjecture.
P02. All Diophantine equations of a certain form have at
least one (infinitely many) integral (rational) solution.
The twin prime conjecture. e + p is irrational
(transcendental).
S02. A specific Diophantine equation has at most finitely
many solutions. e + p is rational (algebraic).
P03. All Diophantine equations of a certain form have at
most finitely many solutions (Faltings). Certain algebraic
numbers can be approximated by rationals in certain ways
with finitely many exceptions (Roth).
Normally mathematics strives to be P01. Thus when a P02
theorem is proved, people want to provide an upper bound
for the existential quantifier (in terms of the outermost
universal quantifier), thereby putting the theorem into P01 17
form. When a S02 theorem is proved, people want to provide
an example of the existential quantifier in front, again
rendering it in P01 form. Similarly, when a P03 theorem is
proved, people want to provide an upper bound for the
existential quantifier (in terms of the outermost universal
quantifier), which still renders it in P01 form.
Thus in finite combinatorics, we have the classical finite
Ramsey theorem.
FINITE RAMSEY THEOREM. Let n be sufficiently large relative
to k,r,m ≥ 1. Any coloring of the unordered k tuples from an
n element set, with at most m colors, has an r element
monochromatic set. I.e., an r element set all of whose
unordered k tuples have the same color.
Obviously FRT as stated is P02. Ramsey himself gave upper
bounds on n as a function of k,r,m. They are iterated
exponential, with the stack of size roughly k. This
converts FRT to form P01. Ramsey also gave lower bounds of
the same character.
Of course, the finite Ramsey theorem is way below our radar
screen in terms of independence.
Up to very recently, all of the remotely mathematically
natural sentences about finite objects that have been shown
to be independent of even relatively weak systems such as
finite set theory, are P02 or P03. One actually sees that
the independence is tied up with the enormous size of any
realization of the existential quantifier. In such cases,
it is known that one cannot exhibit a bound in the usual
sense, rendering them in P01 form.
At our threshold level of independence, the examples are
our various finite forms of Kruskal’s tree theorem and the
graph minor theorem.
Kruskal’s tree theorem was proved by Kruskal in more
general form using labeled vertices. In particular, we can
use vertices labeled from a finite set.
FINITE KRUSKAL TREE THEOREM (one of many). 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. 18 Here by a truncation of T, we mean the subtree of vertices
at or below a certain height.
The above theorem also requires the construction of sets of
natural numbers by means of quantification over all sets of
natural numbers – the principle that Poincare and Weyl
objected to.
The growth rate associated with this finite form
corresponds exactly to its level of independence.
8. THE MOST CONCRETE: COMPLETELY FINITE COMBINATORICS.
Completely finite combinatorics is required to be in P01
form.
By far the most significant independence results are very
recent, and will be presented in Lecture 2. ...
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.
 Fall '08
 JOSHUA
 Math

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