the importance of incompleteness tour de force....

What is the significance of the incompleteness results?JHintikka.0211.032111 1

Jaakko Hintikka

WHAT IS THE SIGNIFICANCE OF INCOMPLETENESS RESULTS?

1. The importance of incompleteness

The famous incompleteness (unprovability) and undefinability results by Kurt Gödel (Gödel

1931, 1986-2003, and 2002) and Alfred Tarski (Tarski 1933, 1935, 1956 and 1986) are generally

considered as one of the greatest intellectual achievements of the twentieth century. They are by

any criterion an impressive tour de force. Their significance for the foundations of mathematics

has been discussed extensively. These results, especially Gödel‘s, have also been perceived

sometimes as having even wider philosophical and other general theoretical implications. They

have been taken to reveal important limitations of logic, mathematics, axiomatic method and

even the human reason can accomplish. A study of their significance is therefore an important

task.

Originally, Gödel.s and Tarski‘s results were discussed in the framework of different

philosophies of mathematics current at the time, such as logicism, formalism and intuitionism.

For one thing, Gödel‘s second incompleteness theorem has generally been taken to demonstrate

the impossibility of carrying out Hilbert‘s metamathematical program. (See here sections 16-17

of this essay.) The use of this framework is nevertheless largely a historical accident, and is no

longer the appropriate perspective. On the one hand, what has happened since in logic put all the

main issues in a new light. On the other hand, the issues that Gödel‘s results were addressed to

could even at his own time have been approached differently. Georg Kreisel has borrowed a

phrase from Wittgenstein and spoken of the unhealthy influence of logic on the philosophy of

mathematics. (See Kreisel 1967 and cf. Cellucci 1996. In this essay the precise interpretation of

Kreisel‘s phrase will not be discussed.) Without taking a stand on his main thesis, it is

instructive to realize that within mathematics there existed well before Gödel tendencies that

might have put an altogether different spin on the entire discussion of the completeness and

incompleteness of mathematical theories.


2. What did Gödel prove?

But what did Gödel actually prove? Let us begin with Gödel‘s first incompleteness (meta)

theorem. In discussions about its wider significance, it is sometimes said that Gödel proved that

there are true but logically unprovable mathematical propositions. However, this simply is not

at all what he actually did. What he proved is, somewhat more accurately expressed, that in any

formalized system of elementary arithmetic there are propositions that are true but not provable

in that particular system. Whatever Gödelian unprovability there is, is relative to some formal

system of arithmetic.

But what is the sense of formal system intended here? A Pandora‘s Box of issues is

implicit in that question, both systematically and historically. A historical perspective perhaps

helps us to find our bearings here. The meaning of Gödel‘s theorem can be clarified by

comparing the result and the line of thought that led to it with other actual and possible

developments in the foundations of mathematics.

As a thought-experiment in the spirit of Kreisel‘s we can ask: What could have happened

if the leading mathematicians engaged in the examination of the foundations of their subject had

disregarded the development of symbolic logic and continued to do what for instance Hilbert did

in his Grundlagen der Geometrie (1899), that is, continued to discuss mathematical reasoning

and mathematical theorizing by means of the Sprachlogik implicit in ordinary nonsymbolic

language? A thought-experiment along these lines is not unrealistic, especially in the case of

Hilbert. For Hilbert was an axiomatist for whom the important thing about a mathematical

theory was the class of its models, not the expressibility of the axiom system in any particular

symbolism. He did not think that Frege‘s logic was adequate for the purposes of set theory. (See

his letter to Frege reproduced in Frege 1980, pp. 51-52.) What if Hilbert had not been so

generous as to indulge that unsuccessful professor from Jena who thought that he could express

all of arithmetic in a singularly clumsy logical formalism even though he obviously had no idea

what the live issues at the cutting edge of mathematical research were at the time?

One thing is clear. Such a switch would not have meant any reduction in mathematicians‘

interest in questions of completeness and incompleteness, though not necessarily in the same

sense as Gödel. Hilbert and others struggled with the problem as to make an axiom system


complete, as witnessed by the story of his so-called axiom of completeness. (See Hilbert 1903,

Freudenthal 1957.)

3. A Weiertrassian counterexample?

As an exercise in such a alternative history of mathematics, imagine that someone had claimed

to such a logically illiterate mathematician that there cannot be an axiomatization of elementary

arithmetic that is not complete. Quite possibly someone in the tradition of Weierstrass,

Dedekind and Hilbert could have replied: What is the difficulty? I can perfectly well formulate

a complete axiom system for elementary arithmetic. Indeed, Peano already did a large part of the

work. Most of his axioms can be used, as they are. The only remaining problem is to find a

formulation of the principle of mathematical induction that restricts the models of the axioms to

the single intended one, viz. the familiar structure of natural numbers. But we know how to do

that. It suffices to postulate that there does not exist infinite descending sequences of natural

numbers. This is simply the negation of a statement that asserts that there exists one. But how

can it be formalized? By means of Weierstrass‘s epsilon-delta technique. Here‘s what we can

say, in terms of variables ε, δ ranging over natural numbers:

(3.1) For any ε1 one can find δ1< ε1 and for any ε2 one can find (independently of ε1)

δ2< ε2 such that (δ1= δ2) if and only if (ε1 = ε2).

A moment‘s thought shows that a descending chain envisaged in (3.1) does not branch (because

((δ1= δ2)⊃(ε 1= ε2) )) are split (because of ((ε1= ε2)⊃(δ1= δ2))). Hence it must extend ad

infinitum.

What is historically crucial, all the ingredients of this reply, including (3.1) were

available already to Weierstrass. This is well known in the case of the epsilon-delta jargon,

which from a logical point of view is nothing but a natural way of talking about the logic of

quantifiers. Existential quantification is even in several natural languages expressed by

locutions like “one can find” or “one can choose”. It is in this way that quantifiers are

expressed in the prose of traditional mathematicians. And there is nothing wrong with such talk.

The semantics of ordinary quantifiers can be (and has in fact been; see Rantala 1975)

developed in analogy with probability theorists‘ locutions of ―draws‖ from an urn. If such

draws are to show the truth or falsity of a proposition about the totality of the balls in the urn,


they must not be random, but must search for an optimal draw. One has to be sure to find the

right balls in the urn. In what is known as game-theoretical semantics these searches for the

right individuals are (literally) dramatized as moves in a game where the ―Verifier‖ chooses the

values of quantifiers with an existential force and the ―Falsifier‖ those of a universal force. (For

game-theoretical semantics, in brief GTS, see Hintikka and Sandu, 1996. More about it later)

The naturalness of the resulting game-theoretical semantics is illustrated by the historical

fact that logicians have spontaneously resorted to a game-theoretical interpretation of quantifiers

when they did not have access to a Tarski-type truth definition. Indeed, the co-inventor of

modern logic, C.S. Peirce already did so. (See Hilpinen 1983, Pietarinen 2006) Later examples

are found in the so-called game quantifiers and in the theory of infinitely deep languages.

Moreover, the notion of independent ε-δ choices was also already familiar to Weierstrass.

He knew the idea from what are known as uniformity concepts. (See e.g. Bann 1980.) They had

begun to play an important role in analysis where their need had been dramatized by Cauchy‘s

notorious false theorem whose fallaciousness was due to his use of the notion of plain

convergence where uniform convergence was needed. (See e.g. Bottazzini 1986.) For instance,

the function f is (simply) continuous at x if and only if for any real number Σ>Ο there exists

(―one can find‖) a real number δ such that

(3.2) |f(x+y) – f(x)| <ε if | y| < δ

Here δ depends of course on both ε and x. The function f is uniformly continuous on the

interval of x1≤x≤x if and only if the choice of δ in (3.) can be made for all x1 ≤x≤ x2

independently of x. Uniformity is thus nothing more and nothing less than a simple

feature of the logical behavior of some quantifiers, viz. their independence of certain

others.

Yet the importance of uniformity has not always been duly appreciated. (See Bann

1980) On the rare occasions when their uniformity has been noted by logicians, it has

typically been dismissed as a matter of “quantifier ordering”. This is a mistake. The

independence involved in uniformity concepts cannot be accommodated in Frege’s

allegedly universal conceptual notation. The kind of quantifier independence exemplified

in the definitions of uniformity concepts is not expressible in Frege’s Begriffsschrift. (Frege


1879).Nor has this oversight been corrected in subsequent treatments of the received

first-order logic, and with dire consequences for set theory. (See Hintikka forthcoming (a).)

Even though this oversight is presumably not what Kreisel had in mind when he spoke of

the unhealthy influence of logic, it can serve to illustrate what he meant.

In any case, the axiom system so defined — it will be called the anti-Gödel

axiomatization (alias AG) — of elementary arithmetic involves only ingredients that were

available to all mathematicians since at least Weierstasss. Moreover, in the obvious

model-theoretical sense AG is complete in the sense of being categorical. Hence all true

arithmetical propositions are logically implied in it. As one can thus easily ascertain, it is a

finite axiomatization of elementary arithmetic in which all arithmetical truths are logical

consequences of the axioms.

4. The notion of logical consequence

Here we are confronted with a major philosophical and foundational problem situation. Is

AG a counterexample to Gödel’s first incompleteness theorem (which we will call G)? No,

technically G is a perfectly valid result. But so is AG. They do not contradict each other,

because G speaks of provability whereas AG speaks of logical consequence relations.

Hence, the problem is not inconsistency. Rather, the problem lies in the philosophical and

other general theoretical conclusions we should draw here. Should we take our cue from

Gödel and emphasize the limitations of what logical proofs can accomplish in mathematics

or is the right response to extol the power of logical axiomatizations which can even give us

categorical theories of arithmetic and heaven knows what else? Obviously, much sharper

analysis of the conceptual situation than what is offered by most of the extant discussions

of Gödel’s results.

A philosophical logician’s first reaction might perhaps be to allege that AG is not a

formal system in her or his sense. In particular, it might be pointed out that so far the

notion of logical consequence has been used only informally. But this does not touch the

real issues. For the notion of logical consequence presupposed in A is a perfectly formal

one in an obvious and obviously relevant sense. A proposition S2 (in the language of

arithmetic, say) is logically implied by S1 iff every model of S1 is also a model of S2,


independently of the interpretation of all their nonlogical constants (including arithmetical

ones). This relation is a completely formal one. It depends only on the logical form of S1

and S2. Any such consequence relation can in principle be used as a rule of purely formal

logical inference.

This sense of logical consequence is the one that matters most for the applications of

logic, for instance for the use of logic in axiomatic theorizing. The main purpose of an

axiom system in mathematics and in science is to capture some interesting class of

structures as the models of the axioms. If this model-theoretical capture is successful, this

class of structures can be studied purely logically without resorting to any empirical

results. This can be done as soon as the means of studying the axiom system are purely

formal, ascertainable on the basis of the logical form of the propositions in question. As

Hilbert might have put it, the logical forms can be the same no matter whether we are

speaking of points, lines and circles or of tables , chairs and beermugs.

5. Gödel’s notion of deductive consequence

Now did Gödel show that an axiom system of arithmetic could not have all arithmetical

truths as its logical consequences in the sense just explained? No, he did not. What Gödel

did was in effect to require that the definition of a formal system of arithmetic must include

a finite list of rules of inference. In other words, Gödel restricted the logical consequence

relations that a mathematician is allowed to use in any particular axiom system of

arithmetic (or in any other formalized mathematical theory) to some finite assortment of

logical rules.

From a mathematical point of view, this is completely arbitrary. The purpose of a

mathematical or scientific axiom system is not to legislate the reasoning that a

mathematician uses. Rather, the point is to study the class of structures that an axiom

system is calculated to capture — in Gödel ‘s case, the structure of natural numbers — by

any available logical means as long as they are purely formal, so that we do not have to go

beyond the axiom itself in applying them.


Thus the idea of a formal axiom system of arithmetic that Gödel was relying on is

not the only possible one, and looked upon in a wider perspective it involves an arbitrary

restriction on the number of rules of inference that may be used. Why, then, did Gödel in

effect adopt his notion of formal system?

One answer is that he trusted a wrong logic. In some cases, formal rules of inference

(and the logical truths on which they are based) can be reduced to others. When they can

be so reduced to a finite number, the branch of logic in question is called complete

(completely axiomatizable). A case in point is the received first-order logic originally

formulated by Frege and Russell as a part of their more comprehensive logic and separated

from the set by Hilbert and Ackermann (1928). Gödel proved this logic complete in 1930,

and it was the logic he was employing in the axiomatizations of arithmetic that he

considered in his incompleteness theorems. But there is no reason to believe that this

logic is adequate for the task of dealing with arithmetical truth. Gödel’s first

incompleteness theorem holds only on the assumption that the logic used is the received

first-order logic, and in this sense relative to it. It is better justified to look upon Gödel’s

theorem as showing the failure of the Frege-Russell logic to deal with arithmetical truth

than to interpret it as showing that elementary arithmetic is incomplete in a systematically

relevant sense.

One can put a somewhat different spin on the matter. Needless to say, a complete

logic would offer practical and theoretical advantages. If a complete logic were adequate

and only a finite number of formal rules of inference need to be used in arithmetic, we

could recursively enumerate all (and only) the true sentences of arithmetic. Thus the true

import of Gödel’s first incompleteness theorem can be seen to be essentially combinatorial,

viz. the insight that the set of arithmetical truths is not recursively enumerable.

This is an extremely important result. However, it is now seen that its implications

are different from what they are often taken to be. If the class of arithmetical truths were

recursively enumerable, we could program a computer to give out these truths one by one.

Hence the limitations that Gödel’s first incompleteness theorem brings out are not


limitations on logic, but limitations on what computers can do in arithmetic and by

implication in mathematical theorizing in general.

8. Incompleteness vs. undefinability of truth

At this point, we can perhaps draw an interim conclusion from what has been found. It has been

seen that the perspective in Gödel‘s first incompleteness theorem in which it is perceived as

marking a limitation on what logic can do in mathematics is not the only possible one and in any

case is in need of qualifications. Its direct impact is not to bring out any boundaries as to what

logic can do only on what can be done mechanically. It need not worry philosophers or

mathematicians, only hackers.

We have seen that there some of Gödel‘s predecessors on the mathematical side had

access to means of forming a complete (in a sense) system of arithmetic. It looks as if logicians

have been obsessed by what can be done by mechanically following rules while mathematicians

are thinking in terms of what constructive human thought activity can accomplish. However,

we have not seen truly what distinguishes the two perspectives nor how they could perhaps be

integrated into a wider perspective that would enable us to compare their relative merits.

Obviously, further questions have to be asked here.

Moreover, what has been found does not imply that Gödel’s first incompleteness

theorem could not have some other foundational or other general theoretical implications.

For one important thing, Gödel’s result has often been bracketed with Tarski’s

undefinability theorem. According to this theorem, it is an impossibility to define truth for

a first-order language by the sole means of the same language. (Tarski 1933, 1935.) It is

assumed in this theorem that the logic used is the traditional (Frege-Russell) first-order

logic.

The undefinability of truth would indeed have major repercussions. Truth is the pivotal

concept in all model theory. Tarski-type undefinability would therefore imply that first-order

theories cannot usually be used to formulate or to discuss their own model theory. This would

seriously handicap the use of first-order axiomatic set theories, such as the familiar ‗Zermelo-

Fraenkel‘ theory, as their own model theory and more generally their own metatheory. Set

theory would be inadequate as a universal ―working mathematicians model theory.‖


Historically, Gödel’s first incompleteness theory is closely intertwined with the

question of the definability of truth (Cf. here Feferman (1984) and the references are given

there.) Gödel’s original project was to construct a relative consistency proof for analysis

assuming the consistency proof for analysis, assuming the consistency of arithmetic. Such a

result would have exorcised one of the scepters of the Grundlagenkrisis most relevant to

the concerns of working mathematicians . Gödel discovered that he needed for this

purpose in the arithmetical target language the notion of arithmetical truth. But when he

assumed its availability, he ran into paradoxes reminiscent of the Liar Paradox. Deftly, he

turned these paradoxes into an undefinability result, in other words, into the special case

of Tarski’s undefinability theorem for first-order arithmetical theories.

Because his colleagues in the Vienna Circle were suspicious of semantical notions

like truth, Gödel in the end formulated his result as an incompleteness (unprovability)

theorem, rather than an undefinability result. He nevertheless continued to think that the

gist of his first incompleteness theorem is the undefinability of arithmetical truth in

arithmetic itself. He even mentioned the Liar Paradox in his original exposition of his

theorem.

9. The irrelevance of the Liar

In this Gödel nevertheless was wrong. The kind of deductive incompleteness of arithmetic

that Gödel proved simply is not the same notion as the definability of arithmetical truth. As

was shown, Gödel’s result is at bottom combinatorial. It applies to the class of

arithmetically true sentences as purely syntactical objects independently of their meaning.

It holds independently not only of the particular notation used in arithmetic but

independently of any particular axiomatization, including the choice of the logic used in it.

In contrast, arithmetical truth is a semantical notion. Its definability is relative to

the language in question. Among other things such definability depends on the logic used,

for instance on which logical constants are available and on their interpretation. For

example, is the notion of negation used the dual negation ~ or the contradictory negation

?


It was unfortunate even in a proof-theoretical perspective that Gödel mentioned the

Liar Paradox in his original paper, and not only because this paradox is a semantical one,

not a matter of the combinatorial structure of the class of true sentences. As was

indicated, what Gödel had in mind was obviously the way in which he had reached his first

incompleteness theorem. He had realized the undefinability of arithmetical truth by means

of the received (Frege-Russell) first-order logic. (In his very title, Gödel refers to the

Principia Mathematica.) The way he came to realize this undefinability suggested to him a

strategy for his own proof.

However, on a stricter scrutiny the analogy between Liar-type and Gödel’s actual

proof argument is superficial, and encourages simplistic accounts of Gödel’s argumentation

(Cf here Hintikka 2000.) Unlike the traditional forms of the Liar Paradox, Gödel’s proof does

not involve self-reference, There are no first-person pronouns or other vehicles of self-

reference in the conceptual repertoire of arithmetic. Rather, the paradox arises from the

fact that the technic of Gödel’s number system assigns to numbers a double identity, as

numbers simpliciter and as codifications of numerical expressions. A paradoxical Gödelian

sentence is not like Epimenides uttering ―I am now lying‖ but like Dirty Harry claiming (as a

replique in a movie) ―In this situation even Clint Eastwood could not help smiling.‖(If he

maintains a stiff upper lip, he could be falsifying his own statement.)

10. Undefinability and circularity

What is the basic reason for the undefinability of truth for a language using the received first-

order logic in the same language? (We can assume here that the language in question is rich

enough to allow a formulation of its own syntax e.g. and a Gödel numbering.) Tarski does not

tell us. Yet an answer is available: threat of circularity. In order to see this, consider an

attempted explicit definition of the truth predicate T(x) applicable to the Gödel numbers

(and others, if need be):

(10.1) (∀x)(T(x) ⟷ D[x])

or in a slightly more explicit form

(10.2) (∀x)((x=g(S)) ⊃(T(x) ↔ D[x]))


Here the definiens D{x} is a complex predicate expressing the truth condition of the

sentence S with the Gödel number x. In D[x] this complex predicate is attributed to some

number x. This number depends on g(S) in a way that in a conventional first-order

language can be expressed by means of quantifiers (and propositional connectives.). These

quantifiers (Qy) codify certain choice functions. In order for the definition not to be

circular, these choices and a fortiori the quantifiers codifying them must be independent of

(∀x) because x=g(S).

When the semantics of quantifiers is formulated in game-theoretical terms ( as in

what is known as the game-theoretical semantics or GTS), this independence simply

means the informational independence of the corresponding game moves and hence)

amounts to a well-known game-theoretical notions. But in the conventional first-order logic

such independence cannot always be expressed, the reason being that any quantifier (Q1y)

occurring within the formal (syntactical) scope of another quantifier (Q2x) unavoidable depends

on it.

Similar observations apply by the same token to dependence relations between

propositional connectives in relation to each other and to quantifiers. They come into play if

someone tries to consider quantified sentences as long disjunctions and conjunctions.

Hence the real reason for the kind of undefinability of truth for a first-order language that

Tarski established is no deep feature of the concept of truth, nor does it lie in any unavoidable

limitation of symbolic logic. It is not a consequence of excessive deductive power of the

received first-order logic, either, contrary to what has sometimes been alleged. It is due to the

unnecessary rigidity of the notation of the conventional Frege-Russell first-order logic which

forces all quantifiers within the formal (syntactical) scope of another one to depend on it. The

output is not the deductive richness of the received first-order logic, but its expressive poverty,

more fully expressed, its notational rigidity. This rigidity is illustrated by the fact that a richer

logic is in principle obtainable without any additional symbols merely by liberating the

conventions governing the use of those all-important punctuation marks of logic, viz. brackets

(parentheses).


11. A Kreiselian contrast?

In addition to bringing out the reason for Tarskian undefinability of truth, these observations

provide us with a wider perspective in which we can view and compare with each other the two

valid results, Gödel‘s theorem and the anti-Gödel completeness result AG. The inconspicuous

notion that is ultimately responsible for the contrast between G and AG is the independence of

the two choices (choice functions) in the anti-Gödelian completeness result. This notion is

inexpressible in the traditional (Frege-Russell) first-order logic which not only gives rise to

Tarskian undefinability but would also have made it impossible for Gödel to realize the

possibility of a complete (in terms of logically valid formal consequence) axiom system of

arithmetic, as long as he was putting his entire bet on the received Frege-Russell Logic. Thus G

and AG do not contradict each other, but they are incommensurable in the sense of relying

literally on a different logic.

12. What do quantifiers do?

More generally, we are now in a position to look at the entire nature and history of symbolic

logic in terms of a distinction that is unavoidable but has not been commonly recognized in it.

To speak of independence of a choice codified in a quantifier means to recognize implicitly or

explicitly an aspect in the semantical function of quantifiers that has not always received

adequate attention. Logicians in the tradition of Frege, Russell and their ilk emphasize the role

of quantifiers as expressing nonemptyness (existence) and exceptionlessness (universality) of

predicates. Frege proposed to construe quantifiers as higher-order predicates whose attribution

to a lower-level one expresses precisely its instantiation or exhaustiveness in the relevant

universe of discourse. This way of thinking encourages a construal of quantified sentences as

long disjunctions or long conjunctions. Many prominent logicians and philosophers have

indulged in such thinking, for instance Ludwig Wittgenstein in his Tractatus (Wittgenstein

1922). (Admittedly, later he came to recognize this as his ―biggest mistake‖ in Tractatus.)

But quantifiers have in fact another semantical task. By the formal dependence of a

quantifier (Q2y) on another quantifier (Q1x) one can express the dependence of the variable y

bound to it on the variable x of the other one. Here formal dependence means a purely

syntactical relation that in the received first-order logic is expressed by the nesting of scopes. In


contrast, the intended dependence between variables is a material (―real-life‖) dependence, for

instance the kind of dependence that the equations of mathematical physics express.

In a rough-and-ready fashion, we can thus distinguish two preferential traditions in the

foundations of mathematics and mathematical logic. Both have to cope somehow with both

tasks of quantifiers, but they free different kinds of problems and options and use different

locutions.

For instance, the Fregean idea of quantifiers expressing nonemptyness and exceptionless

applied most happily to sentence-initial quantifiers. Hence what happens in the most natural

deduction methods in treating a quantifier inside a formula is that it is brought out to the

beginning of a sentence by applying instantiation rules to the quantifiers within whose scope it

occurs. There is nothing wrong with such a procedure, except that questions of dependence and

independence will have to be understood and dealt with by reference to those instantiation rules,

which accordingly deserve closer attention than they typically receive. In fact, the conceptual

situation becomes clearer if this procedure is relaxed and instantiation rules are generalized so as

to apply also to embedded quantifiers. For instance, their true instantiating term replacing an

existentially quantified variable x in (∃x)(F[x] must not normally be a singular constant but a

constant function term of the form f(y1, y2, …) where (Q1y1), (Q2y2)… are all the quantifiers on

which (∃x) depends. (Dependencies on constants may have to be treated similarly.) And f is a

new function chosen differently for different quantifiers (∃x). Functions like f have a name.

They are what are called Skolem functions. They can in fact serve to eliminate all quantifiers

with an existential force. The selection of their arguments brings out all the dependence relations

between quantifiers. This is in some ways a preferable way of organizing one‘s first-order logic

anyway. By so doing logical inferences become manipulations of equations. Moreover, the so-

called axiom of choice turns out to be a principle of first-order logic.

This is in fact a brief recipe for formulating the so-called independence friendly (IF) first-

order logic by means of which (among many other things) truth can be defined for a first-order

language in the same language. In spite of these advantages, most logicians and most

philosophers are still clinging to the unfortunately constrained Frege-Russell kind of first-order

logic.


13. Anti-Gödelian completeness

In terms of IF logic (or equivalent), the anti-Gödel theorem AG can be explicitly formulated in

logical terms. This involves more novelties than expressing quantifier independence. At first it

might seem that that is all that is required. AG might easily be expressed as soon as we can

express quantifier independence, for instance by means of the stroke notation of IF logic, where

the independence of (∃y) of (∀x) can be expressed by writing it (∃y/∀x). We can in fact write

the strengthened principle of induction as follows:

(13.1) ~(∀x1)(∀x2)(∃y1/∀x2)(∃2/∀x1)(((x1=x2)↔(y1=y2)) & (y1<x1) & (y2 < x2))

This is in fact correct, assuming that we can interpret the negation sign ~ in the intended way.

This requires in fact an explanation.

In (a suitably extended) IF first-order logic there are two different notions of negation

unavoidably present. The contradictory one or ¬S in the semantical game with S there exists

no winning strategy for the verifier. Since semantical games are not in general discriminate , this

does not imply that the falsifier has one. The existence of such a falsifying strategy is expressed

by ~S. Now the existence of an infinite descending sequence of natural numbers is expressed by

~S. Now the existence of an infinite descending sequence of natural numbers is expressed by the

following sentence S.

(13.2) (∀x1)(∀x2)(∃y1/∀x2)(∃y2/∀x1)(((x1=x2)↔(y1 = y2)) & (y1<x1) & (y2 < x2)))

In order to be able to use mathematical induction, we have to assume ~S, not ¬S. Now IF

logic shows that ~S can be written as

(13.3) (∃x1), (∃x2)(∀y1/∃x2) (∀y/∃x1)

(((x1 = x2) ↔ (y1=y2)) ⊃

((y1≤ x1) ∨ (y2 ≤ x2))

What is especially interesting here is that in (13.3) the independence indicator

slashes are uneliminable. If they are omitted we obtain ¬S. This throws further light on

the reasons for the kind of incompleteness that Gödel proved. Axiomatizations of


elementary arithmetic using traditional first-order logic (as Gödel’s does) are incomplete

because the principle of mathematical induction cannot be expressed in this type.

It is now seen that G and AG cannot be compared in any direct way. Gödel ‘s G says

at bottom that arithmetical truths cannot be recursively enumerated. AG says that all

truths of elementary arithmetic are formal logical consequences of a finite set of axioms

formulated in a generalized IF first-order logic. This implies that this generalized IF logic

itself does not allow of complete logical axiomatization in the misleading sense of

“axiomatizing” logic that was discussed earlier. This is nevertheless an unavoidable fact of

logician’s lif and does not have anything to do specifically with arithmetic or with any other

application of logic. Since there is a complete disproof procedure in IF logic, one can even

suggest that the deductive incompleteness of IF logic is only a consequence of inevitable

failure of the tertium non datur in IF logic.

In a wider context, the anti-Gödel argument is much more intuitive than the

Gödelian incompleteness proof. It brings out connections between some of the most

important concepts in the foundations of mathematics, including the meaning of quantifiers

as dependence indicators, the limitations of traditional first –order logic and the

presuppositions of reasoning by mathematical induction.

14. Traditional first-order logic is unsatisfactory

Above all, a closer study of the anti-Gödel argument brings out the massive fact that logics

cannot in general be expected to be deductively complete in the sense of having all its

logical truths recursively enumerable. That this completeness holds in the received Frege-

Russell first-order logic is a lucky accident — or, rather, an unlucky one, in that it has

suggested to logicians a misleading picture of the relationship of logic to its applications.

Such applications cannot be restricted to mechanizable inferences.

15. Quantifiers and choice terms

But why have traditional mathematicians not done better in the foundations than the logicians

who have been handicapped by their reliance on the received Frege-Russell first-order logic? In


contrast to logicians, working mathematicians are typically paying close attention to

dependencies of variables and as the example of Weierstrass shows, also their independencies.

However, their ways of thinking have not been examined in logical terms as carefully as the

more restricted Frege-Russell first-order logic which has for instance been used as the working

logic of set theory.

One spontaneous way for foundationalists in this traditions is to use choice terms and

choice functions. Such a treatment of quantifiers as proxies for choice functions would fit well

into Hilbert‘s overall project. The objects produced by the relevant choice functions might

constitute the models that he needed for his consistency proofs. Now IF logic yields a way of

accomplishing this with Skolem functions which are choice functions, and in IF logic all

quantifiers with existential force can be eliminated in favor of Skolem function terms. This

approach can in fact be formalized by means of Skolem functions. Appearances

notwithstanding, their use does not involve any higher-order logic in that they only employ

function terms, but not any quantification over functions.

For instance, Hilbert and Bernays tried to deal with those modes of mathematical

reasoning that go beyond the received first-order logic by means of certain choice terms that they

called epsilon terms. (Hilbert and Bernays 1934-39) The result was Hilbert‘s epsilon calculus.

In it, quantifiers are in fact eliminated by means of epsilon terms. However, Hilbert and Bernays

did not fully capture the dependence indicating role of quantifiers. They did not explicitly

indicate what an epsilon choice depends on. If they had done so, they could have formulated IF

sixty years ahead of the actual schedule. They could have realized Hilbert‘s striking prediction

that an appropriate approach to logic could make the ―axiom‖ of choice as plain a truth as

2+2=4. (See Hilbert 1917.) More generally speaking, if Hilbert and Bernays had indicated

explicitly what an epsilon term depends on, they could by the same token have expressed the

independence of one choice from another. This would have inoculated their logic against

Gödel-type incompleteness results.

6. Hilbert’s “formalistic” strategy

Or could it really have done so? The usual view of Gödel‘s second incompleteness theorem is

that it showed the impossibility of carrying out Hilbert‘s overall metamathematical program.


(For this program, see, besides Hilbert himself, also Kreisel 1958.) Now did it? The most

definite answer is a resolute ―yes and no‖ — obviously in two different senses.

In order to sort out these senses, we have to ask. What was Hilbert trying to do, anyway?

As an axiomatist, he was interested primarily to understand and to safeguard what happens in a

major mathematical theory, which he thought of as being presented axiomatically. Now an

axiom system is calculated to study a certain class of structures, viz. its models. Logic does not

dictate the choice of the axioms. That choice is merely an attempt to catch some antecedently

interesting subject matter, suggested perhaps by empirical information or by intuition. Given

such an axiom system, a mathematician has today a legitimate and more or less important task,

viz to study those models logically. As long as this logic is sound, a mathematician need not

have any foundational fears.

Or rather, no fears except one. Such an axiomatic enterprise is predicated on the

assumption that there is indeed a class of structures to be studied. These structures are exhibited

by the models of the axiom system, which in logical terms means that the axiom system must be

consistent. This is the motivation of Hilbert‘s preoccupation with consistency and consistency

proofs. Consistency proofs were to be his way of cutting the Gordian knot of the entire

Grundlagenkrisis, not so much by solving all the logical problems in the foundations of

mathematics as by making an end run around them.

Accordingly, for Hilbert the relevant sense of consistency was model-theoretical, not

deductive. So why did he get involved in the formalization of logic and mathematics? The

answer lies in his insight that even the relevant model-theoretical relations are purely formal,

depending only on the logical forms of propositions and not on their subject matter. The models

of a mathematical theory could equally well consist of tables, chairs and beer mugs as of points,

lines and circles, without affecting any questions of logical structure.

Hilbert boldly extended this insight one step further. (See here Hilbert 1922.) The

models we need to consider could by the same token consist of symbols and symbol

combinations — of verbs , nouns and adjectives or of variables, connectives and predicates,

instead of chairs, tables and beermugs. Hilbert expressed his idea in so many words, but

unfortunately not in model-theoretical terms. This idea is the gist of Hilbert‘s almost universally


misunderstood ―formalism‖. Hilbert is not maintaining that all there is to mathematical activity

is operating with symbols. He is suggesting that all the structures we need to consider in

mathematics can be reproduced as structures of symbols.

In its own right, this is a sound idea. It has since been used by several logicians for the

first time apparently by Leon Henkin in his 1950 completeness proof for the traditional first-

order logic. The models Henkin constructs for consistent formulas are literally sets of logical

expressions. The same idea is utilized in the notion of model set. (See Hintikka 1955, Smullyan

19 .).

This possibility of using symbols themselves as models of the propositions they express

was taken by Hilbert to imply that he could prove the model theoretical consistency of axiomatic

theories by proving their deductive consistency. This presupposes that the logic used is

deductively complete. For this reason, Hilbert‘s project must be evaluated on the historically

obvious assumption that he is using the received Frege-Russell logic. If so, Gödel‘s argument

applies, and his so-called second incompleteness theorem shows that the deductive consistency

of elementary arithmetic cannot be proved in the same axiomatic theory. This generally

acknowledged conclusion of Gödel‘s result therefore in a valid one.

17. Hilbert vindicated

This is not the end of the story of Gödel and Hilbert‘s program (cf. here Peckhaus 1990.) Hilbert

was interested in the model-theoretical consistency of axiomatic theories, in the sense of the

existence of models for the axiom system. There are many other ways to approach the quest ion

of their existence besides the one Hilbert tried. In fact, what has been found here provides a

general method for proving such consistency for a class of important mathematical theories.

Indeed, IF logic allows for new opportunities for approaching questions of consistency.

In all an extended IF logic, we have two negations, on the on hand the strong (dual) negation ~

which game-theoretically means the existence of a strategy (winning strategy for the falsifier),

and on the other hand the old-fashioned contradictory negation ¬ which means the absence of a

meaning strategy for the verifier. Now a natural way of expressing the consistency of S is ¬ ~ S.

This means that S is not false. Model-theoretically this amounts to the existence of at least one

model in which S is not false, that is, is either true or indefinite, neither true nor false.


Hence to prove the consistency of S is to prove ¬ ~ S. Now in the IF first order logic

there exists a complete disproof procedure. Thus a relative consistency proof of a new axiom X

with respect to an old theory T can be considered as a proof of

(17.1) T⊃ ¬~X

Now one way of proving the consistency of elementary arithmetic could be a consistency proof

of the critical indicator principle relative to the rest of the Peano-type axiom P. The induction

principle can be taken to be of the form ~S, where S is the IF sentence that asserts the existence

of an infinite descending sequence of natural numbers (Cf. sec.13 above) . Hence the relative

consistency proof would mean proving

(17.2) P⊃ ¬ ~ ~S

In other words, the following conjunction has to be shown to be disproved:

(17.3) P & S

(which is equivalent to (P &~~S)) But (17.3) is an IF sentence and hence disprovable if false.

Hence the relative consistency of arithmetic in the model-theoretical sense (with respect

to an axiomatization of arithmetic in IF first order language) can be proved logically. In brief,

the consistency of arithmetic can be proved arithmetically, if we use IF logic instead of

traditional first-order logic. Such a proof has been presented in (Hintikka and Kalakadilar 2006).

Such a proof would have satisfied Hilbert. What he wanted was to show that the relevant

axiomatic theories have models, for those models are what the theory is calculated to explore.

Admittedly, what has been shown by the argument just concluded is that there are models in

which the axioms of arithmetic are not false, not necessary models in which they are true. But

both kinds of models are legitimate subjects of a mathematician‘s investigation.

This becomes more obvious when we look at the situation in terms of the construction of

new axiom systems in mathematics. The relative consistency proofs. The choice of a new

arithmetical axiom can always be legitimized by proving logically its consistency relative to the

earlier theorem.


All this can be extended from elementary arithmetic to any mathematical axiom system

that can be formulated in terms of IF logic. This applies to a large part of classical analysis, thus

bringing us back to the beginning of Gödel‘s Odyssey. For instance, the assumption that every

set of reals with an upper bound can be so expressed and shown to be relatively consistent.

Hilbert‘s program is thus alive and well after Gödel‘s second incompleteness theorem. How far

it can be carried out remains to be investigated.

So what follows for the wider questions prompted by independence results? Was Kreisel

right? Or should he have been more specific and blamed the unhealthy influence on the idea of

logical reasoning as an instance of mechanizable computation? Or blamed it on the hegemony of

the received Frege-Russell first-order logic? I will let my readers draw their own conclusions.

And are the implications of independence results worth the extensive discussion they

have prompted? Yes, because only through a thorough critical sorting can we find out how to

overcome the limitations many seem to place on the prospects of logic and mathematics. The

fortunes of Hilbert‘s rightly understood project can illustrate the need of such an examination.


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the importance of incompleteness tour de force....

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