Godelian sentences and semantic arguments Текст научной статьи по специальности «Математика»

Логические исследования 2020. Т. 26. № 1. С. 60-77 УДК 16

Logical Investigations 2020, Vol. 26, No. 1, pp. 60-77 DOI: 10.21146/2074-1472-2020-26-1-60-77

Gabriel Sandu Godelian sentences and semantic arguments

Gabriel Sandu

University of Helsinki,

P.O. Box 24 (Unioninkatu 40 A), 00014 Helsinki, Finland. E-mail: [email protected]

Abstract: This paper contains some philosophical reflections on Godelian (undecidable) sentences and the recognition of their truth using semantic arguments. These reflections are not new, similar matters have been extensively addressed in the philosophical literature. The matter is rather one of emphasis.

Keywords: Godelian sentences, Godel's incompleteness theorem, semantical argument, truth theory, arithmetic, proof, provability

For citation: Sandu G. "Godelian sentences and semantic arguments", Logicheskie Issledo-vaniya / Logical Investigations, 2020, Vol. 26, No. 1, pp. 60-77. DOI: 10.21146/2074-14722020-26-1-60-77

To the memory of Alexandr Karpenko, such a great friend

1. Godel incompleteness theorem

Let L be the language of arithmetic, consisting of

- variables, x0,x\,x,y,...

- logical constants: V, 3x, =

- nonlogical constants: 0, S, +, X.

(Here '0' is an individual constant, 'S', is a one-place function symbol and '+', and 'x' are two place function symbols.)

From these items, the terms and formulas of the language of L are formed in the standard way.

As Tarski observed, the object language of a formalized science, comes together with a theory, usually given by listing its axioms and rules of inference. In our case the starting point is the theory Q (minimal arithmetic) which is the set of logical consequences of the following axioms:

© Sandu G.

1. VxVy(Sx = Sy ^ x = y)

2. Vx(Sx = 0)

3. Vx(x = 0 ^ 3y(s = Sy))

4. Vx(x + 0 = x)

5. VxVy(x + Sy = S(x + y))

6. Vx(x x 0 = 0)

7. VxVy(x x Sy = (x x y) + x).

Notice that this theory is finitely axiomatizable. The language of Q is interpreted in a metalanguage in which '0' is assigned the the natural number zero, 'S' is assigned the successor function, '+' is assigned the operation of addition 'x' is assigned multiplication. It is known that Q is a rather strong theory which is able to represent all recursive functions (in a technical sense of the notion of 'representation', which is assumed to be known. It is also known that Q defines (in a technical sense assumed to be known) its own syntax and many semantical notions. This happens, as shown by Godel, via the notion of godel numbering. As a result, each term t in the language L gets associated with a godel number rtn; and each formula A receives its godel number rAn. Recalling that every natural number m has a name m in L, where m is an abbreviation for (the numeral) SSL..0 (m times), we see that every term t and every formula A have names in the arithmetical language, rtZ and rAn, respectively. This fact, together with the ones mentioned earlier, makes possible to introduce, for any formula A in the language of arithmetic, the diagonalization of A, which is the expression

3x(x = rAZ a A).

When A is a formula with one free variable, then we see that asserting the diagonalization of A amounts to predicating A of its own goodel number.

From Godel's results, it follows that for any theory T extending Q, the set of godel numbers of theorems of T is not definable in T, from which it can be further inferred that the set of Godel numbers of true arithmetical sentences ("true in the standard model") is not definable. This last statement is usually known as "Tarski's theorem"; it is somehow debatable in the literature whether Godel himself was aware of this result or not, but this matter will not concern us here. The first statement is standardly proved by reductio using the

diagonalisation lemma which asserts that for any theory T which extends Q, for any formula B(y) there is a sentence A such that

T h A o—B(rAZ).

The second statement follows directly from it, by observing that the set of true arithmetical sentences is an extension of Q.

The variant of the Goodel's incompleteness theorem we are interested in is proved by first showing that for every extension T of Q there is a formula PrT(x) in the language of arithmetic which has the form ElyProvT(x, y) and is such that for any sentence A in the language of arithmetic:

• T h A if and only if ElyProvT(rAZ, y) is true (in the standard model) if and only if for some natural number m, Provy(rAZ, mm) is true if and only if (given the representability of ProvT in Q), Q h ProvT(rAZ,m) for some m.

Here ProvT(x,y) is a primitive recursive formula, that is, a formula which contains only bounded quantifiers and is closed under the standard propos-itional connectives. Thus, from the above we get that if T h A then Q h ProvT(ZAZ,m) for some m, and given that T is an extension of Q we also get T h 3yProvT(rAn,y), i.e., T h Pr(rAn). Now applying the Diagon-alization lemma to the formula 3yProvT(rAZ, y) Godel showed that there is a sentence, usually denoted by G such that

T h G o -ByProvt(rGZ, y)

The sentence G is called a Godel sentence for T. It is taken to say: "I am unprovable".

We recall that a theory T is called w—inconsistent if there is a formula F(x) such that T h 3xF(x) but T h —F(0), T h —F(1), T h —F(2),...(for every natural number 0,1,2,...). T is called w—consistent if it is not w—inconsistent. Now Goodel proved

Theorem 1. (Godel First Incompleteness Theorem). Let T be a consistent, axiomatizable extension of Q and let G be a Godel sentence for T. Then T F G. If T is w—consistent, then T F —G.

The proof is well known but we rehearse it here (we follow Boolos, Jeffrey and Burgess), because it serves as a basis for extracting, later on, a semantic argument. Suppose that T h G. Hence, by our previous comments, 3yProvT(rGZ, y) is true (in the standard model) and by a well known result, Q h 3yProvT(rGZ, y); given that T is an extension of Q we also have

T h ByProvT(rGn,y). From the Diagonalization lemma we also know that T I—<3yProvT(rQn, y). Thus T is inconsistent, a contradition. Hence T F G.

For the second claim, suppose that T I--G. By the diagonalization lemma,

T h 3yProvT(rGl,y). But given that T is consistent and T I--G, we must

have T F G. This implies that for no natural number n, n is the code of a proof of G in T, that is, -ProvT(HGZ,0), -ProvT(HGH, 1), -ProvT(HGZ,2)..., are all true (in the standard model), where each of these formulas are primitive recursive. Hence Q h -Prov(rG2,0), Q h -Prov(rG2, 1), Q h -Prov(rG2,2).... and since T is an extension of Q we also have T h -Prov(rGJ_,0), T h -Prov(rG2,1), T I—~Prov(rC2,2).... Hence T is u—inconsistent, which contradicts our assumption. We conclude T F -G.

After reviewing these results, let us return to the question which is the main concern in this paper, namely Godel's method to produce undecidable sentences such as G, and especially a claim often made in this connection to the effect that these sentences are true and recognized to be true. Here is, for instance, how Dummett describes Godel's result:

By Godel's theorem there exists, for an intuitively correct formal system for elementary arithmetic, a statement [G] expressible in the system but not provable in it, which not only is true but can be recognized by us to be true... [Dummett, 1963].

The puzzling question is: how do we "recognize" that G (or any statement equivalent to it) is true?

The above proof of the theorem does not give an explicit argument about how we come to recognize G as true, neither did Godel provide one. But it is not very difficult to extract one. From the Diagonalization lemma we know that the statement G is equivalent to a universal statement, viz. -3yProvT(rGl,y) (i..e Vy-ProvT(rG2, y)). From the second part of the proof we see that every numerical instance is provable (and true) in the system. Since G is the universal quantification over all these numerical instances, then G is true. Of course in this last step we rely on our grasp of the standard model (this is what the u—consistency is supposed to ensure).

In fact, this is Dummett's argument for the truth of Godel's sentence:

The statement [G] is of the form VxA(x),where each one of the statements A(0),A(1),A(2), ...is true: since A(x) is recursive, the notion of truth for these statements is unproblematic. Since each of the statements A(0),A(1),A(2), ...is true in every model of the formal system, every model of the system in which G is false must be a non-standard model...whenever, for some predicate B(x), we

can recognize all of the statements B(0),B (1),BA(2), ...as true in the standard model, then we can recognize that VxA(x) is true in that model. This fact ...we know on the strength of our clear intuitive conception of the structure of the model [Dummett, 1963, p. 191].

As we see from this quote, we come to appreciate that the undecidable Godel sentence G for Q is true not by working inside the system but rather by conducting a so called semantical argument which makes an essential use of the concept of truth itself. Dummett is not the only one to have seen the importance of semantical arguments. There is another semantical argument which uses the truth predicate, distinct from Dummett's argument, which goes back to Alfred Tarski [Tarski, 1956]. In order to present it, we need to say somehting about arithmetical induction.

The system Q of minimal arithmetic is knowingly defficient in that it fails to prove many universal statements about numbers which are usually proved by mathematical induction. Typically, if we want to prove that every number has a given property, we prove it by showing that 0 has that property, and then we show, from the assumption that an arbitrary number x has that property, that the successor Sx has that property. To accommodate induction one needs a more adequate set of axioms for number theory. To this effect we add to the 7 axioms of the system Q all sentences of the form

8. [A(0) A Vx(A(x) ^ A(S(x))] ^ VxA(x)

(8) is usually known as the Induction axiom scheme. The theory which is the set of all sentences in the language of arithmetic which are logical consequences of (1)-(8) is known as Peano Arithmetic (PA). It is a simple mathematical fact that definability and representability in Q entail definability and repres-entability in any extension of Q and thus in PA in particular. From now on we shall operate with PA. Tarski's semantical argument which proves the truth of the Godelian statement G for PA, uses a universal statement which cannot be proved in Q but needs PA.

1.1. The representability of the syntax in arithmetic

Tarski's truth-definition for arithmetic exploits the representability of the syntax of PA in PA.

It is a mathematical fact that there are functions /-,, /v, /3 defined on the natural numbers such that the following hold:

- /-(rAn) = r—Az, for every formula A in the object language;

- /v(rAn, r£Z) = rA V Bn, for every formulas A, B in the object language;

- /3(rAn,n) = r3xnAn, for every formula A and natural number n.

There is also a function fsub (the susbstitution function) which has the property:

fsub(rA", _xZ, rt") = rA(t)n

for every formula A in the language of arithmetic, variable and term t in the same language.

All these functions are recursive, thus representable in Q and hence in PA which means there are formulas Neg(x,y),Dis(x,y,z),Ex(x,y,z) and Sub(x,y, z,w) in the language of arithmetics so that for all formulas A, B, term t, and natural number n we have

a) PA h Vy (Neg(rAZ, y) o y = r-A")

b) PA h Vy (Dis(rA~z, r_BZ, y) o y = rAVBZ)

c) PA h Vy (Ex(_AZ, n,y) o y = _3xuA")

d) PA h Vy (Sub(rAZ, _xZ, rtZ, y) o y = rA(t)z) Similarly, the function f= on the natural numbers such that

f= (rtz, _sZ) = rt = sn

for all terms t, s in the language of arithmetic is representable in PA by, say, the expression Id(x,g,z), that is,

PA h Vy (Idfr, _sZ, y) o y = rt = sn). If in (a) we instantiate y with _-Az we get

PA h Neg(rAZ, r-AZ) o _-Az = r-AZ

The formula on the right side is a theorem of the predicate calculus (with identity), hence PA proves it. Thus PA h Neg(rAn, r-A"1). We can show that for each formula A of the object language there is exactly one formula B of the object laguage such that PA h Neg^A", rB") and B is -A. Therefore we can take Neg to be a function and write Neg(rAZ) = r-A".

In a similar way we can also take Dis, Ex, Sub, Id, Less to be also functions. Thus we shall have

a*) PA h Neg(rAZ) = r-A", for every formula A in the object language.

b*) PA h Dis(rA", rB") = rA V B", for every formulas A,B in the object language

c*) PA h Ex(rAn, n) = r3xnAn, for every formula A in the object language and natural number n.

d*) PA h Sub(rAZ, rxZ, rtZ) = rA(t)n, for every formula A and term t of the object language and every natural number i.

e*) PA h Id(rtn, rsn) = rt = sn, for all terms t, s of the object language.

In a similar way it can be shown that PA defines its own syntax: being a closed term, a variable, a formula and a sentence (of the language of arithmetic). That is, there are formulas ct(x), var(x), /orm(x) and sen(x) in the object language such that the following holds:

f) PA h ct(rtZ), for every closed term t.

g) PA h var(rxin), for every natural number i.

h) PA h /orm(rAZ), for every formula A.

j) PA h sen(rAZ), for every closed sentence A.

PA also defines some semantical properties. There is a formula Den(x) in the object language (that we can take to be a function) such that

k) PA h t = s o Den(rtZ) = Den(rsn), for all terms t, s in the object language.

2. Tarski's truth theory

In the case of Tarski's truth theory for arithmetic we do not need to go via the notion of satisfaction but use directly the truth-predicate Tr. The reason for this is that each natural number has a name in the object language.

The axioms of the truth-definition are given in the metalanguage containing Tr is a predicate symbol:

Ax1 Vx(Tr(x) ^ sen(x))

(If x is true, then x is of a sentence)

Ax2 VxVy(ct(x) A ct(y) ^ (Tr(Id(x,y)) o Den(x) = Den(y)))

(The identity between two closed terms x and y is true iff their denotations are the same)

Ax3 Vx(Sen(x) ^ (Tr(Neg(x)) o —Tr(x)))

(The negation of the sentence is true iff the sentence is not true)

Ax4 VxVy(sen(x) A sen(y) ^ (Tr(Dis(x, y)) o Tr(x) V Tr(y))) (A disjunction is true iff either sentence is true)

Ax5 VxiVx2(/orm(xi) A var(x2) ^ (Tr(Ex(xi, x2)) o 3t(Tr(Sub(xi,x2,t))))

(An existential sentence is true iff there is a closed term t such that the sentence which is the result of the substitution of the free variable x2 in x1 by t is true.)

Let PA(Tr) be the set of sentences which are the logical consequences of the 7 axioms of PA, the five axioms (Ax1)-(Ax5), and plus the Induction schema (8) which allows occurrences of the truth-predicate in the formulas A(x). It can be shown that PA(Tr) is materially adequate, that is,

PA(Tr) h Tr(rAZ) o A,

for any sentence A in the language of arithmetic.

It is well known that the Tarskian truth theory proves the following universal statements:

• The principle of noncontradiction (consistency). For every sentence y of the object language it is not the case that both y and its negation are true:

PA(Tr) h Vy (Sen(y) ^ — (Tr(y) A Tr(neg(y)))). This property follows directly from Ax3.

• The principle of excludded middle. Every sentence of the object language is true ot its negation is true:

PA(Tr) h Vy (Sen(y) ^ Tr(y) V Tr(neg(y))).

This property follows from the other direction of Ax3.

• The principle of soundness. All theorems are true:

PA(Tr) h Vx(PrpA(x) ^ Tr(x)).

This principle fully exploits the occurrence of the truth-predicate in the Induction scheme. We omit its proof but it consists, informally, of the following steps:

1. All the axioms of PA are true.

2. The rules of inference of PA preserve truth.

3. Hence every theorem of PA is true (i.e. PA(Tr) h Vx(PrP^(x) ^ Tr(x)).

2.1. Tarski's semantical argument

In the postscript to the English translation of his seminal article, Tarski adds some interesting parallels between his results and those of Godel:

Moreover Godel has given a method for constructing sentences which- assuming the theory concerned to be consistent- cannot be decided in any direct way in this theory. All sentences constructed according to Godel's method possess the property it can be established whether they are true or false on the basis of the metatheory of higher order having a correct definition of truth [Tarski, 1956, p. 274].

To establish the truth of such a Goodelian sentence Tarksi uses the principle of soundness listed in the previous sesction. We present Tarski's semantical argument (Tarski, 1936, Theorem 5) for the Godelian sentence -PrPA(r-0 = 0") (that we shall abbreviate by ConPA) which is taken to express the consistency of PA. The semantical argument for G is similar. There is nothing original in my presentation, this argument has been rehearsed many times [Ketland, 1999] and [Shapiro, 1998].

Godel's second incompleteness theorem shows that PA F ConPA and PA F -ConPA. But Tarski shows

PA(Tr) h ConpA.

The argument is straightforward. From the soundness principle we get (i) PA(Tr) h Prpa(r-0 = 0") ^ Tr(r-0 = 0").

We also know that the theory of truth proves all the T-instances, i.e., (ii) PA(Tr) h Tr(r-0 = 0") o -0 = 0.

But PA proves 0 = 0, and thus PA(Tr) h 0 = 0, which together with (ii) entails

(iii) PA(Tr) h -Tr(r-0 = 0").

From (i) and (iii) we get

(iv) PA(Tr) h -Prpa(r-0 = 0")

that is, PA(Tr) h ConpA.

Tarski's semantical argument is usually expressed in words, in order to enhance its explanatory power:

• In a first step we establish the principle of soundness as we showed earlier:

1. All the axioms of PA are true.

2. The rules of inference of PA preserve truth.

3. Hence every theorem of PA is true,

PA(Tr) h Vx(Prpa(x) ^ Tr(x)).

• A second step established that the sentence '—0 = 0' is not true:

PA(Tr) h —Tr(r—0 = 0n)

(see (iii))

• In a third step we combined the conclusion of the first and of the second step and concluded that '—0 = 0' is not a theorem:

PA(Tr) h —Prp4(r—0 = 0"1)

(see (iv))

• Finally we note that —PrPA(r—0 = 0n) is the Consistency statement C on P A.

The crucial role in this argument is the universal generalization which is the Principle of soundness. It confers the semantic argument the form of a nomo-logical argument which shows the explanatory role of the truth predicate:

Let us return to the Godelian statement G (or Conpa). Let us suppose a logic teacher asserts that Conpa is true, and the puzzled student asks for an explanation. The student believes the teacher's word that Conpa is true, but he wants to be shown why Conpa is true. The student wants something like a convincing proof or an explanatory proof. The natural answer is to remark that all the axioms of PA are true and the rules of inference preserve truth. Thus every theorem of PA is true. It follows that '—0 = 0' is not a theorem and thus PA is consistent.... It seems to me that this informal version of the derivability of Conpa is as good an explanation as there is. The argument shows why Conpa is true or why Conpa is a consequence- and the move through the notion of truth provides the explanation [Shapiro, 1998, p. 505].

3. Feferman's program

Tennant [Tennant, 2002] argues against Ketland [Ketland, 1999] and Shapiro [Shapiro, 1998] that Tarski's theory of truth is not the only way we can come to recognize the truth of the Godel sentence. In particular, Tennant claims, the generalization "All theorems are true" is not the only way to express the soundness of an arithmetical system S. There is, instead, another way to express it, viz., using reflection principles of the form

(pa) If f is a primitive recursive sentence and f is provable in S, then f.

As we see, this reflection principle does not use the truth-predicate. Tennnat follows here Feferman [Feferman, 1962], who emphasizes that "Reflection principles are axioms schemata ...which express, insofar as is possible without use of the formal notion of truth, that whatever is derivable in S is true".

Let us take stock. We have discussed two semantic arguments invoked in how we come to recognize that Godelain sentences are true.

One such argument, due to Tarski, and explicitly described in Shapiro's quote in the last section, uses the generalization "All theorems are true" and can be run in an extension PA(Tr) of PA which, in addition to the truth axioms, allows occurrences of the truth predicate in the induction scheme.

The other semantic argument, described earlier in the second quote from Dummett also uses the truth-predicate. However, Tennant [Tennant, 2002] rephrases it, so that the reference to "the structure of the model" is deleted and the truth-predicate lifted out as required by Feferman's reflection principles. Here is Tennant's formulation of his own semantic argument:

G is a universally quantified sentence (as it happens, one of Goldbach type, that is, a universal quantification of a primitive-recursive predicate). Every numerical instance of that predicate is provable in the system S. (This claim requires a subargument exploiting Godel numbering and the representability in S of recursive properties.) Proof in S guarantees truth. Hence every numerical instance of G is true. So, since G is simply the universal quantification over those numerical instances, it too must be true [Tennant, 2002, p. 556].

Tennant shows that this argument can be faithfully represented in a "sufficiently strong" arithmetical system S enriched with reflection principles (with no occurrence of the truth-predicate) in Feferman's style.

I will now describe shortly the main lines of Tennant's argument. Before doing that let me mention what it means for a formal system of arithmetic S to be "sufficiently strong": S represents recursive properties and proves the Diagonal-ization lemma (i.e., there is a proof in S leading from G to -3yProvT(rG", y);

and there is a proof in the other direction too), and S also proves the equivalence between the Godelian sentence G and the consistency sentence ConS. It is known that there are several systems which satisfy this requirement, e.g. PA.

Tennant proposes an extension of S with Feferman's principle of uniform primitive recursive reflection (which is more general than the principle (pa) mentioned above):

(UR) Add to S all sentences of the form

Vn(PrS(r0(n)n)) ^ Vm^(m)

where 0 is a primitive recursive formula and n is, as before the numeral corresponding to the natural number nand Prs(r0n) is, like before, an abbreviation for 3yProvS(0, y)

He then shows that in this extension a faithful formalization of the semantical argument described above can be run. The proof goes like this [Tennant, 2002, p. 577]. (We let S* denote the system S plus (UR)).

Suppose m codes a proof of G in S. Hence by representability (a natural number being the code of a proof in S of a formula is a primitive recursive relation), S h ProvS(rGn, m), where ProvS is a primitive recursive formula. But S proves also, from the assumption G, the sentence Vy—ProvS(rGn, y) (i.e. the diagonalization lemma), which by universal instantiation implies —ProvS(rGn,m). Given our assumption that S is consistent, we have a contradiction, from which we conclude that m does not code a proof of G in S. Again by representability we get S I—iProvS(rGn, m). But n has been chosen arbitrarily, hence for every n, there is some proof of —Provs(rGn, n) in S, from which with the help of (UR) we derive (in S*) that Vy—ProvS(rGn, y). Finally, by the Diagonalization Lemma, we get G (in S*).

The penultinate steps requires perhaps some additional clarification. If I understood correctly, "for every n, there is some proof of —ProvS(rGn, n) in S" is just the sentence VnPrS(r0(n)n) in the antecedent of (UR), where 0(n) is the primitive recursive sentence — Provy(rGn,n).

We are then told:

The foregoing proof justifies the assertion of G. The stronger system S* contains methods for reflecting on the justification resources of the weaker system S. These methods can be seen at work, in the application, in the proof just give, of various rules of inference that are available in S* but not in S [Tennant, 2002, p. 577].

The thing which I find somehow problematic in the proof are the penultimate steps:

...But n has been chosen arbitrarily, hence for every n, there is some proof of -ProvS(rG", n) in S, from which, with the help of (UR), we derive (in S*) that Vy-ProvS(rG", y).

I take them to correspond to the informal steps of Tennant's own semantic argument listed earlier in this section. It seems to me that we can justify these steps only on the basis of our intuitive understanding of the standard model, as Dummett pointed out. The principle of uniform recursive reflection (UR) just expresses this understanding in a formal way. We may have eliminated the truth-predicate as required by a minimlist conception of truth, but the justification of (UR) is still grounded in such understanding. This matter is orthogonal to the goal of this essay, so I will not dwell on it.

One can still perhaps argue that Tarski's truth-definition is more general, because it can also account for the intuition that all S—theorems are true (sound), and not just the primitive recursive ones. Tennant's response to this objection is that we could add as well to S* the schema (soundness principle)

ProvS(_i") ^ f,

where f is any sentence in the language of arithmetic. It is known from Lob's theorem that this principle cannot be derivable in S without making S inconsistent. But in the present case we add the soundness principle not to S directly but to S extended with the principle of uniform primitive recursive reflection, and this avoids the inconsistency.

To sum up, I agree with Tennant that the difference between the two semantic arguments is that between saying (Tarski) and showing (Feferman). That is, Tarski's truth theory can state the principle of soundness in one single universal statement "All theorems are true". In this case the "recognition" of the truth of the Godelian sentence takes the form of a nomological explanation which uses that universal statement [Ketland, 1999; Shapiro, 1998]. On the other side, the Feferman-Tennant framework (S* extended with the soundness axiom scheme) uses an axiom scheme which can be seen as a list of the infinitely many instances of the universal statement Vx(PrS(x) ^ Tr(x):

Prs(_1i") ^ Tr(rfi") PrS(rf2") ^ Tr(_i2")

in which the truth-predicate has been eliminated in virtue of the equivalences

Tr(^T) o pi Tr(r^2n) o P2

In this case the recognition of the truth of G does not take the form of a nomological argument (because there is no collection of all these instances into one universal statement). It consists in the apprehension of the proof of G in the extension of e.g. PA with the soundness principle. Truth does not "transcend" proof, truth is just proof (in the extended system).

4. The justification of the extensions

A question which arises quite naturally at this stage is about the justification of different extensions which settle the Godelian statements, and about the nature of these statements themselves. Is a given extension more justified than another? This question revives an older discussion which goes back to Godel concerning intrinsic versus extrinsic extensions of a theory which has been the inspiring source for the Feferman program.

Godel's reflections took place in the context of set theory (What is Cantor's continuum problem? [Godel, 1947]) but they also apply mutatis mutandis to arithmetic. Godel introduced a distinction between an intrinsic and extrinsic extension of an axiom system. An intrinsic extension, unlike an extrinsic one, is justified on the basis of one grasping the concepts of the base theory. Godel gave as an example the Axiom of Determinacy in set theory that he regarded as an extrinsic axiom because it is not justified by our understanding of sets, in contrast to Mahlo's axioms for big cardinals. In addition, Godel also mentioned intrinsic extensions with undecidable statements (Godelian sentences) that one recognizes as true in virtue of their meaning, that is, by reflecting on their undecidability.

Godel's remarks suggest the idea to treat the truth axioms of Tarski's theory of truth as examples of intrinsic extensions of the base theories, whose justification is grounded in our grasping of the concepts of the base theory, that is, natural numbers and operations on natural numbers. In fact this suggestion, which was not made by Godel, has been explicitly advocated later on by Koellner in his reflections on Godel's distinctions:

Let us consider first our conception of natural numbers which is underlying PA. This conception of natural numbers not only justifies the principle of mathematical induction for the language of PA, but for any other extension of the language of PA which has

a sense. For instance if we extend the language of PA by adding the tarskian truth-predicate and we extend the axioms of PA by adding the tarskan axioms for truth, then, on the basis of our conception of natural numbers, we are justified in accepting the instances of the induction scheme in which the truth-predicate occurs. In the resulting system one can prove ConPA....By contrast, the Axiom of determinacy AD is not justified by our understanding of natural numbers [Koellner, 2006].

Similar ideas have been expressed by Feferman. Starting with the 60's and inspired by Godel, he addressed the question of the extensions of schematic formal systems (formal systems which contain axiom shemes, like ZFC and PA) with new axioms. He started looking for the possibility to generate systematically extensions of such systems whose acceptance was already implicit in the base theory. One of the mechanisms Feferman proposed is reflection principles. We saw an illustration of this mechanism when presenting Tennant's ideas. Little by little Feferman also came to consider extensions which contains explicitly a truth-predicate and developed the notion of reflexive closure of a schematic theory [Feferman, 1991], which allows for the Induction scheme to range over the truth-predicate. In this case the extended system can prove statements of the form Vx(PrPA(x) ^ Tr(x)). This has been, as we saw, Tarski's way.

I think there is an important difference between Godel's notion of intrinsic extension where the new axioms display or unfold the content of the notions of the base theory, and the two extensions of PA introduced in this paper. It seems to me that neither Tarski's extension of PA with his theory of truth, nor Tennant's extension of a sufficiently strong arithmetical system S (e.g. PA) with reflection principles ProvS(rf") ^ f, "unfold" the content of the notion of natural number. None of this extensions is, in my opinion, grounded in our knowledge and understanding of natural numbers but rather "reflect" on the properties of certain methods of proof that have been adopted. That is, although these methods of proof operate on arithmetical and logical resources, they also possess certain properties confered to them by certain philosophical positions which are constitutive of their definitions. The extension axioms or schemata are about these properties (e.g. soundness, truth, consistency) and not about the content of the notion of natural number. Goodelian arithmetical statements as well as their analogues in set theory contain explicit references to these methods of proof, as a consequence of which they inherit an additional content which is not purely arithmetical, or set-theoretical, for that matter. One can find a partial recognition of this point in [Horsten, 2011]:

Godelian proofs of GZFC and ConZFC are certainly partly mathematical in nature. The proof cited above, for example, involves an instance of the principle of mathematical induction, which is a mathematical principle if there ever was one. It is just that such Godelian proofs are not purely mathematical proofs. For they essentially contain the notion of truth, which is itself not a mathematical but a philosophical notion. This is not to deny that mathematics can be applied to produce interesting theories of truth. It is just that mathematical theories of truth do, on this view, belong not to pure mathematics but at least to applied mathematics, or to the more mathematical part of philosophy [Horsten, 2011].

Horsten refers here to the philosophical notion of truth, and to Godelian proofs using a truth-predicate, but my main point in this paper is slightly different. It concerns the notions of proof and provability. It is a metamathematical notion which reflects a certain finitistic, philosophical standpoint. By making explicit reference to such notions, Godelian sentences acquire also a higher-order, not purely numerical content, which depends on the properties of these notions and cannot be reduced to the concept of natural number. One possible way to be more explicit about the higher-order content of Godelian sentences is through some remarks made by Isaacson [Isaacson, 1991; Isaacson, 1996]. He contrasts arithmetical sentences provable in PA with the Godelian sentences: the former have a pure arithmetical content, and the system PA which proves them arises out of our undertanding of natural numbers. On the other side, the meaning of Godelian statements involve our reflections on our understanding of natural numbers.

The ideas discussed in this paper have been debated many times in the post Godelian era. The contribution of the paper is simply one of emphasis. Myhill, for instance expresses similar ideas in an often quoted passage:

Indeed it seems to me that the use of the word 'proof' in ordinary non-philosophical mathematical discussion is rather clearly neither a syntactical nor a semantical term. It is as self-contradictory to use methods of proof without admitting their correctness, as it is to make statements without admitting their truth. (I am not using 'self-contradictory' in the sense of formal logic, but roughly as a synonym for 'irrational'.) Therefore if a person who has been using certain methods for proving arithmetical theorems succeeds in making these methods explicit, he is ipso facto committed to the perfectly definite proposition that the use of those methods cannot lead to a false arithmetical statement, for example the statement

that 0 is equal to 1. By Godel's technique of arithmetization, which translates every statement of formal deducibility into a statement of arithmetic, any such person is compelled to admit a new arithmetical statement, namely the arithmetized version of the statement that his methods cannot lead to a proof of the statement that 0 is equal to 1. By Godel's theorem, he could not have established this statement by his previous methods. Hence, as soon as a person makes explicit the tools which he has been using in the construction of arithmetical proofs, he is ipso facto in a position to obtain new arithmetical proofs which he could not have obtained by using those tools alone. The whole process is closely related to what the British philosophical logician W.E. Johnson called 'intuitive induction'; we find ourselves making certain inferences and we thereupon realize that the pattern of those inferences is such as to confer validity on arguments in which they occur. This realization is a demonstrative and rational step quite apart from any question of formalization, though of course the results of an intuitive induction can be formalized after the induction has taken place [Myhill, 1960, p. 461].

It is difficult to disagree with these remarks. Myhill, like other commentators I discussed (Horsten) is concerned with the distinction between different kind of proofs. My concern in this paper was, however, with the other side of the coin: the meaning of the Godelian sentences which are settled by these proofs. The minor point I tried to make was that, by making reference to notions like proof (provability), these sentences have a content which transcend the arithmetical content of purely numerical statements. This is the internal, conceptual reason for which, in some cases (not all; there are Godelian statements like "I am provable" which are provable), their proof has to mobilize higher-order (meta-theoretical) resources, be they in the form of a truth-theory, a la Tarski, or reflection principles, a la Feferman. I think that Godel was aware of this fact when he made a distinction between intrinsic extensions with Godelian sentences and intrinsic extensions with other kind of axioms which unfold the content of the basic notions like natural numbers.

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