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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
Revised: June 8, 2007
Revised: June 19, 2007
Revised June 21, 2007
Revised September 28, 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 Relations.
12. Incompleteness in the Future.
We wish to thank Warren Goldfarb and Hilary Putnam for help with several historical
points. 2 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,
(Wang 1987, 1996), (Dawson 2005), and the historically comprehensive five volume set
(Gödel 19862003).
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 phenomena in new
critical directions.
The incompleteness phenomena lie at the heart of the Gödel legacy. Some careful
formulations, informed by some post Gödelian developments, are presented in sections
3,4,5.
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. 3
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 implied by
the consistency statement  hence the consistency statement is not provable. It was later
established that the two are in fact demonstrably equivalent.)
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 (Gödel 1940), (Cohen 196364).
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 mathematicians.
Specifically, the reference to unrestricted uncountable sets (of real numbers) in
CH readily distinguishes CH from “normal” mathematics, which relies, almost 4
exclusively, on the “essentially countable”, (e.g., the continuous or piecewise
continuous).
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 (due independently to H.G.
Dales and J. Esterle), and later shown to be not refutable without the continuum
hypothesis; i.e., not refutable in the usual ZFC axioms (due to R. Solovay). See (Dales
2001) for the refutation, and (Dales, Woodin 1987) for the consistency (non refutability)
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 5
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.1
We now know that the incompleteness phenomena do 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. 1 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
(Avigad 2003). 6
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 phenomena 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, (Gödel 1929), 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 (Gödel 19862003 4459). These Introductory notes were written by Burton Dreben and Jean van Heijenoort.
On page 52, the following passage from a letter from Gödel to Hao Wang, is
quoted (December 7, 1967): 7
“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.”2
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 (Feferman 2004), surprisingly the first clear statement
in Tarski’s work of the formal semantics for predicate calculus did not appear until
(Tarski 1952) and (Tarski, Vaught 1957).
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. 2 Skolem 1923a above is (Skolem 1922) in our list of references. 8
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 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 (Rayo, Williamson 2003), and
(Williamson 2000, 2003, 2006).
We have not yet published on this topic, but unpublished reports on our results are
available on the web. Specifically, in (Friedman 1999), and in (Friedman 2002a 6599). 9
We plan to publish a monograph on this topic in the not too distant future. 3. THE FIRST INCOMPLETENESS THEOREM.
The Gödel first incompleteness theorem is first proved in (Gödel 1931). It is
proved there in detail for a specific variant of what is now known as the simple theory of
types (going back to Bertrand Russell), with natural numbers at the lowest type. This is a
rather strong system, nearly as strong as Zermelo set theory.
It asserts that there is a sentence that is neither provable nor refutable in this
system.
In (Gödel 1932b), Gödel formulates his incompleteness theorems for extensions
of a variant of what is now known as PA = Peano arithmetic.
(Gödel 1934) gives another treatment of the results in [Gödel 31], but also, most
importantly, introduces the notion of recursive functions and relations.
At the end of (Gödel 1931 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 10
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 (Gödel 1931).
Today, Gödel is credited for quite general forms of the first incompleteness
theorem. There are already claims of generality in the original paper, (Gödel 31). 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.
(Rosser 1936) is credited for significant additional generality, using a clever
modification of Gödel’s original formal self referential construction. It is shown there
that the hypothesis of 1consistency can be replaced 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 (Robinson 1952),
and (Tarski, Mostowski, Robinson 1953), with the use of the formal system Q.
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 11
1. Sx ≠ 0.
2. Sx = Sy Æ x = y.
3. x ≠ 0 Æ ($y)(x = Sy).
4. x + 0 = x.
5. x + Sy = S(x + y).
6. x • 0 = 0.
7. x • Sy = (x • y) + x.
8. x £ y ´ ($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 (Robinson 1952), and (Tarski, Mostowski, Robinson 1953). 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 12
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 negative 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 asked for a decision procedure for determining whether a given
polynomial with integer coefficients in several integer variables has a zero.
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 (Davis 1973),
(Matiyasevich 1993). The MRDP theorem was shown to be provable in the weak
fragment of arithmetic, EFA = IS0(exp), in (Dimitracopoulus, Gaifman 1982).
We can use (Dimitracopoulus, Gaifman 1982) 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 ($x1,...,xn)(s = t) in L(Q) that is neither provable nor refutable in T.
It is not clear whether EFA can be replaced by a weaker system in Theorem 3.3
such as Q. 13
An important issue is whether there is a “reasonable” existential equation
($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 (additions and multiplications).
In 1992, Matiyasevich showed that nine unknowns over the nonnegative integers
suffices for recursive unsolvability. One form of the result (not the strongest form) says
that the problem of deciding whether or not a polynomial with integer coefficients in nine
unknowns has a zero in the nonnegative integers, is recursively unsolvable. A detailed
proof of this result (in sharper form) was given in (Jones 1982).
Also (Jones 1982) proves that, e.g., the problem of deciding whether or not a
polynomial with integer coefficients defined by at most 100 operations (additions and
multiplications with integer constants) has a zero in the nonnegative integers, is
recursively unsolvable.
It is well known that degree 4 suffices for recursive unsolvability. In (Jones
1982), it is shown that degree 4 and 58 nonnegative integer unknowns suffice for 14
recursive unsolvability. I.e, the problem of deciding whether or not a polynomial with
integer coefficients, of degree 4 with at most 58 unknowns, is recursively unsolvable.
In fact, (Jones 1982) 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 (Siegel 72). See (Grunewald,
Segal 1981), (Masser 1998).
For degree 3, the existence of an algorithm is wide open, even for three variables
(over the integers, the nonnegative integers, or the rationals). For degree 3 in two integer
variables, an algorithm is known, but it is wide open for degree 3 in two rational
variables.
It is clear from this discussion that the gap between what is known and what could
be the case is enormous, just in this original context of deciding whether polynomials
with integer coefficients have a zero in the (nonnegative) integers. Specifically, <3,3>
could conceivably be on this list of pairs. 15
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 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 (Gödel 1931), 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. 16
Gödel promised a part 2 of (Gödel 1931), but this never appeared. There is some
difference of opinion as to whether Gödel planned to provide detailed proofs of his
second incompleteness theorem in part 2, or whether Gödel planned to let others carry
out the details.
In any case, the necessary details were carried out in (Hilbert, Bernays
1934,1939), and later in (Feferman 1960), and most recently, in (Boolos 1993).
In (Hilbert, Bernays 1934,1939), the so called Hilbert Bernays derivability
conditions were isolated in connection with a detailed proof of Gödel’s second
incompleteness theorem given in (Hilbert, Bernays 1934,1939). Later, these conditions
were streamlined in (Jerosolow 1973).
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: 17
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.
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)]. 18
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]).
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 19
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 given above
using derivability conditions are rather subtle and involved.
We recently addressed this problem in (Friedman 2007a), 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. 20
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.
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. 21
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.
We use the system EFA = exponential arithmetic for this more general treatment.
EFA is the system of arithmetic based on addition, multiplication and exponentiation,
with induction applied only to formulas all of whose quantifiers are bounded to terms.
This is the same as the system IS0(exp) in (Hajek, Pudlak 1993 p. 37).
INFORMAL SECOND INCOMPLETENESS (general many sorted, EFA). 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, EFA). 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 22
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.
Here, 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.
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. Precisely this approach was adopted by (Quine, 1940, 1951, Chapter
7).
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 23
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. This is the choice made in (Quine, 1940, 1951, Chapter 7).
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 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. This
was also done in (Quine, 1940, 1951, Chapter 7). 24
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 (Gödel 1936), 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 (Mostowski
1952). Similar results were also proved in (Ehrenfeucht, Mycielski 1971) and (Parikh 25
1971). Also see (Parikh 1973) for results going in the opposite direction concerning the
number of lines in proofs in certain systems.
In (Friedman 79), 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 (Pudlak
1985), which establishes an (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 direction to take finite second incompleteness is to give some versions
which are not asymptotic. I.e., they involve specific numbers of symbols that are argued
to be related to actual mathematical practice.
Although the very good upper bound of O(n) is given in (Pudlak 1985) for a proof
of Conn(T) in T, at least for some reasonable systems T, the situation seems quite 26
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. It seems plausible that there is no subexponential upper bound here.
But obviously, if there is some algorithm and polynomial P that PA can prove is
an algorithm for testing satisfiability of Boolean expressions whose run time is bounded
by P, then PA proves Conn(ZF) using a polynomial number of symbols in n (assuming
Con(ZF) is in fact true). So in order to show that there is no subexponential upper bound
here, we will have to refute this strong version of P ≠ NP. However, this appears to be as
challenging as proving P ≠ NP.
There are some other aspects of lengths of proofs that seem important. One is the
issue of overhead.
Gödel established in [Gödel 1940] that any proof of an arithmetic 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 arithmetic 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 27
and ZF, and also to ZFC + ÿCH and ZFC. Also they apply equally well to the Cohen
forcing method (Cohen 19631964), 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 (Tarski 1951), 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 axioms consist of the usual field axioms, together with 1 is not the sum of squares, x or x is a square, and every polynomial of odd degree with
leading coefficient 1 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.
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 (Cohen 1969), (Renegar 1982ac), (Basu, Pollack, Roy 2006). In terms of computational complexity, the set of true first 28
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 (Goodstein 1944) and (Paris, Harrington 29
1977), and are proved just beyond PA. We discovered many examples in connection with
theorems of J.B. Kruskal (Kruskal 1960), and Robertson, Seymour (Roberton, Seymour
1985, 2004), which are far stronger, with no predicative proofs. See (Friedman 2002b).
None of these three references discusses the connection with sizes of proofs. This
connection is discussed in (Smith 1985 132135), and in the unpublished abstracts
(Friedman 2006ag) from the FOM Archives.3
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 (Gödel 1936).
3 See http://cs.nyu.edu/pipermail/fom/ 30 6. THE NEGATIVE INTERPRETATION.
Gödel wrote four fundamental papers concerning formal systems based on
intuitionistic logic: (Gödel 1932a), (Gödel 1933a), (Gödel 1933b), (Gödel 1958). (Gödel
1972) is a revised version of (Gödel 1958).
In (Gödel 32a), 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 (Hosoi, Ono 1973) and (Minari 1983).
In (Gödel 1933a), Gödel introduces his negative interpretation in the form of an
interpretation of PA = Peano arithmetic in HA = Heyting arithmetic. Here HA is the
corresponding 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,⁄,Ÿ,Æ. The interpretation for
propositional calculus inductively interprets 31
^ as ^.
ÿ as ÿ.
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.
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). 32
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 (Kreisel 68a 344), (Kreisel 68b Section 5), (Myhill 74),
(Friedman 73), (Leivant 85).
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 (Gentzen 1969), (Schütte 1977)), and from Gödel’s so called Dialectica or
functional interpretation, in (Gödel 1958), (Gödel 1972).
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 required proof theory. Nevertheless, in this case, the
Dialectica interpretation has been extended by Spector in (Spector 1962), and the fact that
these two systems have the same provable P02 sentences then 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 (Friedman 1978), we broke this impasse by modifying Gödel’s negative
interpretation via what is now called the A translation. Also see (Dragalin 1980). 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 33
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.
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 (Friedman
1973) and (Leivant 1985).
(Godel 1958) and (Godel 1972) 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. 34
In Gödel’s Dialectica interpretation, theorems of HA are interpreted as derivations
in a quantifier free system T of primitive recursive functionals of finite type that is based
on quantifier free axioms and rules, including a rule of induction.
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 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
(Gödel 58) in (Gödel 19862003 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 (Spector 1962). 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 and highly impredicative system Z2. However, in 35
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 the five
references to Kohlenbach (and joint authors) in the list of references. 7. THE AXIOM OF CHOICE AND THE CONTINUUM
HYPOTHESIS.
Gödel wrote six manuscripts directly concerned with the continuum hypothesis:
Two abstracts, (Gödel 1938), (Gödel 1939a). One paper with sketches of proofs, (Gödel
1939b). One research monograph with fully detailed proofs, (Gödel 1940). One
philosophical paper, (Gödel 1947,1964), 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 formulation 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. 36
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 (Solovay 1970).
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 verified proofs  necessarily incorporate very 37
substantial abbreviation mechanisms. See, e.g., (Barendregt, Wiedijk 2005), (Wiedijk
2006).
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 (Cohen 19631964). 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 (Feferman 1960)).
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 (Gogol 1979), (Friedman 2003a).
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 (Friedman 2003b). It is also known that
AxC can be written with five quantifiers in Œ,=, over ZFC. See (Maes 2007). 38
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. They take the view that the continuum
hypothesis is a well defined 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 (Gödel 1947,1964).
The late P.J. Cohen led a panel discussion at the Gödel Centenary called On
Unknowability, where he 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. 39
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 respective 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
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. 40
A qo (quasi order) is a reflexive transitive relation (A,£). A wqo (well quasi
order) is a qo (A,£) such that for all infinite sequences 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.
(Kruskal 1960), treats finite trees as finite posets, and studies the qo
there exists an inf preserving embedding from T1 into T2.
THEOREM 8.1. (Kruskal 1960). The above qo of finite trees as posets is a wqo.
The simplest proof of Theorem 8.1 and some extensions, is in (NashWilliams
1963), 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 (Feferman 1964,1968), (Feferman 1998). Poincare, Weyl,
and others railed against impredicative mathematics. See (Weyl 1910), (Weyl 1987),
(Feferman 1998 289291), and (Foline 1992).
THEOREM 8.2. (Friedman 2002b). Kruskal’s tree theorem cannot be proved in FS.
KT goes considerably beyond FS, and an exact measure of KT is known. See
(Rathjen, Weiermann 1993). 41
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)).
THEOREM 8.3. (Kruskal 1960). 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 (Friedman, Montalban, Weiermann in
preparation).
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. 42
Note that Corollary 2.6 is P02.
THEOREM 8.7. Corollary 8.5 cannot be proved in FS. This holds even for linear bounds
n+k with variable n and constant k.
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,...
With this kind of strong uniformity, we can obviously strip Theorem 8.9 of
infinite sequences of trees. Using the linear bounds n+k, k fixed, we obtain:
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 (using bounds n+k, variable
n, k constant), 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 (Friedman 2002b). 43
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 well known graph minor
theorem of Robertson, Seymour (Robertson, Seymour 1985, 2004).
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 and Seymour to prove a form of EKT that we knew
implied full EKT, just from GMT. They complied, and we wrote the triple paper
(Friedman, Robertson, Seymour 1987).
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 (Friedman, Robertson, Seymour 1987) 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). 44
Robertson, Seymour also claims to be able to use the subcubic graph theorem for
linkage to EKT (see (Robertson, Seymour 1985), (Friedman, Robertson, Seymour 1987)).
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 (Friedman 2006ag).
*) 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. 45
My proof of Theorem 9.1 in (Friedman 1981) relied heavily on Borel
determinacy, due to D.A. Martin. See (Martin 1975), (Martin 1985), and (Kechris 1994
137148).
THEOREM 9.2. (Friedman 1981). 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 the five
papers of Debs and Saint Raymond in the references.
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.
THEOREM 9.5. (Friedman 2005). 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. (Friedman 2005). The existence of the cumulative hierarchy up through
every countable ordinal is sufficient to prove Theorems 9.1 and 9.3. However, the 46
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 (Friedman 2005). ZFC + DOM
implies Theorem 3.4 (Debs, Saint Raymond 2007). 10. BOOLEAN RELATION THEORY.
The principal reference for this section is the forthcoming book (Friedman in
preparation]. 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.
We now write them in BRT form.
THIN SET THEOREM. For all f Œ MF there exists A Œ INF such that fA ≠ N. 47
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. 48
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.
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: 49
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.
Furthermore, the exotic case comes out true in the second classification.
THEOREM 10.1. 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 RELATIONS.
Here we present an explicitly P01 sentence that is independent of ZFC involving
finite relations. This is intensively ongoing research, and Proposition 11.3 represents the
current state of the art (unpublished).
We consider binary relations R Õ [t]k ¥ [t]k = [t]2k. Here k,t ≥ 1 and [t] = {1,...,t}.
We say that R is strictly dominating if and only if for all x,y,
R(x,y) Æ max(x) < max(y).
For A Õ [t], we write 50
A’ = [t]k\A.
RA = {y: ($x Œ A)(R(x,y))}.
THEOREM 11.1. Every strictly dominating R Õ [t]2k has an A Õ [t] with RA = A’. A is
unique.
There is an equivalent form of Theorem 11.1 using antichains. An R antichain is
an A Õ [t] such that for all x,y Œ A, ÿR(x,y). This is analogous to independent sets in
graph theory.
THEOREM 11.2. Every strictly dominating R Õ [t]2k has an antichain A with A¢ Õ RA.
A is unique.
Let x,y Œ Zr, where Z is the set of all integers. We say that x,y are order
equivalent if and only if for all 1 £ i,j £ n, xi < xj iff yi < yj.
We say that S Õ [t]n is order invariant if and only if for all order equivalent x,y Œ
[t]n, x Œ S ´ y Œ S.
For x Œ Nk and m Œ N, we write mx = (mx1,...,mxn), and xm =
(x1m,...,xkm).
PROPOSITION 11.3. Every strictly dominating order invariant R Õ [t]2k has an antichain
A where every (x,(8k)!x,(8k)!x1) Œ A¢9 is order equivalent to some (y,(8k)!x,(8k)!x2) Œ
(RA)9. 51
THEOREM 11.4. Proposition 11.3 is provably equivalent to Con(SMAH) over ACA.
Proposition 11.2 follows immediately from Theorem 11.2, if we replace 2 with 1 (set y =
x).
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 11.3 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 deep, and rich in possibilities, that we
expect the future to eclipse 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 (Gödel 19862003 Vol. V, letter 21, 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 52
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 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 in the presence of some arithmetic.
(Gödel 1931).
ii. Second Incompleteness. Incompleteness concerning the most basic
metamathematical property  consistency. (Gödel 31), (Hilbert Bernays 1934,1939),
(Feferman 1960), (Boolos 93).
iii. Consistency of the AxC. Consistency of the most basic, and once
controversial, early candidate for a new axiom of set theory. (Gödel 1940).
iv. Consistency of the CH. Consistency of the most basic set theoretic 53
mathematical problem highlighted by Cantor. (Gödel 1940).
v. Œ0 consistency proof. Consistency proof of PA using quantifier free reasoning
on the fundamental combinatorial structure, Œ0. (Gentzen 1969).
vi. Functional recursion consistency proof. Consistency proof of PA using higher
type primitive recursion, without quantifiers. (Gödel 1958), (Gödel 1972).
vii. Independence of AxC. Independence of CH (over AxC). Complements iii,iv.
(Cohen 19631964). Forcing.
viii. Open set theoretic problems in core areas shown independent. Starting soon
after (Cohen 19631964), starting dramatically with R.M. Solovay (e.g., his work on
Lebesgue measurability (Solovay 1970), and his independence proof of Kaplansky’s
Conjecture (Dales, Woodin 1987)), and continuing with many others. See the rather
comprehensive (Jech 2006). Also see the many set theory papers in (Shelah, 19692007).
Core mathematicians have learned to avoid raising new set theoretic problems,
and the area is greatly mined.
ix. Large cardinals necessarily used to prove independent set theoretic statements.
Starting dramatically with measurable cardinals implies V ≠ L (Scott 1961). Continuing
with solutions to open problems in the theory of projective sets (using large cardinals),
culminating with the proof of projective determinacy, (Martin, Steel 1989).
x. Large cardinals necessarily used to prove the consistency of set theoretic
statements. See (Jech 2006).
xi. Uncountably many iterations of the power set operation necessarily used to
prove statements in and around Borel mathematics. See (Friedman 1971), (Martin 1975), 54
(Friedman 2005), (Friedman 2007b). 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. (Friedman 1981), (Stanley 85), (Friedman 2005), (Friedman 2007b).
Includes some Borel selection theorems of Debs and Saint Raymond (see section 9 above
and the references to Debs and Saint Raymond).
xiii. Independence of finite statements in or around existing combinatorics from
PA and subsystems of second order arithmetic. Starting with (Goodstein 1944), (Paris,
Harrington 1977), and, most recently, with (Friedman 2002b), and (Friedman 2006ag).
Uses extensions of v) above, (Gentzen, 1969), from (Buchholz, Feferman, Pohlers, Seig
1981). Includes Kruskal’s theorem, the graph minor theorem of Robertson, Seymour
(Robertson, Seymour, 1985, 2004), and the trivalent graph theorem of Robertson,
Seymour (Robertson, Seymour, 1985).
xiv. Large cardinals necessarily used to prove sentences in discrete mathematics,
as part of a wider theory (Boolean Relation Theory). (Friedman 1998), and (Friedman in
preparation).
xv. Large cardinals necessarily used to prove explicitly P01 sentences. See section
11 above for the current state of the art.
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 55
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, telecommunications, or computer software and hardware.
Through my efforts over 40 years, I can see, touch, and feel a certain
combinatorial structure that keeps arising  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 express it. I likely need some richer
context than the completely primitive combinatorial settings that I currently use. This
difficulty will definitely 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 all of which can be decided 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 56
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 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 merely 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 can know what that will be like. But I am
convinced that the Gödel legacy will remain very much alive – at least as long as there is
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*This research was partially supported by NSF Grant DMS 0245349. ...
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