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Unformatted text preview: 1 MY FORTY YEARS ON HIS SHOULDERS
by
Harvey M. Friedman*
Ohio State University
Gödel Centenary
Delivered: April 29, 2007
Expanded: May 24, 2007
1. General Remarks.
2. The Completeness Theorem.
3. The First Incompleteness Theorem.
4. The Second Incompleteness Theorem.
5. Lengths of Proofs.
6. The Negative Interpretation.
7. The Axiom of Choice and the Continuum Hypothesis.
8. Wqo Theory.
9. Borel Selection.
10. Boolean Relation Theory.
11. Finite Graphs.
12. Incompleteness in the Future. 1. GENERAL REMARKS.
Gödel's legacy is still very much in evidence. We will not
attempt to properly discuss the full impact of his work and
all of the ongoing important research programs that it
suggests. This would require a book length manuscript.
Indeed, there are several books discussing the Gödel legacy
from many points of view, including, for example, [Wa87],
[Wa96], [Da05], and the historically comprehensive five
volume set [Go,8603].
In sections 27 we briefly discuss some research projects
that are suggested by some of his most famous
contributions.
In sections 811 we discuss some highlights of a main
recurrent theme in our own research, which amounts to an
expansion of the Gödel incompleteness phenomenon in a
critical direction.
The incompleteness phenomenon lies at the heart of the
Gödel legacy. Some careful formulations, informed by some
post Gödelian developments, are presented in sections
3,4,5. 2
One particular issue that arises with regard to
incompleteness has been a driving force for a considerable
portion of my work over the last forty years. This has been
the ongoing search for necessary uses of set theoretic
methods in normal mathematics.
By way of background, Gödel’s first incompleteness theorem
is an existence theorem not intended to provide a
mathematically intelligible example of an unprovable
sentence.
Gödel’s second incompleteness theorem does provide an
entirely intelligible example of an unprovable sentence specifically, the crucially important consistency
statement. (Remarkably, Gödel demonstrates by a brief
semiformal argument, that the sentence he constructs for
his first incompleteness theorem is demonstrably equivalent
to the consistency statement  hence the consistency
statement is not provable.)
Nevertheless, the consistency statement is obviously of a
logical nature rather than of a mathematical nature. This
is a distinction that is readily noticed by the general
mathematical community, which naturally resists the notion
that the incompleteness theorem will have practical
consequences for their own research.
Genuinely mathematical examples of incompleteness from
substantial set theoretic systems had to wait until the
well known work on the axiom of choice and the continuum
hypothesis by Kurt Gödel and Paul Cohen. See [Go40],
[Co63].
Here, the statement being shown to be independent of ZFC the continuum hypothesis  is of crucial importance for
abstract set theory.
However, mathematicians generally find it easy to recognize
an essential difference between overtly set theoretic
statements like the continuum hypothesis (CH) and “normal”
mathematical statements. Again, this is a particularly
useful observation for the mathematician who resists the
idea that incompleteness may impact their own research.
Specifically, the use of unrestricted uncountable sets (of
real numbers) in CH readily distinguishes CH from “normal” 3
mathematics, which relies, almost exclusively, on the
“essentially countable”.
A more subtle example of an overtly set theoretic statement
that requires a second look to see its overtly set
theoretic character, is Kaplansky’s Conjecture concerning
automatic continuity. In one of its more concrete special
forms, it asserts that
*) every homomorphism from the Banach algebra c0 of infinite
sequences of reals converging to 0 (under the sup norm) to
any separable Banach algebra, is continuous.
Now *) was refuted using the continuum hypothesis, and
later shown to be not refutable without the continuum
hypothesis; i.e., not refutable in the usual ZFC axioms.
See [Dal01] for the refutation, and [DW87] for the
consistency result.
It is, of course, much easier for mathematicians to
recognize the overtly set theoretic character after they
learn that there are set theoretic difficulties. By taking
the negation,
**) there exists a discontinuous homomorphism from the
Banach algebra c0 of infinite sequences of reals converging
to 0 (under the sup norm) to some separable Banach algebra.
It is clear that one is asking about the existence of an
object that was well known, even at the time, to
necessarily have rather pathological properties. This is
the case even for discontinuous group homomorphisms from ¬
into ¬ (which can be shown to exist without the continuum
hypothesis). For instance, it is well known that there are
no discontinuous group homomorphisms from ¬ into ¬ that are
Borel measurable.
At the outer limits, normal mathematics is conducted within
complete separable metric spaces. (Of course, we grant that
it is sometimes convenient to use fluff  as long as it
doesn’t cause any trouble). Functions and sets are normally
Borel measurable within such so called Polish spaces. In
fact, the sets and functions normally considered in
mathematics are substantially nicer than Borel measurable,
generally being continuous or at least piecewise continuous
 if not outright countable or even finite. 4
NOTE: Apparently, nonseparable arguments are being used in
the proofs of certain number theoretic results such as
Fermat’s Last Theorem. We have been suggesting strongly
that this is an area where logicians and number theorists
should collaborate in order to see just how necessary such
appeals to nonseparable arguments are. We have conjectured
that they are not, and that EFA = IS0(exp) = exponential
function arithmetic suffices. See [Av03].
We now know that the incompleteness phenomena does
penetrate the barrier into the relatively concrete world of
Borel measurability  and even into the countable and the
finite world  with independence results of a mathematical
character.
In sections 811 we discuss my efforts concerning such
concrete incompleteness, establishing the necessary use of
abstract set theoretic methods in a number of contexts,
some of which go well beyond the ZFC axioms.
Yet it must be said that our results to date are very
limited in scope, and demand considerable improvement. We
are only at the very beginnings of being able to assess the
full impact of the Gödel incompleteness phenomena.
In particular, it is not yet clear how strongly and in what
way the Gödel incompleteness phenomenon will penetrate
normal mathematical activity. Progress along these lines is
steady but painfully slow. We are confident that a much
clearer assessment will be possible by the end of this
century  and probably not much earlier.
In section 12, we take the opportunity to speculate far
into the future. 2. THE COMPLETENESS THEOREM.
In his Ph.D. dissertation, [Go29], Gödel proved his
celebrated completeness theorem for a standard version of
the axioms and rules of first order predicate calculus with
equality.
This result of Gödel was anticipated, in various senses, by
earlier work of T. Skolem as discussed in detail in the
Introductory notes in Vol. I of [Go,8603], 4459. These
Introductory notes are written by Burton Dreben and Jean
van Heijenoort. 5 On page 52, the following passage from a letter from Gödel
to Hao Wang, is quoted (December 7, 1967):
“The completeness theorem, mathematically, is indeed an
almost trivial consequence of Skolem 1923a. However, the
fact is that, at the time, nobody (including Skolem
himself) drew this conclusion (neither from Skolem 1923a
nor, as I did, from similar considerations).”
According to these Introductory Notes, page 52, the
situation is properly summarized as follows:
“Thus, according to Gödel, the only significant difference
between Skolem 1923a and Gödel 19291930 lies in the
replacement of an informal notion of “provable” by a formal
one ... and the explicit recognition that there is a
question to be answered.”
NOTE: Skolem 1923a is [Sk22] in our list of references.
To this, we would add that Gödel himself relied on a
semiformal notion of “valid” or “valid in all set theoretic
structures”. The appropriate fully formal treatment of the
semantics of first order predicate calculus with equality
is credited to Alfred Tarski. However, as discussed in
detail in [Fe04], surprisingly the first clear statement in
Tarski’s work of the formal semantics for predicate
calculus did not appear until [Ta52] and [TV57].
Let us return to the fundamental setup for the completeness
theorem. The notion of structure is taken in the sense most
relevant to mathematics, and in particular, general
algebra: a nonempty domain, together with a system of
constants, relations, and functions, with equality as
understood.
It is well known that the completeness proof is so robust
that no analysis of the notion of structure need be given.
The proof requires only that we at least admit the
structures whose domain is an initial segment of the
natural numbers (finite or infinite). In fact, we need only
admit structures whose relations and functions are
arithmetically defined; i.e., first order defined in the
ring of integers.
However, the axioms and rules of logic are meant to be so 6
generally applicable as to transcend their application in
mathematics. Accordingly, it is important to interpret
logic with structures that may lie outside the realm of
ordinary mathematics. A particularly important type of
structure is a structure whose domain includes absolutely
everything.
Indeed, it can be argued that the original Fregean
conception of logic demands that quantifiers range over
absolutely everything. From this viewpoint, quantification
over mathematical domains is a special case, as “being in a
given mathematical domain” is treated as (the extensions
of) a unary predicate on everything.
These general philosophical considerations were sufficient
for an applied philosopher like me to begin reworking logic
using structures whose domain consists of absolutely
everything.
The topic of logic in the universal domain has been taken
up in the philosophy community, and in particular, by T.
Williamson in [RW03], [Wi00], [Wi03], [Wi06].
We did not publish on this topic, but reports on our
results are available on the web. Specifically, in [Fr99],
and in [Fr02], 6599. Here we describe some highlights from
[Fr99]
and [Fr02] to give the reader a sense of the development.
We plan to publish a monograph on this topic in the not too
distant future.
There is a preliminary result, which we already knew in the
early 1970’s (see Theorem 2.2 below). Let us consider the
completeness theorem for structures whose domain is a given
fixed nonempty set D. As a preliminary to considering
domains D that lie outside the scope of mathematics, we
will view D as a nonempty set in set theory without the
axiom of choice.
We begin with the following well known result. We use PC(=)
for the usual first order predicate calculus with equality.
THEOREM 2.1. (ZF). Let D be an infinite well ordered set,
and j be a sentence in PC(=). The following are equivalent.
i. j is true in all infinite structures.
ii. j is true in all structures with domain N.
iii. j is true in all structures with domain D. 7
iv. j is provable with the usual axioms and rules augmented
with the scheme “there exists at least n objects”, n Œ N.
But what properties do we need about D in order that we can
use only structures with domain D? Let INF be the axiom
scheme used in iv above. We have known the answer since the
early 1970’s (unpublished).
THEOREM 2.2. (ZF). Let D be a nonempty set. The following
are equivalent.
i. A sentence is true in all structures with domain D if
and only if it is provable from INF. (D completeness of
INF).
ii. D has at least two elements, there is a oneone
function from D2 into D, and D has a linear ordering.
The proof of Theorem 2.2 uses the Ehrenfeucht Mostowski
method of stretching indiscernibles, in an essential way.
See [EMo56]. Thus we can motivate the development of that
machinery for a particularly fundamental purpose connected
with completeness. Instead, it was originally introduced to
provide a fundamental technique for building models with
lots of automorphisms.
We can move closer to our intention of dealing with
structures whose domain includes absolutely everything by
working within class theory, and considering structures
whose domain is the class V of all sets. But bear in mind
that V is a rather small part of the universe W of
everything!
Below, NBG refers to the von Neuumann Bernays Gödel theory
of classes used by Gödel in [Go40]. MK is the stronger
Morse Kelley theory of classes.
THEOREM 2.3. (VB). The following are equivalent.
i. V completeness of INF.
ii. V has a linear ordering.
Furthermore, the V completeness of INF is neither provable
nor refutable in MK + AxC (the axiom of choice for sets
only).
We now come to the universal domain W. To begin with, we do
not expect to have the W completeness of INF. This is
because we do not expect to be able to linearly order W.
In particular, the sentence 8 < is a linear ordering
is not W satisfiable (satisfiable with domain W). I.e., the
sentence
< is not a linear ordering
is W valid; i.e., holds in all structures with domain W.
The idea is that W is much too varied (as varied as
possible!) to support any criterion for strictly comparing
any two things in it. I.e., the world is much too varied to
support the idea of strictly comparing any two entities.
However, once we decide to go down this path of accepting
such ideas on what relations exist on W, the challenge is
to obtain a robust completeness theorem.
Let us examine the basis behind rejecting the existence of
a linear ordering of W.
We ought to have two distinct entities which cannot be told
apart in the following sense. There exists x ≠ y such that
for all binary predicates R, R(x,y) ´ R(y,x).
But this raises an important issue as to the nature of
predication. In mathematical contexts, we customarily use a
notion of predication that allows the free use of any
finite number of objects. These are often called
parameters. Under this notion of predication, which we call
general predication, we cannot have such x ≠ y, since we
can take R to hold of exactly the pairs u,v such that u =
x.
However, the notion of pure predication is much more
restrictive, and is arguably conceptually prior to general
predication. Here one cannot refer to specific objects in
forming a predicate. One must operate purely conceptually.
We write a superscript p to indicate pureness, and a
superscript g to indicate generalness. Thus we have the
principle
($x ≠ y)(" binary Rp)(Rp(x,y) ´ Rp(y,x)).
Is there a corresponding binary indiscernibility principle 9
for general predication? Yes.
(" binary Rg)($x ≠ y)(Rg(x,y) ´ Rg(y,x)).
It is clear that we are going to have to distinguish
between two notions of W structures. A pure (general) W
structure is a structure whose domain is W and whose
constants, relations, and functions are given by pure
(general) predication on W.
In the case of constants, this means that the unary
predicate picking out the constant is pure (general). Of
course, there is obviously a general predicate picking out
any given object  but not necessarily a pure predicate.
Functions are treated in terms of their corresponding
relation of one more argument.
We now need a basic theory of predication on W, which will
serve as an unimpeachable base theory for the development
of logic on the universal domain W.
The language of BTPpg is three sorted (BTP is read “basic
theory of predication”). The first sort consists of
objects, the second sort consists of pure unary predicates
on W, and the third sort consists of general unary
predicates on W. We have a constant symbol 0 of the first
sort, and a binary function symbol < > from and into the
first sort. We view 0 and < > as “pure”. The atomic
formulas are s = t, Rp(t), Rg(t), where s,t are terms of the
first sort, Rp is a variable ranging over the second sort (a
pure variable), and Rg is a variable ranging over the third
sort (a general variable). We will not use = for the second
and third sorts.
Note that in BTPpg, we have unary predicates only, where
predicates of more arguments are treated as unary
predicates using < >.
The axioms of BTPpg are as follows.
1.
2.
3.
in
is
4.
in <x,y> ≠ 0.
<x,y> = <z,w> ´ (x = z Ÿ y = w).
($Rp)("x)(Rp(x) ´ j), where j is a formula in L(BTPpg)
which Rp is not free, and where every free variable in j
either a pure variable or x.
($Rg)("x)(Rg(x) ´ j), where j is a formula in L(BTPpg)
which Rg is not free. 10 The restrictions in 3 reflect the idea of pure predication,
and the (lack of) restrictions in 4 reflect the idea of
general predication.
It is clear that BTPpg proves that every pure predicate is
equivalent to a general predicate. We shall see that the
converse is not provable.
There is an important principle relating general and pure
predication that we shall see is not provable in BTPpg.
This asserts that every general unary predicate is
equivalent to the cross section of a pure binary predicate.
(Binary predication is identified with unary predication
using the pairing function < >). If we accept this
principle, then we could eliminate use of general
predication, as it would be reduced to pure predication.
It is useful to separate BTPpg into two parts. Thus BTPp
uses only the object sort and the pure sort, with only
axioms in its restricted language. Also BTPg uses only the
object sort and the general sort, with only axioms in its
restricted language.
It can be shown that BTPpg is a conservative extension of
BTPp and a conservative extension of BTPg.
Both BTPp and BTPg are sufficient to appropriately develop
the natural number system. BTPg and BTPp also support an
adequate theory of finite sequences of objects. BTPg
supports the usual Tarski semantics for L(=), with its
inductive truth definition. BTPp also supports the usual
Tarski semantics for L(=), for pure structures, with its
inductive truth definition.
It is easy to see that the logical strengths of BTPpg,
BTPp, BTPg, are that of Z2, or equivalently, of ZFC\P.
BTPp, BTPg is also sufficient to prove the appropriate
formulations of Theorem 2.2.
THEOREM 2.4. BTPp (BTPg) proves that the following are
equivalent for any pure (general) domain D.
i) the pure (general) sets of sentences of PC(=) that are
pure (general) D satisfiable are exactly the pure (general)
sets of sentences of PC(=) that are pure (general) N
satisfiable; 11
ii) the pure (general) sets of sentences of PC(=) that are
pure (general) D satisfiable are exactly the pure (general)
sets of sentences of PC(=) that are consistent with INF(=);
iii) D has at least two elements, there is a pure (general)
oneone function from D2 into D, and D has a pure (general)
linear ordering.
There is a tricky point in carrying out the proof of
Theorem 2.4 . Since we are working with equality, we expect
to factor out by a suitable equivalence relation at the end
of the proof, in the usual way. However, BTPpg does not
directly support such factor constructions, as sets or
predicates are not objects. So we need to develop a way
around this by creating an appropriate set of unique
associates for the relevant equivalence classes.
At this point, we wish to introduce some preferred models
of BTPp, BTPg, and BTPpg.
The objects of these models will be built up from < > and
the formal letters a1,a2,... . Specifically, < > and each ai
are objects. If s,t are objects, then so are <s,t>. We let
W be the set of all these objects. The constant 0 is
interpreted as < >. Of course, < > is interpreted by < >.
Note that the automorphisms of (W,0,< >) are exactly the
permutations of W induced by permutations of {a1,a2,...}.
The pure predicates are the subsets of W that are fixed
under all automorphisms of (W,0,< >).
We use two interpretations of the general predicates. The
gen(fin) predicates are the S Õ W such that for some finite
A Õ {a1,a2,...}, S is fixed under all automorphisms of
(W,0,< >) that are the identity on A.
The gen(den) predicates are the S Õ W such that for some A
Õ {a1,a2,...} whose set of subscripts is of zero density, S
is fixed under all automorphisms of (W,0,< >) that are the
identity on A.
We thus have the structures
(W,0,< >,pure,Œ).
(W,0,< >,gen(fin),Œ). 12
(W,0,< >,gen(den),Œ).
(W,0,< >,pure,gen(fin),Œ).
(W,0,< >,pure,gen(den),Œ).
THEOREM 2.5. These five structures satisfy, respectively,
BTPp, BTPg, BTPg, BTPpg, BTPpg. In the fourth structure,
the general predicates are the cross sections of the pure
predicates with one object parameter. In the fifth
structure, this fails.
Consider the important principle of pure indiscernibles:
PI. ($x1 ≠ ... ≠ xn)("Rp)(j), where n ≥ 2, and j is the
conjunction of all equivalences Rp(<x1,...,xn>) ´
Rp(<y1,...,yn>), where y1,....,yn is a permutation of the
distinct variables x1,...,xn.
If we replace Rp by Rg in either version of PI, then we get
an obvious refutation from BTPg. Thus we take the principle
of general indiscernibles to be
GI. ("Rg)($x1 ≠ ... ≠ xn)(j), where n ≥ 2, and j is the
conjunction of all equivalences Rg(<x1,...,xn>) ´
Rg(<y1,...,yn>), where y1,....,yn is a permutation of the
distinct variables x1,...,xn.
There is an obvious multiple form of GI where we use more
than one Rg, simultaneously. This can be derived in BTPg +
GI.
THEOREM 2.6. These five structures satisfy, respectively,
PI, GI, GI, PI and GI, PI and GI.
We now come to completeness theorems for the fragment
PC(=,rel) of first order predicate calculus with equality
with relation symbols only, for universal sentences.
THEOREM 2.7. BTPp + PI proves that a universal sentence in
PC(=,rel) is pure W satisfiable if and only if it has a
model with the same cardinality as the number of distinct
variables that appear, whose elements form symmetric atomic
indiscernibles. The same statement holds for BTPg + GI.
There are a number of extensions of the usual axioms and
rules of PC(=,rel) that can be used for Theorem 2.7. Here 13
is one of them.
SYM(=,rel). Let n,k ≥ 1 and j1,...,jk be atomic formulas of
PC(=,rel) whose variables are among x1,...,xn. Take ($x1 ≠
... ≠ xn)(y), where y is the conjunction of all equivalences
ji ´ ji[x1/y1,...,xn/yn], where y1,...,yn is a permutation of
the distinct variables x1,...,xn.
THEOREM 2.8. The following is provable in very weak
systems, and easily in BTPp and BTPg. A universal sentence
in PC(=,rel) is consistent with SYM(=,rel) if and only if
it has a model with the same cardinality as the number of
distinct variables that appear, whose elements form
symmetric atomic indiscernibles for the relation symbols
that appear.
Might there be an alternative view of pure predication on W
that leads to a different axiomatization for the universal
sentences in PC(=,rel)? Of course. The entirely set
theoretic view leads to the usual axiomatization INF(=).
But there is no alternative view that makes it harder to be
pure W satisfiable, (than being consistent with
SYM(=,rel)), as the following indicates.
THEOREM 2.9. Let j be a universal sentence in PC(=,rel).
Then BTPp proves that j is pure W satisfiable if and only
if BTPg proves that j is general W satisfiable if and only
if j has a model with the same cardinality as the number of
distinct variables that appear in j, whose elements form
symmetric atomic indiscernibles for the relation symbols of
j.
We have developed the theory beyond the universal sentences
in PC(=,rel), but there are many issues that have yet to be
resolved. 3. THE FIRST INCOMPLETENESS THEOREM.
The Gödel first incompleteness theorem is proved in [Go31].
It is proved in detail for one specific variant of what is
now known as PA = Peano arithmetic, based on first order
predicate calculus with 0,S,+,•, and equality. It asserts
that there is a sentence that is neither provable nor
refutable in PA. 14 At the end of [Go31], p. 195, Gödel writes that “The
results will be stated and proved in full generality in a
sequel to be published soon.” Also we find, on page 195,
from Gödel:
Note added 28 August 1963. In consequence of later
advances, in particular of the fact that due to A.M
Turing’s work a precise and unquestionably adequate
definition of the general notion of formal system can now
be given, a completely general version of Theorems VI and
XI is now possible. That is, it can be proved rigorously
that in every consistent formal system that contains a
certain amount of finitary number theory there exist
undecidable arithmetic propositions and that, moreover, the
consistency of any such system cannot be proved in the
system.
The sequel was never published at least partly because of
the prompt acceptance of his results after the publication
of [Go31].
Today, Gödel is generally credited for quite general forms
of the first incompleteness theorem. There are already
claims of generality in [Go31]. In modern terms: in every
1consistent recursively enumerable formal system
containing a small amount of arithmetic, there exist
arithmetic sentences that are neither provable nor
refutable.
[Ros36] is credited for significant additional generality,
using a clever modification of Gödel’s original formal self
referential construction. It is shown there that we can
replace the hypothesis of 1consistency with the weaker
hypothesis of consistency.
Later, methods from recursion theory were used to prove yet
more general forms of first incompleteness, and where the
proof avoids use of formal self reference  although even
in the recursion theory, there is, arguably, a trace of
self reference present in the elementary recursion theory
used.
The recursion theory approach, in a powerful form, appears
in [Ro52], [TMR53], with the use of the formal system Q. 15
Q is a single sorted system based on 0,S,+,•,£,=. In
addition to the usual axioms and rules of logic for this
language, we have the nonlogical axioms
1.
2.
3.
4.
5.
6.
7.
8. Sx ≠ 0.
Sx = Sy Æ x = y.
x ≠ 0 Æ ($y)(x = Sy).
x + 0 = x.
x + Sy = S(x + y).
x • 0 = 0.
x • Sy = (x • y) + x.
x £ y ´ ($z)(z + x = y). The last axiom is purely definitional, and is not needed
for present purposes (in fact, we do not need £).
THEOREM 3.1. Let T be a consistent extension of Q in a
relational type in many sorted predicate calculus of
arbitrary cardinality. The sets of all existential
sentences in L(Q), with bounded universal quantifiers
allowed, that are i) provable in T, ii) refutable in T,
iii) provable or refutable in T, are each not recursive.
For the proof, see [Ro52], [TMR53]. It uses the
construction of recursively inseparable recursively
enumerable sets; e.g., {n: jn(n) = 0} and {n: jn(n) = 1}.
One can obtain the following strong form of first
incompleteness as an immediate Corollary.
THEOREM 3.2. Let T be a consistent extension of Q in many
sorted predicate calculus whose relational type and axioms
are recursively enumerable. There is an existential
sentence in L(Q), with bounded universal quantifiers
allowed, that is neither provable nor refutable in T.
We can use the solution to Hilbert’s tenth problem in order
to obtain other forms of first incompleteness that are
stronger in certain respects. In fact, Hilbert’s tenth
problem is still a great source of very difficult problems
on the border between logic and number theory, which we
will discuss below.
Hilbert asks for a decision procedure for determining
whether a given polynomial with integer coefficients in
several integer variables has a zero. 16 The problem received a negative answer in 1970 by Y.
Matiyasevich, building heavily on earlier work of J.
Robinson, M. Davis, and H. Putnam. It is commonly referred
to as the MRDP theorem (in reverse historical order). See
[Da73], [Ma93]. The MRDP theorem was shown to be provable
in the weak fragment of arithmetic, EFA = IS0(exp), in
[DG82].
We can use [DG82] to obtain the following.
THEOREM 3.3. Let T be a consistent extension of EFA in many
sorted predicate calculus whose relational type and axioms
are recursively enumerable. There is a purely existential
equation ($x1,...,xn)(s = t) in L(Q) that is neither
provable nor refutable in T.
It is not clear whether EFA can be replaced by a weaker
system in Theorem 3.3 such as Q.
An important issue is whether there is a “reasonable”
existential equation ($x1,...,xn)(s = t) that can be used in
Theorem 3.3 for, say, T = PA or T = ZFC. Note that
($x1,...,xn)(s = t) corresponds to the Diophantine problem
“does the polynomial st with integer coefficients have a
solution in the nonnegative integers?”
Let us see what can be done on the purely recursion
theoretic side with regards to the complexity of
polynomials with integer coefficients. The most obvious
criteria are
a. The number of unknowns.
b. The degree of the polynomial.
c. The number of operations.
In 1992, Matiyasevich showed that nine unknowns over the
nonnegative integers suffices for recursive unsolvability.
A detailed proof of this result was given in [Jo82].
Also [Jo82] proves that 100 operations suffices for
recursive unsolvability.
It is well known that degree 4 suffices for recursive
unsolvability. In [Jo82], it is shown that degree 4 and 58
nonnegative integer unknowns suffice for recursive
unsolvability. 17 In fact, [Jo82] provides the following sufficient pairs
<degree,unknowns>, where all unknowns range over
nonnegative integers:
<4,58>, <8,38>, <12,32>, <16,29>,
<20,28>, <24,26>, <28,25>, <36,24>,
<96,21>, <2668,19>, <2 ¥ 105,14>,
<6.6 ¥ 1043,13>, <1.3 ¥ 1044,12>,
<4.6 ¥ 1044,11>, <8.6 ¥ 1044,10>,
<1.6 ¥ 1045,9>.
For degree 2 (a single quadratic) we have an algorithm
(over the nonnegative integers, the integers, and the
rationals), going back to [Si72]. See [GS81], [Ma98].
For degree 3, the existence of an algorithm is wide open,
even for two variables (over the integers, the nonnegative
integers, or the rationals).
It is clear from this discussion that the gap between
is known and what could be the case is enormous, just
this original context of deciding whether polynomials
integer coefficients have a zero in the (nonnegative)
integers. Specifically, <3,2> could conceivably be on
list of pairs. what
in
with
this These upper bounds on the complexity sufficient to obtain
recursive unsolvability can be directly imported into
Theorem 3.3, as the underlying number theory and recursion
theory can be done in EFA. Although one obtains upper
bounds on pairs (number of variables,degree) in this way,
this does not address the question of the size of the
coefficients needed in Theorem 3.3.
In particular, let us call a polynomial P a Gödel
polynomial if
i. P is a polynomial in several variables with integer
coefficients.
ii. The question of whether P has a solution in nonnegative
integers is neither provable nor refutable in PA. (We can
also use ZFC here instead of PA).
We have never seen an upper bound on the “size” of a Gödel
polynomial in the literature. In particular, We have never
seen a Gödel polynomial written down fully in base 10 on a 18
small piece of paper.
One interesting theoretical issue is whether one can
establish any relationship between the “size” of a Gödel
polynomial using PA and the “size” of a Gödel polynomial
using ZFC. 4. THE SECOND INCOMPLETENESS THEOREM.
In [Go31], Gödel only sketches a proof of his second
incompleteness theorem, after proving his first
incompleteness theorem in detail. His sketch depends on the
fact that the proof of the first incompleteness theorem,
which is conducted in normal semiformal mathematics, can be
formalized and proved within (systems such as) PA.
Gödel promised a part 2 of [Go31], but this never appeared,
presumably because the logic community was convinced that
this could surely be carried out.
The necessary details were carried out in [HB34,39], and
later in [Fe60], and most recently, in [Bo93].
In [HB34,39], the so called Hilbert Bernays derivability
conditions were isolated in connection with a detailed
proof of Gödel’s second incompleteness theorem given in
[HB34,39]. Later, these conditions were streamlined in
[Je73].
We take the liberty of presenting our own particularly
careful and clear version of the Hilbert Bernays
conditions.
Our starting point is the usual language L = predicate
calculus with equality, with infinitely many constant,
relation, and function symbols. For specificity, we will
use
i) variables xn, n ≥ 1;
ii) constant symbols cn, n ≥ 1;
iii) relation symbols Rnm, n,m ≥ 1;
iv) function symbols Fnm, n,m ≥ 1;
v) connectives ÿ,Ÿ,⁄,Æ,´;
vi) quantifiers $,".
We start with the following data. 19
1. A relational type RT of constant symbols, relation
symbols, and function symbols.
2. A set T of sentences in (the language based on) RT.
3. A oneone function # from formulas of RT into closed
terms of RT.
4. A distinguished unary function symbol NEG in RT, meaning
“negation”.
5. A distinguished unary function symbol SSUB in RT,
meaning “self substitution”.
6. A distinguished unary function symbol PR in RT, meaning
“provability statement”.
7. A distinguished formula PROV with at most the free
variable x1, expressing “provable in T”.
We require the following. Let A be a formula of RT.
8. NEG(#(A)) = #(ÿA) is provable in T.
9. SSUB(#(A)) = #(A[x1/#(A)]) is provable in T.
10. PR(#(A)) = #(PROV[x1/#(A)]) is provable in T.
11. PROV[x1/#(A)] Æ PROV[x1/PR(#(A))] is provable in T.
12. If A is provable in T, then PROV[x1/#(A)] is provable in
T.
Here #(A) is the Gödel number of the formula A, as a closed
term of RT. Also NEG means “negation”, SSUB means “self
substitution”, PR means “provability statement”, PROV means
“is provable”.
THEOREM 4.1. (Self reference lemma). Let A be a formula of
RT. There exists a closed term t of RT such that T proves
t = #(A[x1/t]).
Proof: Let s = #(A[x1/SSUB(x1)]).
Note that the result of replacing x1 by #(A[x1/SSUB(x1)]) in
A[x1/SSUB(x1)] is A[x1/SSUB(s)].
We now apply condition 9 to the formula A[x1/SSUB(x1)]. We
obtain
SSUB(s) = #(A[x1/SSUB(s)])
is provable in T. Thus the closed term SSUB(s) is as
required. QED
LEMMA 4.2. (I am not provable lemma). There exists a closed
term t such that T proves t = #(ÿPROV[x1/t]). 20 Proof: By Theorem 4.1, setting A = ÿPROV. QED
We fix a closed term t provided by Lemma 4.2.
LEMMA 4.3. Suppose T proves ÿPROV[x1/t]. Then T is
inconsistent.
Proof: Assume T proves ÿPROV[x1/t]. By condition 12,
PROV[x1/#(ÿPROV[x1/t])]
is provable in T. Hence T proves PROV[x1/t]. Hence T is
inconsistent. QED
LEMMA 4.4. T proves PROV[x1/t] Æ PROV[x1/PR(t)]. T proves
PROV[x1/t] Æ PROV[x1/NEG(PR(t))].
Proof: By the definition of t, write t = #(B). Then the
first claim follows immediately from condition 11.
By condition 10, T proves
PR(t) = #(PROV[x1/t]).
By condition 8, T proves
NEG(PR(t)) = NEG(#(PROV[x1/t])) = #(ÿPROV[x1/t]) = t.
The second claim follows immediately. QED
We let CON be the sentence
("x1)(ÿ(PROV Ÿ PROV[x1/NEG(x1)])).
THEOREM 4.5. (Abstract second incompleteness). Let T obey
conditions 112. Suppose T proves CON. Then T is
inconsistent.
Proof: Suppose T is as given. By Lemma 4.4, T proves
ÿPROV[x1/t]. By Lemma 4.3, T is inconsistent. QED
Informal statements of Gödel's Second Incompleteness
Theorem are simple and dramatic. However, current versions
of the Formal Second Incompleteness are complicated and
awkward. Even the abstract form of second incompleteness 21
given above using derivability conditions are rather subtle
and involved.
We recently addressed this problem in [Fr07], where we
present new versions of Formal Second Incompleteness that
are simple, and informally imply Informal Second
Incompleteness.
These results rest on the isolation of simple formal
properties shared by consistency statements. Here we do not
address any issues concerning proofs of Second
Incompleteness.
We start with the most commonly quoted form of Gödel's
Second Incompleteness Theorem  for the system PA = Peano
Arithmetic.
PA can be formulated in a number of languages. Of these,
L(prim) is the most suitable for supporting formalizations
of the consistency of Peano Arithmetic.
We write L(prim) for the language based on 0,S and all
primitive recursive function symbols. We let PA(prim) be
the formulation of Peano Arithmetic for the language
L(prim). I.e., the nonlogical axioms of PA(prim) consist of
the
axioms for successor, primitive recursive defining
equations, and the induction scheme applied to all formulas
in L(prim).
INFORMAL SECOND INCOMPLETENESS (PA(prim)). Let A be a
sentence in L(prim) that adequately formalizes the
consistency of PA(prim), in the informal sense. Then
PA(prim) does not prove A.
We have discovered the following result. We let PRA be the
important subsystem of PA(prim), based on the same language
L(prim), where we require that the induction scheme be
applied only to quantifier free formulas of L(prim).
FORMAL SECOND INCOMPLETENESS (PA(prim)). Let A be a
sentence in L(prim) such that every equation in L(prim)
that is provable in PA(prim), is also provable in PRA + A.
Then PA(prim) does not prove A. 22
Informal second incompleteness for PA(prim) can be derived
in the usual semiformal way from the above formal second
incompleteness for PA(prim).
FORMAL CRITERION THEOREM 1. Let A be a sentence in L(prim)
such that every equation in L(prim) that is provable in
PA(prim), is also provable in PRA + A. Then for all n, PRA
+ A proves the consistency of PA(prim)n.
Here PA(prim)n consists of the axioms of PA(prim) in prenex
form with at most n quantifiers.
The above development can be appropriately carried out for
systems with full induction. However, there is a more
general treatment which covers finitely axiomatized
theories as well.
INFORMAL SECOND INCOMPLETENESS (general many sorted,
exparith). Let L be a fragment of L(many) containing
L(EFA). Let T be a consistent extension of EFA in L. Let A
be a sentence in L that adequately formalizes the
consistency of T, in the informal sense. Then T does not
prove A.
FORMAL SECOND INCOMPLETENESS (general many sorted,
exparith). Let L be a fragment of L(many) containing
L(EFA). Let T be a consistent extension of EFA in L. Let A
be a sentence in L such that every universalized inequation
in L(EFA) with a relativization in T, is provable in EFA +
A. Then T does not prove A.
FORMAL CRITERION THEOREM II. Let L be a fragment of L(many)
containing L(EFA). Let T be a consistent extension of EFA
in L. Let A be a sentence in L such that every
universalized inequation in L(EFA) with a relativization in
T, is provable in EFA + A. Then EFA proves the consistency
of every finite fragment of T.
A relativization of a sentence j of L(EFA), in T, is an
interpretation of j in T which leaves the meaning of all
symbols unchanged, but where the domain is allowed to
consist of only some of the nonnegative integers from the
point of view of T.
Finally, we mention an interesting issue that we are
somewhat unclear about, but which can be gotten around in a
satisfactory way. 23 It can be said that Gödel’s second incompleteness theorem
has a defect in that one is relying on a formalization of
Con(T) within T via the indirect method of Gödel numbers.
Not only is the assignment of Gödel numbers to formulas
(and the relevant syntactic objects) ad hoc, but one is
still being indirect and not directly dealing with the
objects at hand  which are syntactic and not numerical.
It would be preferable to directly formalize Con(T) within
T, without use of any indirection. Thus in such an
approach, one would add new sorts for the relevant
syntactic objects, and introduce the various relevant
relations and function symbols, together with the relevant
axioms.
However, in so doing, one has expanded the language of T.
Accordingly, two choices are apparent.
The first choice is to make sure that as one adds new sorts
and new relevant relations and function symbols and new
axioms to T, associated with syntax, one also somehow has
already appropriately treated, directly, the new syntactic
objects and axioms beyond T that arise when one is
performing this addition to T.
The second choice is to be content with adding the new
sorts and new relevant relations and function symbols and
new axioms to T, associated with the syntax of T only  and
not try to deal in this manner with the extended syntax
that arises from this very process.
We lean towards the opinion that the first choice is
impossible to realize in an appropriate way. Some level of
indirection will remain. Perhaps the level of indirection
can be made rather weak and subtle. Thus we lean towards
the opinion that it is impossible to construct extensions
of, say, PA that directly and adequately formalize their
entire syntax. We have not tried to prove such an
impossibility result, but it seems possible to do so.
In any case, the second choice, upon reflection, turns out
to be wholly adequate for casting what may be called
“direct second incompleteness”. This formulation asserts
that for any suitable theory T, if T’ is the (or any)
extension of T through the addition of appropriate sorts,
relations, functions, and axioms, directly formalizing the 24
syntax of T, including a direct formalization of the
consistency of T, then T’ does not prove the consistency of
T (so expressed).
We can recover the usual second incompleteness theorem for
T from the above direct second incompleteness, by proving
that there is an interpretation of T’ in T.
Thus under this view of second incompleteness, one does not
view Con(T) as a sentence in the language of T, but instead
as a sentence in the language of an extension T’ of T.
Con(T) only becomes a sentence in the language of T through
an interpretation (in the sense of Tarski) of T’ in T.
There are many such interpretations, all of which are ad
hoc. This view would then eliminate ad hoc features in the
formulation of second incompleteness, while preserving the
foundational implications. 5. LENGTHS OF PROOFS.
In [Go36], Gödel discusses a result which, in modern
terminology, asserts the following. Let RTT be Russell’s
simple theory of types with the axiom of infinity. Let RTTn
be the fragment of RTT using only the first n types. Let
f:N Æ N be a recursive function. For each n ≥ 0 there are
infinitely many sentences j such that
f(n) < m
where n is the least Gödel number of a proof of j in RTTn+1
and m is the least Gödel number of a proof of j in RTTn.
Gödel expressed the result in terms of lengths of proofs
rather than Gödel numbers or total number of symbols. Gödel
did not publish any proofs of this result or results of a
similar nature. As can be surmised from the Introductory
remarks by R. Parikh, it is likely that Gödel had
inadvertently used lengths, and probably intended Gödel
numbers or numbers of symbols.
In any case, the analogous result with Gödel numbers was
proved in [Mo52]. Similar results were also proved in
[EM71] and [Pa71]. Also see [Pa73] for results going in
the opposite direction concerning the number of lines in
proofs in certain systems. 25
In [Fr79], we considered, for any reasonable system T, and
positive integer n, the finite consistency statement Conn(T)
expressing that “every inconsistency in T uses at least n
symbols”. We gave a lower bound of n1/4 on the number of
symbols required to prove in Conn(T) in T, provided n is
sufficiently large. A more careful version of the argument
gives the lower bound of n1/2 for sufficiently large n. We
called this “finite second incompleteness”.
A much more careful analysis of finite second
incompleteness is in [Pu85], which establishes a
W(n(log(n))1/2) lower bound and an O(n) upper bound, for
systems T satisfying certain reasonable conditions.
It would be very interesting to extend finite second
incompleteness in several directions. One direction is to
give a treatment of a good lower bound for a proof of
Conn(T) in T, which is along the lines of the Hilbert
Bernays derivability conditions, adapted carefully for
finite second incompleteness. We offer our treatment of the
derivability conditions in section 4 above as a launching
point. A number of issues arise as to the best way to set
this up, and what level of generality is appropriate.
Another
to give
involve
related direction to take finite second incompleteness is
some versions which are not asymptotic. I.e., they
specific numbers of symbols that are argued to be
to actual mathematical practice. Although the very good upper bound of O(n) is given for a
proof of Conn(T) in T, at least for some reasonable systems
T, the situation is quite different if we are talking about
proofs in S of Conn(T), where S is significantly weaker than
T. For specificity, consider how many symbols it takes to
prove Conn(ZF) in PA, where n is large. Obviously, if PA can
prove that some specific algorithm for testing
satisfiability of Boolean expressions works and runs in a
polynomial computer time, then PA will be able to prove
Conn(ZF) using a polynomial number of symbols in n (assuming
Conn(ZF) is in fact true). So this makes it very unlikely
that a polynomial bound can be given in n for proving
Conn(ZF) in PA.
There may be a clever way to establish an exponential type
lower bound here without attacking the famous P = NP
problem. Perhaps someone can refute what we just said. 26
There are some other aspects of lengths of proofs that seem
important. One is the issue of overhead.
Gödel established in [Go40] that any proof of a sentence A
in NBG + ÿAxC can be converted to a proof of A in NBG. He
used the method of relativization. Thus one obtains
constants c,d such that if A is provable in NBG + ÿAxC
using n symbols, then A is provable in NBG using at most
cn+d symbols.
What is not at all clear here is whether c,d can be made
reasonably small. There is clearly a lot of overhead
involved on two counts. One is in the execution of the
actual relativization, which involves relativizing to the
constructible sets. The other overhead is that one must
insert the proofs of various facts about the constructible
sets including that they form a model of NBG.
The same remarks can be made with regard to NBG + GC + ÿCH
and NBG + GC, where GC is the global axiom of choice. Also,
these remarks apply to ZF + ÿAxC and ZF, and also to ZFC +
ÿCH and ZFC. Also they apply equally well to the Cohen
forcing method [Co63], and proofs from AxC, CH.
We close with another issue regarding lengths of proofs in
a context that is often considered immune to incompleteness
phenomena. Finite incompleteness phenomena is very much in
evidence here.
Alfred Tarski, in [Ta51], proved the completeness of the
usual axioms for real closed fields using quantifier
elimination. This also provides a decision procedure for
recognizing the first order sentences in (¬,<,0,1,+,,•).
His method applies to the following three fundamental axiom
systems.
1) The language is 0,1,+,,•. The
usual field axioms, together with
squares, x or x is a square, and
degree with leading coefficient 1 axioms consist of the
1 is not the sum of
every polynomial of odd
has a zero. 2) The language is 0,1,+,,•,<. The axioms consist of the
usual ordered field axioms, together with every positive
element has a square root, and every polynomial of odd
degree with leading coefficient 1 has a zero. 27
3) The language is 0,1,+,,•,<. The axioms consist of the
usual ordered field axioms, together with the axiom scheme
asserting that if a first order property holds of
something, and there is an upper bound to what it holds of,
then there is a least upper bound to what it holds of.
For reworking and improvements on Tarski, see [Co69],
[Re82a], [Re92b], [Re92c], [BPR06]. In terms of
computational complexity, the set of true first order
sentences in (¬,<,0,1,+,,•) is exponential space easy and
nondeterministic exponential time hard. The gap has not
been filled. Even the first order theory of (¬,+) is
nondeterministic exponential time hard.
The work just cited concerns mainly the computational
complexity of the set of true sentences in the reals
(sometimes with only addition). It does not directly deal
with the lengths of proofs in systems 1),2),3).
What can we say about number of symbols in proofs in
systems 1),2),3)? We conjecture that with the usual axioms
and rules of logic, in all three cases, there is a double
exponential lower and upper bound on the number of symbols
required in a proof of any true sentence in each of
1),2),3).
What is the relationship between sizes of proofs of the
same sentence (without <) in 1),2),3)? We conjecture that,
asymptotically, there are infinitely many true sentences
without < such that there is a double exponential reduction
in the number of symbols needed to prove it when passing
from system 1) to system 3).
These issues concerning sizes of proofs are particularly
interesting when the quantifier structure of the sentence
is restricted. For instance, the cases of purely universal,
purely existential are particularly interesting,
particularly when the matrix is particularly simple. Other
cases of clear interest are "..."$...$, and $...$"...",
with the obviously related conditions of surjectivity and
nonsurjectivity being of particular interest.
Another aspect of sizes of proofs comes out of strong
mathematical P02 sentences. The earliest ones were presented
in [Goo44] and [PH77], and are proved just beyond PA. We
discovered many examples in connection with theorems of
J.B. Kruskal and Robertson/Seymour, which are far stronger, 28
with no predicative proofs. See [Fr02a].
None of these three references discusses the connection
with sizes of proofs. This connection is discussed in
[Sm85], 132135, and in the unpublished abstracts [Fr06ag]
from the FOM Archives.
The basic idea is this. There are now a number of
mathematically natural P02 sentences ("n)($m)(R(n,m)) which
are provably equivalent to the 1consistency of various
systems T. One normally gets, as a consequence, that the
Skolem function m of n grows very fast, asymptotically, so
that it dominates the provably recursive functions of T.
However, we have observed that in many cases, one can
essentially remove the asymptotics. I.e., in many cases, we
have verified that we can fix n to be very small (numbers
like 3 or 9 or 15), and consider the resulting S01 sentence
($m)(R(n,m)). The result is that any proof in T (or certain
strong fragments of T) of this S01 sentence must have an
absurd number of symbols  e.g., an exponential stack of
100 2’s. Yet if we go a little beyond T, we can prove the
full P02 sentence ("n)($m)(R(n,m)) in a normal size
mathematics manuscript, thereby yielding a proof just
beyond T of the resulting S01 sentence R(n,m) with n fixed
to be a small (or remotely reasonable) number. This
provides a myriad of mathematical examples of Gödel’s
original length of proof phenomena from [Go36]. 6. THE NEGATIVE INTERPRETATION.
Gödel wrote four fundamental papers concerning formal
systems based on intuitionistic logic: [Go32], [Go33],
[Go33a], [Go58]. [Go72] is a revised version of [Go58].
In [Go32], Gödel proves that the intuitionistic
propositional calculus cannot be viewed as a classical
system with finitely many truth values. He shows this by
constructing an infinite descending chain of logics
intermediate in strength between classical propositional
calculus and intuitionistic propositional calculus. For
more on intermediate logics, see [HO73] and [Mi83].
In [Go33], Gödel introduces his negative interpretation in
the form of an interpretation of PA = Peano arithmetic in
HA = Heyting arithmetic. Here HA is the corresponding 29
version of PA = Peano arithmetic based on intuitionistic
logic. It can be axiomatized by taking the usual axioms and
rules of intuitionistic predicate logic, together with the
axioms of PA as usual given. Of course, one must be careful
to present ordinary induction in the usual way, and not use
the least number principle.
It is natural to isolate his negative interpretation in
these two ways:
a. An interpretation of classical propositional calculus in
intuitionistic propositional calculus.
b. An interpretation of classical predicate calculus in
intuitionistic predicate calculus.
In modern terms, it is convenient to use ^,⁄,Ÿ,Æ. The
interpretation for propositional calculus inductively
interprets
^ as ^.
Ÿ as Ÿ.
Æ as Æ.
⁄ as ÿÿ⁄.
For predicate calculus,
" as ".
$ as ÿÿ$.
j as ÿÿj, where j is atomic.
Now in HA, we can prove n = m ⁄ ÿn = m. It is then easy to
see that the successor axioms and the defining equations of
PA are sent to theorems of HA, and also each induction
axiom of PA is sent to a theorem of HA.
Also the axioms of classical predicate calculus become
theorems of intuitionistic predicate calculus, and the
rules of classical predicate calculus become rules of
intuitionsitic predicate calculus.
So under the negative interpretation, theorems of classical
propositional calculus become theorems of intuitionsitic
propositional calculus, theorems of classical predicate
calculus become theorems of intuitionistic predicate
calculus, and theorems of PA become theorems of HA. 30
Also, any P01 sentence ("n)(F(n) = 0), where F is a
primitive recursive function symbol of PA, is sent to a
sentence that is provably equivalent to ("n)(F(n) = 0).
It is then easy to conclude that every P01 theorem of PA is
a theorem of HA.
Gödel’s negative interpretation has been extended to many
pairs of systems, most of them of the form T,T’, where T,T’
have the same nonlogical axioms, and where T is based on
classical predicate calculus, whereas T’ is based on
intuitionistic predicate calculus. For example, see [Kr68],
p. 344, [Kr68a], Section 5, [My74], [Fr73], [Le85].
A much stronger result holds for PA over HA. Every P02
sentence provable in PA is provable in HA. The first proofs
of this result were from the proof theory of PA via Gentzen
(see [Ge69], [Sc77]), and from Gödel’s so called Dialectica
or functional interpretation, in [Go58]. [Go72] is a
revised version of [Go58]..
However, for other pairs for which the negative
interpretation shows that they have the same provable P01
sentences  say classical and intuitionistic second order
arithmetic  one does not have the proof theory. However,
in this case, the Dialectica interpretation has been
extended by Spector in [Sp62], and the fact that these two
systems have the same provable P02 sentences follows.
Nevertheless, there are many appropriate pairs for which
the negative interpretation works, yet there is no proof
theory and there is no functional interpretation.
In [Fr78], we broke this impasse by modifying Gödel’s
negative interpretation via what is now called the A
translation. Also see [Dr80]. We illustrate the technique
for PA over HA, formulated with primitive recursive
function symbols.
Let A be any formula in L(HA) = L(PA). We define the Atranslation jA of the formula j in L(HA), in case no free
variable of A is bound in j. Take jA to be the result of
simultaneously replacing every atomic subformula y of j by
(y ⁄ A). In particular, ^ gets replaced by what amounts to
A. 31
The A translation is an interpretation of HA in HA. I.e.,
if jA is defined, and HA proves A, then HA proves jA. Also,
obviously HA proves A Æ jA.
Now suppose ($n)(F(n,m) = 0) is provable in PA, where F is a
primitive recursive function symbol. By Gödel’s negative
interpretation, ÿÿ($n)(F(n,m) = 0) is provable in HA. Write
this as (($n)(F(n,m) = 0) Æ ^) Æ ^.
By taking the A translation, with A = ($n)(F(n,m) = 0), we
obtain that HA proves
(($n)(F(n,m) = 0 ⁄ ($n)(F(n,m) = 0)) Æ ($n)(F(n,m) = 0)) Æ
($n)(F(n,m) = 0.
(($n)(F(n,m) = 0) Æ ($n)(F(n,m) = 0)) Æ ($n)(F(n,m) = 0.
($n)(F(n,m) = 0).
This method applies to a large number of pairs T/T’ as
indicated in [Fr73] and [Le85].
[Go58] and [Go72] present Gödel’s so called Dialectica
interpretation, or functional interpretation, of HA. Here
HA = Heyting arithmetic, is the corresponding version of PA
= Peano arithmetic with intuitionistic logic. It can be
axiomatized by taking the usual axioms and rules of
intuitionistic predicate logic, together with the axioms of
PA as usual given. Of course, one must be careful to
present ordinary induction in the usual way, and not use
the least number principle.
In Gödel’s Dialectica interpretation, theorems of HA are
interpreted as derivations in a quantifier free system T of
functionals of finite type that is based on quantifier free
axioms and rules, including a rule of induction. The
functionals include the recursion functionals at higher
types.
The Dialectica interpretation has had several applications
in different directions. There are applications to
programming languages and category theory which we will not
discuss.
To begin with, the Dialectica interpretation can be
combined with Godel’s negative interpretation of PA in HA
32
to form an interpretation of PA in Gödel’s quantifier free
system T.
One obvious application, and motivation, is philosophical,
and Gödel discusses this aspect in both papers, especially
the second. The idea is that the quantifiers in HA or PA,
ranging over all natural numbers, are not finitary, whereas
T is arguably finitary  at least in the sense that T is
quantifier free. However, the objects of T are at least
prima facie infinitary, and so there is the difficult
question of how to gauge this tradeoff. One idea is that
the objects of T should not be construed as infinite
completed totalities, but rather as rules. We refer the
interested reader to the rather extensive Introductory
notes to [Go58] in [Go,8603], Vol. II.
Another application is to extend the interpretation to the
two sorted first order system known as second order
arithmetic, or Z2. This was carried out by Clifford Spector
in [Sp62]. Here the idea is that one may construe such a
powerful extension of Gödel’s Dialectica interpretation as
some sort of constructive consistency proof for the rather
metamathematically strong highly impredicative system Z2.
However, in various communications, Gödel was not entirely
satisfied that the quantifier free system Spector used was
truly constructive.
We believe that the Spector development has not been fully
exploited. In particular, it ought to give rather striking
mathematically interesting characterizations of the
provably recursive functions and provable ordinals of Z2 and
various fragments of Z2.
Another fairly recent application is to use the Dialectica
interpretation, and extensions of it to systems involving
functions and real numbers, in order to obtain sharper
uniformities in certain areas of functional analysis that
had been obtained before by the specialists. This work has
been pioneered by U. Kohlenbach. See [Ko05], [Ko•1],
[Ko•2], [Ko•3], [KO03]. 7. THE AXIOM OF CHOICE AND THE CONTINUUM
HYPOTHESIS.
Gödel wrote six manuscripts directly concerned with the
continuum hypothesis: Two abstracts, [Go38], [Go39]. One 33
paper with sketches of proofs, [Go39a]. One research
monograph with fully detailed proofs, [Go40]. One
philosophical paper, [Go47,64], in two versions.
The normal abbreviations for the axiom of choice is AxC.
The normal abbreviation for the continuum hypothesis is CH.
A particularly attractive statement of CH asserts that
every set of real numbers is either in oneone
correspondence with a set of natural numbers, or in oneone
correspondence with the set of real numbers.
Normally, one follows Gödel in considering CH only in the
presence of AxC. However, note that in this form, CH can be
naturally considered without the presence of AxC. However,
Solovay’s model satisfying ZFCD + “all sets are Lebesgue
measurable” also satisfies CH in the strong form that every
set of reals is countable or has a perfect subset (this
strong form is incompatible with AxC). See [So70].
The statement of CH is due to Cantor. Gödel also considers
the generalized continuum hypothesis, GCH, whose statement
is credited to Hausdorff. The GCH asserts that
for all sets A, every subset of ℘(A) is either in oneone
correspondence with a subset of A, or in oneone
correspondence with ℘(A).
Here ℘ is the power set operation.
Gödel’s work establishes an interpretation of ZFC + GCH in
ZF. This provides a very explicit way of converting any
inconsistency in ZFC + GCH to an inconsistency in ZF.
We can attempt to quantify these results. In particular, it
is clear that the interpretation given by Gödel of ZFC +
GCH in ZF, by relativizing to the constructible sets, is
rather large, in the sense that when fully formalized,
results in a lot of symbols. It also seems to result in a
lot of quantifiers. How many?
So far we have been talking about the crudest formulations
in primitive notation, without the benefit of abbreviation
mechanisms. But abbreviation mechanisms are essential for
the actual conduct of mathematics. In fact, current proof
assistants  where humans and computers interact to create
34
verified proofs  necessarily incorporate very substantial
abbreviation mechanisms. See, e.g., [BW05], [Wie06].
So the question arises as to how simple can an
interpretation be of ZFC + GCH in ZF, with abbreviations
allowed in the presentation of the interpretation? This is
far from clear.
P.J. Cohen proved that if ZF is consistent then so is ZF +
ÿAxC and ZFC + ÿCH, thus complementing Gödel’s results. See
[Co6364]. The proof does not readily give an
interpretation of ZF + ÿAxC, or of ZFC + ÿCH in ZF. It can
be converted into such an interpretation by a general
method whereby under certain conditions (met here), if the
consistency of every given finite subsystem of one system
is provable in another, then the first system is
interpretable in the other (see [Fe60]).
Again, the question arises as to how simple can an
interpretation be of ZF + ÿAxC or of ZFC + ÿCH, in ZF, with
abbreviations allowed in the presentation of the
interpretation? Again this is far from clear. And how does
this question compare with the previous question?
There is another kind of complexity issue associated with
the CH that is of interest. First some background. It is
known that every 3 quantifier sentence in primitive
notation Œ,=, is decided in a weak fragment of ZF. See
[Gog79], [Fr03]. Also there is a 5 quantifier sentence in
Œ,= that is not decided in ZFC (it is equivalent to the
existence of a subtle cardinal over ZFC). See [Fr03a]. It
is also known that AxC can be written with five quantifiers
in Œ,=, over ZFC. See [Maexx].
The question is: how many quantifiers are needed to express
CH over ZFC, in Œ,=? We can also ask this and related
questions where abbreviations are allowed.
Most mathematicians instinctively take the view that since
CH is neither provable nor refutable from the standard
axioms for mathematics (ZFC), the ultimate status of CH has
been settled and there is nothing left to ponder.
However, many mathematical logicians, particularly those in
set theory, take a quite different view. This includes Kurt
Gödel. That the continuum hypothesis is a well defined 35
mathematical assertion with a definite truth value. The
problem is to determine just what this truth value is.
The idea here is that there is a definite system of objects
that exists independently of human minds, and that human
minds can no more manipulate the truth value of statements
of set theory than they can manipulate the truth value of
statements about electrons and stars and galaxies.
This is the so called Platonist point of view that is
argued so forcefully and explicitly in [Go,47,64].
The late P.J. Cohen led a panel discussion at the Gödel
Centenary called On Unknowability, conducted a poll roughly
along these lines. The question he asked was, roughly,
“does the continuum hypothesis have a definite answer”, or
“does the continuum hypothesis have a definite truth
value”.
The response from the audience appeared quite divided on
the issue.
Of the panelists, the ones who have expressed very clear
views on this topic were most notably Cohen and Woodin.
Cohen took a formalist viewpoint, whereas Woodin takes a
Platonist one. See their contributions to this volume.
My own view is that we simply do not know enough in the
foundations of mathematics to decide the truth or
appropriateness of the formalist versus the Platonist
viewpoint  or, for that matter, what mixture of the two is
true or appropriate.
But then it is reasonable to place the burden on me to
explain what kind of additional knowledge could be relevant
for this issue.
My ideas are not very well developed, but I will offer at
least something for people to consider.
It may be possible to develop a theory of ‘fundamental
mental pictures’ which is so powerful and compelling that
it supplants any discussion of formalism/Platonism in
anything like its present terms. What may come out is a
fundamental mental picture for the axioms of ZFC, even with
some large cardinals, along with a theorem to the effect 36
that there is no fundamental mental picture for CH and no
fundamental mental picture for ÿCH. 8. 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.
The highlights of wqo theory are that certain qo’s are
wqo’s, and certain operations on wqo’s produce wqo’s.
[Kru60], treats finite trees as finite posets, and studies
the qo
there exists an inf preserving embedding from T1 into T2.
THEOREM 8.1. [Kru60]. The above qo of finite trees as
posets is a wqo.
The simplest proof of Theorem 8.1 and some extensions, is
in [NW63], with the introduction of minimal bad sequences.
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. See [Fe64,68], [Fe98], 249298. Poincare, Weyl, and others railed against
impredicative mathematics. See [We10], [We87], [Fe98], 289291, and [Fo92].
THEOREM 8.2. [Fr02a]. Kruskal’s tree theorem cannot be
proved in FS.
KT goes considerably beyond FS, and an exact measure of KT
is known. See [RW93].
J.B. Kruskal actually considered finite trees whose
vertices are labeled from a wqo £*. The additional
requirement on embeddings is that
label(v) £* label(h(v)). 37 THEOREM 8.3. [Kru60]. 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. We 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 8.4. Labeled KT does not hold in the
hyperarithmetic sets. In fact, RCA0 + KT implies ATR0.
A proof of Theorem 8.4 will appear in [FMW].
It is natural to impose a growth rate in KT in terms of the
number of vertices of Ti.
COROLALRY 8.5. (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 8.6. (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 that Corollary 2.6 is P02.
THEOREM 8.7. Corollary 8.5 cannot be proved in FS. This
holds even for linear bounds with nonconstant coefficient
1.
THEOREM 8.8. 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 8.9. (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,... 38
With this kind of strong uniformity, we can obviously strip
the statement of infinite sequences of trees.
For nonconstant coefficient 1, we have:
THEOREM 8.10. (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 8.10 Æ Theorem 8.9 Æ Corollary 8.5
(nonconstant coefficient 1), we see that Theorem 8.10 is
not provable in FS.
Other P02 forms of KT involving only the internal structure
of a single finite tree can be found in [Fr02a].
We 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.
We 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 8.11. Let G1,G2,... be finite graphs. There exists i
< j such that Gi is minor included in Gj.
We then asked Robertson/Seymour to prove a form of EKT that
We knew implied full EKT, just from GMT. They complied, and
we wrote the triple paper [FRS].
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. We disavow remarks in
[FRS] 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 8.12. 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). 39 Robertson/Seymour also claims to be able to use the
subcubic graph theorem for linkage to EKT (see [FRS87]).
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. See [Fr06a]  [Fr06g].
*) 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 8.13. Every proof of *) in FS uses at least 2[1000]
symbols. Every proof of **) in P11CA0 uses at least 2[1000]
symbols. 9. 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 9.1. Let S Õ ¬2 be a symmetric Borel set. Then S or
¬2\S has a Borel selection.
My proof of Theorem 9.1, [Fr81], relied heavily on Borel
determinacy, due to D.A. Martin. See [Ma75], [Ma85],
[Ke94], 137148.
THEOREM 9.2. [Fr81]. Theorem 9.1 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. See [DSR96], [DSR99], [DSR02],
[DSR01X]. 40 THEOREM 9.3. 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 9.4. 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 9.5. [Fr05]. Theorem 9.3 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 9.6. [Fr05]. The existence of the cumulative
hierarchy up through every countable ordinal is sufficient
to prove Theorems 9.1 and 9.3. However, the existence of
the cumulative hierarchy up through any suitably defined
countable ordinal is not sufficient to prove Theorem 9.1 or
9.3.
DOM: The f:N Æ N constructible in any given x Õ N are
eventually dominated by some g:N Æ N.
THEOREM 9.7. ZFC + Theorem 9.4 implies DOM (Fr05]). ZFC +
DOM implies Theorem 3.4 ([DRS07]). 10. BOOLEAN RELATION THEORY.
The principal reference for this section is the forthcoming
[Fr•]. An advanced draft should appear on the web during
2007.
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. 41 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.
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
SMAH+ but not in SMAH, even with the axiom of
constructibility.
Here SMAH+ = ZFC + ("n)($k)(k is a strongly kMahlo
cardinal). SMAH = ZFC + {($k)(k is a strongly kMahlo
cardinal}k.
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 We
can decide all statements in the fragment within SMAH+.
We 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. 42
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.
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 SMAH, 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. 43
Furthermore, the exotic case comes out true in the second
classification.
THEOREM 4.1. The following is provable in SMAH+ but not in
SMAH, 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”. 11. FINITE GRAPHS.
Here we present an explicitly P01 sentence that is
independent of ZFC involving finite graphs. This is
intensively ongoing research, and Proposition 11.2
represents the current state of the art.
A simple graph G is a pair (V,E), where V = V(G) is a
nonempty set (the vertices), and E = E(G) is a set of
subsets of V of cardinality 2 (the edges).
We say that A Õ V(G) is a G independent set if and only if
there is no {x,y} Œ E(G) with x,y Œ A.
We will consider graphs on any set [t]k, where k,n ≥ 1.
I.e., where V(G) = [t]k. Here [t] = {1,...,t}.
For A Õ [t]k, the neighborhood of A consists of the y such
that {x,y} Œ E(G). The upper neighborhood of A consists of
the y >lex x such that {x,y} Œ E(G).
THEOREM 11.1. Every simple graph on any [t]k has an
independent set, where every vertex outside the set lies in
its upper neighborhood. The independent set is unique.
Let x,y Œ [t]k. We say that x,y are order equivalent if and
only if for all 1 £ i,j £ k, xi < xj iff yi < yj.
We say that G on [t]k is order invariant if and only if for
all x,y,x’,y’ Œ V(G), if (x,x’),(y,y’) are order equivalent
then {x,x’} Œ E(G) ´ (y,y’) Œ E(G). Thus connections are
made in G only according to the relative size of the
coordinates involved.
For x Œ [t]k, we write 2x = (2x1,...,2xk), and x1 =
(x11,...,xk1).
PROPOSITION 11.2. Every simple order invariant graph on any
[t]k has an independent set, where any 28k!x lying on a 4 44
clique outside the set, also lies on a 4 clique in its
upper neighborhood, with 28k!x1 absent.
THEOREM 11.3. Proposition 11.2 is provably equivalent to
Con(SMAH) over ACA. Proposition 11.2 follows immediately
from Theorem 11.1, if we remove “with 28k!x1 absent”.
Here ACA is the arithmetic comprehension axiom scheme with
full induction. SMAH = ZFC + {there exists a strongly nMahlo cardinal}n. ACA can be weakened somewhat.
Note that Proposition 11.2 is explicitly P01. 12. INCOMPLETENESS IN THE FUTURE.
The Incompleteness Phenomena, the centerpiece of Gödel’s
legacy, has come a long way. The same is true of the
related phenomenon of recursive unsolvability, also part of
the Gödel legacy. The phenomena is so rich. and so deep in
possibilities, that we expect the future to dwarf the
present and past.
Yet continued substantial progress is expected to be
painfully slow, requiring considerably more than the
present investment of mathematical and conceptual power
devoted to the extension and expansion of the phenomena.
In fact, this assessment can be justified if, as is common
today, one considers the P = NP problem as part of the
Gödel legacy, on the basis of his letter of March 20, 1956,
to John von Neumann (see [Go,8603], vol. V, letter 21, p.
373377).
Also consider the recursive unsolvability phenomena.
Perhaps the most striking example of this for the working
mathematician is the recursive unsolvability of Diophantine
problems over the integers (Hilbert’s tenth problem), as
discussed in section 3. We have, at present, no idea of the
boundary between recursive decidability and recursive
undecidability in this realm. Yet I conjecture that we will
understand this in the future, and that we will find,
perhaps, that recursive undecidability kicks in already for
degree 4 with 4 variables. However, this would require a
complete overhaul of the current solution to Hilbert’s
tenth problem, replete with new deep ideas. This would
result in a sharp increase in the level of interest for the
working mathematician who is not particularly concerned 45
with issues in the foundations of mathematics.
In addition, we still do not know if there is an algorithm
to decide whether a Diophantine problem has a solution over
the rationals. I conjecture that this will be answered in
the negative, and that the solution will involve some
clever number theoretic constructions of independent
interest for number theory.
We now come to the future of the Incompleteness Phenomena.
We have seen how far this has developed thus far.
i. First Incompleteness. Some incompleteness. [Go31].
ii. Second Incompleteness. Incompleteness concerning the
most basic metamathematical property. [Go31], [HB34,39],
[Bo93].
iii. Consistency of the AxC. Consistency of the most basic,
and then controversial, candidate for a new axiom. [Go40].
iv. Consistency of the CH. Consistency of the most basic
set theoretic problem highlighted by Cantor. [Go40].
v. Œ0 consistency proof. Consistency proof of PA using
quantifier free reasoning on the fundamental combinatorial
structure, Œ0. [Ge69].
vi. Functional recursion consistency proof. Consistency
proof of PA using higher type primitive recursion, without
quantifiers. [Go58], [Go72].
vii. Independence of AxC. Independence of CH (over AxC).
Complements iii,iv. [Co6364]. Forcing.
viii. Open set theoretic problems in core areas shown
independent. Starting soon after [Co6364], starting most
dramatically with R.M. Solovay (e.g., [So70], and his
independence proof of Kaplansky’s Conjecture [Dal01]), and
continuing with many others, notably Shelah (see [Sh•]).
Core mathematicians have learned to avoid raising new set
theoretic problems, and the area is greatly mined. See
[Jec06].
ix. Large cardinals necessarily used to prove independent
set theoretic statements. Starting most dramatically with
measurable cardinals implies V ≠ L ([Sco61]). Also for open 46
problems in the theory of projective sets, most recently
with [MS89] (proof of projective determinacy).
x. Large cardinals necessarily used to prove the
independence of set theoretic statements. See [Jec06].
xi. Uncountably many iterations of the power set operation
necessarily used to prove statements in and around Borel
mathematics. See [Fr71], [Ma75], [Fr05], [Fr07a]. Includes
Borel determinacy, and some Borel selection theorems of
Debs and Saint Raymond (see section 9 above).
xii. Large cardinals necessarily used to prove statements
around Borel mathematics. [Fr81], [St85], [Fr05], [Fr07a].
Includes some Borel selection theorems of Debs and Saint
Raymond (see section 9 above).
xiii. Independence of finite statements in or around
existing combinatorics from PA and subsystems of second
order arithmetic. Starting with [Goo44], [PH77], and, most
recently, with [Fr02a], and [Fr06a]  [Fr06g]. Uses
extensions of v) above (Gentzen), from [BFPS81]. Includes
Kruskal’s theorem, the graph minor theorem
(Robertson/Seymour), and the trivalent graph theorem
(Robertson/Seymour).
xiv. Large cardinals necessarily used to prove sentences in
discrete mathematics, as part of a wider theory (Boolean
Relation Theory). [Fr98], and [Fr•].
xv. Large cardinals necessarily used to prove explicitly P01
sentences. Section 11 above.
Yet this development of the Incompleteness Phenomena has a
long way to go before it realizes its potential to
dramatically penetrate core mathematics.
However, I am convinced that this is a matter of a lot of
time and resources. The quality man/woman hours devoted to
expansion of the incompleteness phenomena is trivial when
compared with other pursuits. Even the creative (and high
quality) study of U.S. tax law dwarfs the effort devoted to
expansion of the incompleteness phenomena by orders of
magnitude  let alone any major sector of technology,
particularly the development of air travel,
telecommuncations, or computer software/hardware. 47
Through my efforts over 40 years, I can see, touch, and
feel a certain ubiquitous combinatorial structure that
keeps arising, that is a demonstrably indelible footprint
of large cardinals. I am able to display this combinatorial
structure through Borel, and discrete, and finitary
statements that are increasingly compelling mathematically.
But I don’t quite have the right way to say it. I likely
need some richer context than the completely primitive
combinatorial settings that I currently use. This
difficulty will be overcome in the future, and that will
make a huge difference in the quality, force, and relevance
of the results to mathematical practice.
In fact, I will go so far as to make the following dramatic
conjecture. It’s not that the incompleteness phenomena is a
freak occurrence. Rather, it is everywhere. Every interesting substantial
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.
Recasting of mathematical theorems as elements of natural
finite sets of statements represents an inevitable general
expansion of mathematical activity. This, I conjecture,
will apply to any standard mathematical context.
This program has been carried out, to some very limited
extent, by BRT – as can be seen in section 10 above.
This may seem like a ridiculously ambitious conjecture,
which goes totally against the current conventional wisdom
of mathematicians  who think that they are immune to the
incompleteness phenomena.
But I submit that even fundamental features of current
mathematics are not likely to bear much resemblance to the
mathematics of the future.
Mathematics as a professional activity with serious numbers 48
of workers, is quite new. Let’s say 100 years old –
although even that is a stretch.
Assuming the human race thrives, what is this compared to,
say, 1000 more years? Probably a bunch of simple
observations in comparison.
Of course, 1000 years is absolutely nothing in evolutionary
or geological time. A more reasonable number is 1M years.
And what does our present mathematics look like compared to
that in 1M years time? These considerations should apply to
our present understanding of the Gödel phenomena.
We can of course take this even further. 1M years time is
absolutely nothing in astronomical time. This Sun has
several billion good years left (although the Sun will
cause a lot of global warming!).
Mathematics in 1B years time?? Who knows. But I am
convinced that the Gödel legacy will still be very much
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*This research was partially supported by NSF Grant DMS
0245349. ...
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