THE SIGNIFICANCE OF INCOMPLETENESS
THEOREMS *

R . L . GOODSTEIN

I WANT to start by considering certain fundamental differences between
the major incompleteness theorems which have been discovered in
researches in the foundations of mathematics during the past thirty years
and the incomplete axiom systems which were found in the study ofpro-
jective geometry during the last century. A confusion between these
forms of incompleteness has led some mathematicians to underestimate
the significance of the newer results and some philosophers to seek to
understand the meaning of the new discoveries by reference to the
technically simpler older work.

When it was discovered, for instance, that a system of projective
geometry in two dimensions which postulated the axioms of incidence
and Pappus' Theorem was incomplete because Desargues' Theorem on
perspective triangles was not derivable in the system, then this incom-
pleteness could be interpreted as a proof of the independence of
Desargues' Theorem, postulated as a new axiom, from the other
axioms. The impossibility of proving Desargues' Theorem is sur-
prising in view of the fact that the corresponding theorem in three
dimensions is derivable from the three-dimensional axioms of inci-
dence, but it is nevertheless, I think, without philosophical significance
because it throws no light on the nature of formal systems as such and
imposes no limitations upon the axiomatic method.

The great modern incompleteness theorems which I shall consider
are those due to Skolem and Godel. Skolem's incompleteness theorem
was discovered in an attempt to explain a paradox which Skolem
himself found in the theory of sets.

The paradox out of which Skolem's incompleteness theorem arises,
is produced by applying a result of Lowenheim's to a formalised set
theory. Lowenheim showed that every consistent set of statements
has a denumerable model, and so any formal system which admits
some model (of the power of the continuum perhaps) has also a
denumerable model. That is to say, for a consistent theory, we can
find an interpretation in which all the objects, of which the theory

* Received 25.vii.62
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SIGNIFICANCE OF INCOMPLETENESS THEOREMS

treats, may be taken to be the natural numbers. Consider now some
formalisation of set theory; according to the Lowenheim theorem we
can find an interpretation of the membership relation of the theory in
which all the sets of the theory are taken to be natural numbers. But in
any adequate formalisation of set theory, using the familiar diagonal
process, we can prove Cantor's theorem that the set of all subsets has a
greater cardinal than the set itself:

Let S be a denumerable set and let a subset of 5 be denoted by a
sequence of zeros and units, a zero in the «th place showing that the
nth member of S is not in the subset, and a unit in the nth place showing
that the nth member of 5 is a member of the subset. Suppose now
that the set of all subsets of 5 is denumerable, and let it be enumerated
as follows:

h = &\, A. a\, . . .
s2 = a\, a\, a\, . . .
h = <A> di2> ^V • • •

where each anT is either o or i.
Define bn = a\ + i (mod 2)
and consider the subset

o- = Oi, b e , b a , . . .

The subset a differs from Sx in respect of the first element of S since
hi is 1 or o according as a\ is 0 or 1, from S2 in respect of the second
element, and so on. Thus the subset a does not occur in the enumera-
tion Slt St, Ss, . . . and the hypothesis that the set of all subsets of 5
is enumerable is disproved.

But by the Lowenheim theorem there is a model of set theory
(supposed consistent) in which each set is associated with a natural
number, so that in defiance of Cantor's theorem, the set of subsets is
denumerable.

This is Skolem's paradox. The conclusion which Skolem himself
drew from the paradox is that a formahsation of set theory can contain
only relatively non-denumerable sets; i.e. sets which are non-denumer-
able only because the formahsation lacks the functions to enumerate
them. In other words every formahsation of set theory must be
incomplete in the sense that there are denumerable sets which cannot
be proved denumerable within the theory. To justify this interpreta-
tion of the paradox one must observe that the proof of Cantor's
theorem in some system starts by assuming that a certain mapping of
a set on its subsets exists and derives a contradiction in the system from
this assumption. Existence here of course means existence in the

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R. L. GOODSTEIN

system, so that the conclusion to be drawn from the contradiction is,
not that the mapping in question does not exist, but only that it does
not exist in the system. Since Lowenheim's theorem assures us that the
mapping does in fact exist, it follows that there is a mapping which is not
contained in the formal system, so that the system is incomplete with
respect to the class of mappings it contains. Unlike the situation in
projective geometry we cannot remedy the deficiency by fortifying
the system with another axiom; we could complete the system only at
the price of rendering it inconsistent. Another way of expressing the
result is to say that Lowenheim's theorem for any particular formalisa-
tion of set theory is not provable by means of the resources of that
formalisation alone.

Of course the notion that there are no absolutely non-denumerable
sets is not a new one. The sole ground we have for believing in the
existence of a non-denumerable set lies in Cantor's theorem itself. But
if we do not assume that the totality of subsets forms a set (and this is
nothing but an assumption) then all that the diagonal process proves is
that from any sequence of subsets we can construct another subset,
just as from any natural number we can construct another, by adding
one. And if we give up the axiom of subsets of course the Skolem
paradox disappears.

There is nothing in the paradox itself to^orce us to give up the axiom
of subsets. A constructivist who already rejects the set of all subsets on
other grounds will not need to reckon with the Skolem paradox; and
a mathematician seeking the greatest possible generality will have to
remain content with relative non-denumerability. At best we can
have a transfrnite hierarchy of systems in each of which there are sets
non-denumerable in a particular system but denumerable in a system
of greater ordinal.

The Gb'del Incompleteness Theorem
I come now to the major incompleteness theorem of mathematical

logic, Godel's Theorem that all sufficiently rich formal systems neces-
sarily contain sentences which are neither provable nor refutable in the
system. There are so many interesting facets to this result that I shall
later consider the proof in some detail, but first I want to observe that
in this theorem, as in the Skolem theorem for sets, the incompleteness
revealed by the theorem cannot be filled by means of a new axiom; it
is true that the particular undecidable sentence constructed in the proof
can itself be postulated as an axiom, but the proof shows that the

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SIGNIFICANCE OF INCOMPLETENESS THEOREMS

system so fortified will still contain undecidable sentences. The un-
decidable sentence in this Theorem of Godel has the form (yx) V(x);
neither (yx) V(x) itself, nor its negation (Ex) —1 P (x) is provable in the
system under consideration, system #" say, but each instance of the
universal sentence P(x), namely P(o),P(i),P(2), . . . is provable in #".
Many attempts have been made to close the gap this theorem is thought
to reveal in the proof structure of formal systems. The most obvious
course to take would be to add to the proof resources of the system a
new derivation scheme permitting the derivation of (yx) P(x) from
the infinite sequence of sentences

P(o), P(i), P(2), . . .;

but this device entirely destroys the finite character of a proof process.
In the Godel Theorem as we shall see, the very possibility of proving an
infinite number of sentences P(o), P(i), . . . without a prior proof of
the universal sentence P(x), was revealed for the first time. This
aspect of Godel construction has been cleverly exploited in a recent
attempt to obtain a (relatively) closed proof system without intro-
ducing non-finitist proof schemata, and is in some respects one of the
most interesting features of the Godel Theorem.

As is well known, the heart of Godel's construction is a one-to-one
mapping of the syntax of a formalised arithmetic upon arithmetic
itself. There are many ways known in which this mapping may be
accomplished but I shall simply suppose that each primitive sign of
some formal system of arithmetic sf has been assigned a number, and
that each sequence of primitive signs with numbers «<,, nu n?, . . ., nt.
whether forming a sentence or not, is given the number 2"° . 3"1 pj*,
where pk is the feth odd prime number. I shall further suppose that
the formal system sf contains all primitive recursive functions, either
directly in the sense of admitting primitive recursive definitions as
axioms or indirectly by having definition resources like existential and
minimal operators. I may mention in passing that it is the failure to
introduce this requirement and to show the fundamental part it plays,
which vitiates most popular accounts of Godfi'" work, ( T shall also
assume that s4 is recursively axiomatisable, so that the predicate ' n is
the number of an axiom ' is expressed by a primitive recursive relation
A(n), i.e. is expressed within s/ itself by this relation. Such relations as
' n is the number of a one variable primitive recursive function ', ' n
is the number of a variable in formula number f', 'n is the number of
a proof of formula f' and ' n is the number of the formula which
results by substituting the numeral representation of number k in

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R. L. GOODSTEIN

formula f' may all be shown to be primitive recursive. The key tools
in establishing these results are the primitive recursiveness of the
relation

{Ex){x <y & R(x, y, z)}

where R itself is primitive recursive, and the reductions to primitive
recursive definition of a related schema of definition known as defini-
tion by course-of-values recursion. To exhibit Godel's undecidable
sentence I denote by

the number of the expression of the formal system si obtained by
substituting the numeral representing the number r for the variable
of number v in the formula with number^ and by

Pr(m, n)

the relation which says that m is the number of a proof of formula
number n. Further let n be the number of the variable n and let a be
the number of the sentence of A which we are denoting by

(V«) -i Pr(m, S(fl(v/m)) (i)
and finally let (Vm) G(m) denote the sentence obtained from (i) by
substituting the numeral a for the variable n. Then neither (vm) G(m)
nor its negation is provable in si. We observe first that since
(ym) G(m) is formed by substituting numeral a for the variable n in
formula number a, its number is therefore

Sf.(v/a).
Hence if (vm) G(m) were provable, and if k were the number of its
proof, then

Pr(k, Sf.(v/a)) (ii)

holds (and the formula in A which this represents is provable in si,
because Pr(m, n) is a primitive recursive relation, and this is one of the
points where this fact is critical to the proof); but this contradicts the
formula which (v m ) G(m) itself represents, i.e. that represented by

(ym) -i Pr(ro, st.(v/a).
I emphasise again that the contradiction is in s( itself, and so if si is
consistent then (vm) G(m) is not provable in si'. To prove that
-~i (v m ) G(m) is also improvable I shall assume rather more than the
consistency of sd, the so-called co-consistency of sd, but this additional
assumption could be dispensed with at the price of taking a rather more
complicated sentence than (vm) G(m). By thecu-consistency of si we
mean that for any formula 1f(m) it is impossible to prove in si that

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SIGNIFICANCE OF INCOMPLETENESS THEOREMS

(Em) - i #(m) and #(o), 9 ( i ) , #(2) . . . all hold simultaneously.
First w e observe that as a consequence of what w e have already estab-
lished, if sd is simply consistent, then none of the numbers 0 , 1 , 2, . . .
is the number of a proof of formula number St,(v/a), and so

-> Pr(m, St,(v/a))

is provable in sd for any m, i.e. G(o), G(i), G{z), . . . are all provable.
Hence b y w-consistency, (Em) —1 G(m) is not provable. Thus w e have
seen that neither ( y i ) (Gm) nor its contrary is provable in sd, although
each instance of the general formula, viz. G(o), G(i), G(2), . . . is
provable. T h e formula (ytn) G(m) is said to be undecidable.

I remarked earlier that Godel's arithmetisation showed for the first
time h o w it is possible in a formal system with finite proof procedure
to prove all the formulae G(o), G(i), . . . without first proving G(m),
(in fact even if G(m) is n o t provable); w e have just seen an instance of
this. In the general case, let N(fe), be the number of the numeral
representing kin. A, and let h be the number of a formula H(n), then
the assertion that all the formulae H(o), H ( i ) , . . . are provable is
expressed by

(Vk)(&») Pr(m, Sfc(v/N(fe))

and this formula may be provable in sd even though H(n) is not. This
constitutes a formalisation of the notion of an arbitrarily assigned integer.

The specific instance of an undecidable formula (y/w) G(m) which
is constructed in Godel's proof is of no particular significance in arith-
metic, but by formalising the proof of undecidability Godel obtained
the remarkable conclusion that the sentence of arithmetic which entails
arithmetic's freedom from contradiction, viz.

(V«)(vr)(v*) "I ( p ( ' . m) & V(s, Neg m)} (C)
(where Neg m is the number of the negation of sentence number m)
is itself undecidable, if arithmetic is consistent. Contrary to a widely
held belief this result does not, however, establish the impossibility of
proving the consistency of a codification of arithmetic by finitist
methods formahsable within the codification. Even though the closed
formula C is not provable in sd, each of its instances

-1 P(r, m) & P(s, Neg m)
is provable in sd for arbitrary r, s, m (in virtue of Gentzen's consistency
proof by transfinite induction), and the general formula expressing
the provability of these instances may itself be provable in s/. But of
course a proof inside sd of s/'s consistency offers no security, for if sd
were inconsistent then every formula in si would be provable in sd.

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R. L. GOODSTEIN

In fact, Godel's result does not really bear upon the problem of con-
sistency itself but affords a means of establishing the independence from
the axioms of si of axioms (like transfinite induction) whose addition to
si suffice to prove the closed formula C above in the enlarged system.

If we add the improvable formula (ym) G(w) to si as a new axiom
forming a system A+ say, then exactly as before we can construct an
undecidable formula in A+, so that A+ is also incomplete; even if we
form a new system B, by adding in all the undecidable formulae of si
as additional axioms, if the axioms of B form a recursive set, then B is
still incomplete. Since (vm) G(/w) is not provable in si the system A~
formed by adding the denial of ( v w ) G(m) is consistent if si is con-
sistent, and so by Lowenheim's Theorem A~ must admit a denumerable
model in which G(o), G(i), . . . are all true but (vm) G(m) is false
and therefore m must take values other than o, i, 2, . . .

It follows from this that A~ must admit what is called a non-
standard model, that is, an interpretation in which a class of objects
which is not ordinally similar to the natural numbers plays the part of
the natural numbers. That this in fact is the case was shown independ-
ently by Skolem in 1934, who proved that a certain class of functions
can play the number role.

The construction of the formula G(m) above can also be carried out
in a system without quantifiers, in some formalisation of recursive
arithmetic, dt say, and we find that G(m) with free variable m is
improvable in 9t, but each of G(o), G(i), . . . is provable. We cannot
therefore explain away Godel's incompleteness theorem as a defect of
quantification theory. St like A is also incomplete. There is how-
ever an important difference between 91 and si since it can be shown
that no non-standard model of 3t can itself be a recursive model.

Discussing Godel's incompleteness theorem in 1934, before
Skolem's result was known here, Wittgenstein was led to the same
interpretation of the theorem, that induction and substitution of
natural numbers for free variables fail to ensure that the natural
numbers are the only values which the variables may take; it is perhaps
surprising that the passages on Godel's Theorem in the recently pub-
lished ' Reflections on the Foundations of Mathematics ' give no hint
of this remarkable insight.

Since every axiom system for the natural numbers is incomplete
and therefore necessarily admits a non-standard model it has been
argued that we must look outside axiom systems for a logical founda-
tion of arithmetic; this view is associated with a neo-realist outlook

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SIGNIFICANCE OF INCOMPLETENESS THEOREMS

in foundation studies, that the elements of mathematics are objects in
a real world with so-to-speak physical properties which the mathe-
matician only partially captures in an axiom system. Certainly in
some of his work the mathematician has an almost overwhelming
conviction that he is uncovering connections which he waiting to be
revealed, but this may only reflect his amazement at the astonishing
way the pieces of a puzzle sometimes fit together; we feel that the
pieces must have been made to fit before we actually handled them,
that we are only reconstructing a puzzle some other mind set for us.
Neo-reahsm is not, however, a return to the Greek standpoint that
formal geometry is an account of the space of our physical sensations.
The case against classical philosophical realism in mathematics is over-
whelmingly strong. In geometry the well-known consistency proofs
of non-Euclidean geometry relative to the Euclidean makes it impos-
sible for only one of the two geometries to be valid and yet both
cannot mirror the real world. The neo-realist argues from this, not
that the elements of geometry are concepts, but that no formal system
can adequately express the whole of geometry, which is something
revealed only to the intuition. This is an attractive thesis; every
mathematician is conscious of possessing an ' inner sight', an inward
short cut. But as a philosophical analysis of mathematics, neo-reahsm
is no more tenable a position than classical realism. Intuition can be
false and misleading and the inward ' short c u t ' just a cul de sac;
intuition is certainly an important element in the creation of mathe-
matics, but to see it as the organ which gives the mathematician access
to mathematical reality is to be deceived by an analogy. When we
say that no formal system can characterise the number concept, we do
not mean that the number concept is something which we already
have independently of the formal system; I may reject every definition
of the meaning of a word, because it fails to characterise what I mean
by the word, and maintain, rightly, that I know well what the meaning
is, and yet my knowing what the meaning is may consist in nothing
more than my rejection of the definitions. Just as I may write a story
and be left with the feeling that this is not the story I meant to write,
although of course I have not already in mind another story with
which I compare it. When we contrast formal mathematics with
intuitive mathematics we are not contrasting an image with reality,
but a game played according to strict rules with a game with rules
which change with the changing situation; a proof in intuitive mathe-
matics may be a particular way of looking at a diagram, i.e. a particular

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fj

R. L. GOODSTEIN

way of using it as a symbol, or it may consist in stepping outside the
particular system with which we are operating.

What we call an intuitive proof of some particular mathematical
relation, is not a proof intelligible only to some special sense, quite the
contrary. An intuitive proof of the relation is a proof which makes
the minimum appeal to esoteric knowledge, which links the relation
most immediately to a familiar background; but in its role as proof
an intuitive proof has the same essential character as a formal proof, it
exposes the connections between one relation and another.

An intuitive proof may for instance be a proof in which generality
is expressed without the use of variables. For instance I may prove the
general theorem ab = ba without introducing variables, by looking at
the array

first as five rows of seven dots and then as seven columns of five dots.
What is general now is the method of proof; to show some one that the
proof is general) it may be necessary to write it out again, with dif-
ferent numbers of dots, but this only means that we seek to draw
attention to certain features of the proof, not that the proof appeals
to a different sense than a proof which uses symbols for generality.
The proof is just as formal as the proof with variables and quantifiers.
What a single formal system is unable to do is to comprehend all
possible partial systems in a single whole—only in this sense are formal
systems necessarily incomplete; the only ' reality ' with which we can
contrast a formal system is another system in a hierarchy of more or
less uniformly formalised systems.

There is another incompleteness theorem for s/ which has no
parallel in St. It can be shown (for instance by means of the well-
known result of classical analysis that a bounded monotonic increasing
sequence is convergent) that there is a formula F(x) in si such that
(Ex) F(x) is provable in sf, but none of F(o), F(i), F(2), . . . is prov-
able. The formula F ii this incompleteness theorem itself contains a
quantifier, and no example has yet been found of a primitive recursive
predicate R such that (Ex) R(x) is provable (in some formalisation of
arithmetic) and yet none of R(o), R(i), R(2), . . . is provable.

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SIGNIFICANCE OF INCOMPLETENESS THEOREMS

Godel's construction of an undecidable sentence in a formal system
sf utilises a detailed knowledge of the proof procedure of the system
and the undecidability may be thought to reflect this structure. Kleene
has however given a uniform process for finding an undecidable for-
mula in every suitable consistent formal system. By means of Godel
arithmetisation one may determine a primitive recursive relation
T(z, x, y) such that for z = o, i, 2, . . ., (By) T(z, x, y) enumerates
(with repetitions) all relations of the form (By) R(x, y) with general
recursive R. Hence given any general recursive R we may determine
r so that

(By) R(x, y) = (By) T(r, x, y).

Let S be a formal system in which every general recursive function
may be expressed and evaluated and such that for any formula F(x)of
S there is a general recursive relation RF(x, y) which holds only when
y is the number of a proof of F(x) in S.

Let t(x, y) be the representation in S of the primitive recursive
relation T(x, x, y), let ^4(x) stand for (yy) —i t(x, y), and let r be the
number given above such that

(Ey)RA(x,y)=(Ey)T(r,x,y) (3)
Then if 5 is consistent and w -consistent, neither y(r) nor ~1 A(r) is
provable in S.

For if A(r) is provable, let p be the number of its proof, so that
RA(r, p) holds, whence by (3) there is an 77 such that T(r, r, 77) holds, and
therefore t(r, 77) is provable; but if A(r) is provable, —1 t(r, 77) is prov-
able. This contradiction in S shows that A(r) is not provable in S,
and therefore —1 (By)RA(r, y) = Y(V) "~I T(r, r, 77) holds; consequently
—1 t(r, y) is provable for each y, and therefore (By) t(r, y) is improvable,
i.e. —1 A(r) is improvable.

Kleene's procedure can be applied only to systems with quantifiers
and in this respect is less general than Godel's original construction
which is applicable also to a free variable system. It is perhaps also
worth noting that even in Kleene's procedure the actual instance of an
undecidable sentence is a function of the formal system being con-
sidered, since the constant r in A(r) is determined by the proof predicate
RA(x, y) and this of course varies from system to system. But the most
interesting feature of the proof is the contrast of the semi-formal pre-
dicate T(x, x, y) with its intended formal counterpart t(x, y). In the
theory of the predicate T(x, x, y) we suppose we have before us a
system of equations from which the value of a function f(x) is derived
by repeated substitution, a certain incompletely defined auxiliary

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R. L. GOODSTEIN

function in the equations yielding a value only when we reach a value
of y such that R(x, y) holds; we then use Godel numbering to replace
the syntactical notion of deriving from a system of equations by a
primitive recursive function (which we may for present purposes
identify with) T(r, x, y), r being the number of the system of equations
and y the number of the derivation of an end equation £(x) = £ from
the system, the final step is to link R(x, y) and T(r, x, y) by the obser-
vation that there is a derivation of an end equation if and only if there
is a y for which R(x, y) holds. We are not concerned here with a
formula in a formal system

(By) R(x. y) = (Ey) T(r, x, y)
but, as we say, with an assertion of existence. What does this mean?
In this case that we have a procedure that enables us to construct the
y for which T(z, x, y) holds from a knowledge of the y for which
R(x, y) holds and conversely. But of course this procedure is a purely
formal procedure. The situation is exactly akin to a familiar applica-
tion of mathematics. If oranges cost 3d. each then I must pay i/3d.
for five; the reason for paying i/3d. is the formal equation

5 X 3 = 1 2 + 3,
but the actual purchase and payment he outside the formal system.
Depicting a computation procedure in this way (without necessarily
using it) is one of the things we mean by an intuitive proof.

It is often said that Godel's formula (v m ) G(m) is frwe but improvable.
The reason for saying that it is true is presumably that since each of
G(o), G(i), G(2), . . . is provable, and so true, therefore G(m) is true
for all m, which is just another way of saying that (v w ) G(m) is true.
Of course if we do mean nothing more by saying that (v m ) G(m) is
true than that G(m) is true for all m then it is certainly true to say that
(ym) G(m) is true but improvable. But the expression is a rather mis-
leading one. The relationship between the formal system and the
metalanguage which is established by recursion assures us that if R(ro)
is a primitive recursive predicate such that R(m) holds for some m then
certainly R(m) is provable in s/, or rather the formula in s/ which
represents R(m) is provable. But it is the essence of Godel's theorem
itself that although G(m) is primitive recursive the formula (v m ) G(m)
does not express the notion ' for all m, G(m) ' in the formal system.
As we have seen there is an interpretation of the system in which
G(o), G(i), G(2), . . . are not all the instances of (v m ) G(w) and there-
fore the truth of these instances is not to be identified with the truth of
the formula (ym) G(m).

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SIGNIFICANCE OF INCOMPLETENESS THEOREMS

Another common mistake is to suppose that (ym) G(m) is true
because it truly affirms of itself that it is non-demonstrable. We
recall that G(m) is an abbreviation for the formula which is obtained by
substituting the number a of the formula

(V«) I Pr(m, &»(v/»))
for the variable n in this formula. .

The number of the resulting formula is of course
Stw(v/a)

as we already have had occasion to remark; hence the fact that (v m )
G(m) is improvable, i.e. that formula number St^v/a) is improvable,
tells us that

- i Pr(m, St^v/a))

is provable for each value of m, but this of course tells us nothing about
the formula

(Vm) -i Pr(m, St,(v/a))
that is, nothing about (vm) G(m). The Godel numbering estabhshes
a code in which each instance of the numerical formula G(m) says
that m is not the number of the proof of (vm) G(m), but (v m ) G(m)
itself says nothing at all in the code. Thus the Godel sentence is
neither an example of self-reference nor of self-description. Even the
sense in which we can say that the formula of si which we are denoting
by

Pr(x,y) (P)
says that x is the number of the proof of formula number y is in need
of clarification. As a formula of s/, P says nothing at all. Asthe
representative in si of a certain arithmetical relation it says that y is the
exponent of the greatest power of the greatest prime number which
divides x; and only as a sentence of the code which the Godel num-
bering estabhshes, does this arithmetical relation say that x is the
number of the proof of formula number y.

Even supposing, which is not in fact the case, that there is a formula
of die formal system (let us call it <f>) such that as a sentence of the code <f>
says something about the formula <f> of the formal system, we still
could not claim that <j> is an example of successful self-reference or
self-description, for as an element of the formal system <j> is just a sign
pattern, and as a sentence of die code^ refers not to itself i.e. not to its
meaning, but to die sign by which it is expressed, in the way the
sentence

' This is written in chalk '
refers to its physical character, not to its sense.

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R. L. GOODSTEIN

What a sentence affirms depends upon die language in which die
sentence is being used as a sentence, and identical sentences may express
difierent propositions in different languages, as the obvious example
of a code in which every English sentence stands for its contrary shows.
But we must not, therefore, suppose, as many logicians do, by analogy
with the specification of the range of a variable in mathematics, that
every sentence p must be qualified by another sentence which names
the language to which p belongs; for this assumption leads to an
infinite hierarchy of languages, without achieving its aim. Whether
we are speaking a common language or not cannot ultimately be
settled by language alone but must show itself in our actions.

The University,
Leicester

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