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Concepts of proof and truth are (even in mathematics) ambiguous. It is commonly accepted that proof is the ultimate warrant for a mathematical proposition, that proof is a source of truth in mathematics. One can say that a proposition A is true if it holds in a considered structure or if we can prove it. But what is a proof? And what is truth?

The axiomatic method was considered (since Plato, Aristotle and Euclid) to be the best method to justify and to organize mathematical knowledge. The first mature and most representative example of its usage in mathematics were Elements of Euclid. They established a pattern of a scientific theory and a paradigm in mathematics. Since Euclid till the end of the 19th century, mathematics was developed as an axiomatic (in fact rather a quasi-axiomatic) theory based on axioms and postulates. Proofs of theorems contained several gaps – in fact the lists of axioms and postulates were not complete, one freely used in proofs various “obvious” truths or referred to the intuition. Proofs were informal and intuitive, they were rather demonstrations; and the very concept of a proof was of a psychological (and not of a logical) nature. Note that almost no attention was paid to the precization and specification of the language of theories – in fact the language of theories was simply the unprecise colloquial language. One should also note here that in fact till the end of the 19th century, mathematicians were convinced that axioms and postulates should be true statements. It seems to be connected with Aristotle’s view that a proposition is demonstrated (proved to be true) by showing that it is a logical consequence of propositions already known to be true. Demonstration was conceived here of as a deduction whose premises are known to be true, and a deduction was conceived of as a chaining of immediate inferences.

Basic concepts underlying the Euclidean paradigm have been clarified on the turn of the 19th century. In particular, the intuitive (and rather psychological in nature) concept of an informal proof (demonstration) was replaced by a precise notion of a formal proof and of a consequence. This was the result of the development of mathematical logic and of a crisis of the foundations of mathematics on the turn of the 19th century which stimulated foundational investigations.

One of the directions of those foundational investigations was the program of David Hilbert and his Beweistheorie. Note at the very beginning that “this program was never intended as a comprehensive philosophy of mathematics; its purpose was instead to legitimate the entire corpus of mathematical knowledge” (cf. Rowe 1989, p. 200).Note also that Hilbert’s views were changing over the years, but always took a formalist direction.

←23 | 24→

Hilbert sought to justify mathematical theories by means of formal systems, i.e., using the axiomatic method. He viewed the latter as holding the key to a systematic organization of any sufficiently developed subject. In “Axiomatisches Denken” (1918, p. 405) Hilbert wrote:

When we put together the facts of a given more or less comprehensive field of our knowledge, then we notice soon that those facts can be ordered. This ordering is always introduced with the help of a certain network of concepts (Fachwerk von Begriffen) in such a way that to every object of the given field corresponds a concept of this network and to every fact within this field corresponds a logical relation between concepts. The network of concepts is nothing else than the theory of the field of knowledge.14

By Hilbert the formal frames were contentually motivated. First-order theories were viewed by him together with suitable non-empty domains, Bereiche, which indicated the range of the individual variables of the theory and the interpretations of the nonlogical vocabulary. Hilbert, as a mathematician, was not interested in establishing precisely the ontological status of mathematical objects. Moreover, one can say that his program was calling on people to turn their mathematical and philosophical attention away from the problem of the object of mathematical theories and turn it toward a critical examination of the methods and assertions of theories. On the other hand, he was aware that once a formal theory has been constructed, it can admit various interpretations. Recall here his famous sentence from a letter to G. Frege of 29th December 1899 (cf. Frege 1976, p. 67):

Yes, it is evident that one can treat any such theory only as a network or schema of concepts besides their necessary interrelations, and to think of basic elements as being any objects. If I think of my points as being any system of objects, for example the system: love, law, chimney-sweep [...], and I treatmy axioms as [expressing] interconnections between those objects, then my theorems, e.g. the theorem of Pythagoras, hold also for those things. In other words: any such theory can always be applied to infinitely many systems of basic elements.15

←24 | 25→

The essence of the axiomatic study of mathematical truths consisted for him in the clarification of the position of a given theorem (truth) within the given axiomatic system and of the logical interconnections between propositions.

Hilbert sought to secure the validity of mathematical knowledge by syntactical considerations without appeal to semantic ones. The basis of his approach was the distinction between the unproblematic “finitistic” part of mathematics and the “infinitistic” part that needed justification. As is well known, Hilbert proposed to base mathematics on finitistic mathematics via proof theory (Beweistheorie). The latter was planned as a new mathematical discipline in which mathematical proofs are studied by mathematical methods. Its main goal was to show that proofs which use ideal elements (in particular actual infinity) in order to prove results in the real part of mathematics always yield correct results. One can distinguish here two aspects: consistency problem and conservation problem. The consistency problem consists in showing (by finitistic methods, of course) that the infinitistic mathematics is consistent; the conservation problem consists in showing by finitistic methods that any real sentence which can be proved in the infinitistic part of mathematics can be proved also in the finitistic part. One should stress here the emphasis on consistency (instead of correctness).

To realize this program, one should formalize mathematical theories (even the whole of mathematics) and then study them as systems of symbols governed by specified and fixed combinatorial rules.

The formal, axiomatic system should satisfy three conditions: it should be complete, consistent and based on independent axioms. The consistency of a given system was the criterion for mathematical truth and for the very existence of mathematical objects. It was also presumed that any consistent theory would be categorical, i.e., would (up to isomorphism) characterize a unique domain of objects. This demand was connected with the completeness.

The meaning and understanding of completeness by Hilbert plays a crucial role from the point of view of our subject. Note at the beginning that in the Grundlagen der Geometrie completeness was postulated as one of the axioms (the axiom was not present in the first edition, but was included first in the French translation and then in the second edition of 1903). In fact the axiom V(2) stated that it is not possible to extend the system of points, lines and planes by adding new entities so that the other axioms are still satisfied. In Hilbert’s lecture at the Congress at Heidelberg in 1904 (cf. 1905b), one finds such an axiom system for the real numbers. Later, there appears completeness as a property of a system. In lectures “Logische Principien des mathematischen Denkens” (1905a, p. 13) Hilbert explains the demand ←25 | 26→of the completeness as the demand that the axioms suffice to prove all “facts” of the theory in question. He says: “We will have to demand that all other facts (Thatsachen) of the given field are consequences of the axioms”. On the other hand, one can say that Hilbert’s early conviction as to the solvability of every mathematical problem – expressed, e.g. in his 1900 Paris lecture (cf.Hilbert 1901) and repeated in his opening address “Naturerkennen und Logik” (cf. Hilbert 1930b) before the Society of German Scientists and Physicians in Königsberg in September 1930 – can be treated as informal reflection of a belief in completeness.

In his 1900 Paris lecture, Hilbert spoke about completeness in the following words (see the second problem): “When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science”.

One can take the “exact and complete description” to be complete enough to decide the truth or falsity of every statement. Semantically such completeness follows from categoricity, i.e., from the fact that any two models of a given axiomatic system are isomorphic; syntactically it means that every sentence or its negation is derivable from the given axioms. Hilbert’s own axiomatizations were complete in the sense of being categorical. But notice that they were not first-order, indeed his axiomatization of geometry from Grundlagen as well as his axiomatization of arithmetic published in 1900 were second-order.

The demand discussed here would imply that a complete (in this sense) system of axioms is possible only for sufficiently advanced theories. On the other hand, Hilbert called for complete systems of axioms also for theories being developed. One should also add here that Hilbert admitted the possibility that a mathematical problem may have a negative solution, i.e., that one can show the impossibility of a positive solution on the basis of a considered axiom system (cf. Hilbert 1901).

In Hilbert’s lectures from 1917–1918 (cf. Hilbert 1917–1918), one finds completeness in the sense of maximal consistency, i.e., a system is complete if and only if for any non-derivable sentence, if it is added to the system then the system becomes inconsistent. In his lecture at the International Congress of Mathematicians in Bologna in 1928, Hilbert stated two problems of completeness: one for the first-order predicate calculus (completeness with respect to validity in all interpretations, hence the semantic completeness) and the second for a system of elementary number theory (formal completeness, in the sense of maximal consistency, i.e., Post-completeness, hence the syntactical completeness) (cf. Hilbert 1930a).

Hilbert’s emphasis on the finitary and syntactical methods together with the demand of (and belief in) the completeness of formal systems seem to be the source and reason of the fact that, as Gödel put it (cf. Wang 1974, p. 9), “[...] formalists←26 | 27→considered formal demonstrability to be an analysis of the concept of mathematical truth and, therefore were of course not in a position to distinguish the two”. Indeed, the informal concept of truth was not commonly accepted as a definite mathematical notion at that time. As Gödel wrote in a crossed-out passage of a draft of his reply to a letter of the student Yossef Balas: “[...] a concept of objective mathematical truth as opposed to demonstrability was viewed with greatest suspicion and widely rejected as meaningless” (cf.Wang 1987, pp. 84–85). Therefore, Hilbert preferred to deal in his metamathematics solely with the forms of the formulas, using only finitary reasonings which were considered to be safe – contrary to semantical reasonings which were non-finitary and consequently not safe. Non-finitary reasonings in mathematics were considered to be meaningful only to the extent to which they could be interpreted or justified in terms of finitary metamathematics.16

On the other hand, there was no clear distinction between syntax and semantics at that time. Recall, e.g., that as indicated earlier, the axiom systems came by Hilbert often with a built-in interpretation. Add also that the very notions necessary to formulate properly the difference syntax-semantics were not available to Hilbert.

The problem of the completeness of the first-order logic, i.e., the fourth problem of Hilbert’s Bologna lecture, was also posed as a question in the book by Hilbert and Ackermann Gnmdzüge der theoretischen Logik (1928). It was solved by Kurt Gödel in his doctoral dissertation (1929, cf. also 1930)where he showed that the first-order logic is complete, i.e., every true statement can be derived from the axioms. Moreover he proved that, in the first-order logic, every consistent axiom system has a model. More exactly, Gödel wrote that by completeness he meant that “every valid formula expressible in the restricted functional calculus [...] can be derived from the axioms by means of a finite sequence of formal inferences”. And added that this is equivalent to the assertion that “Every consistent axiom system [formalized within that restricted calculus] [...] has a realization” and to the statement that “Every logical expression is either satisfiable or refutable” (this is the form in which he actually proved the result). The importance of this result is, according to Gödel, that it justifies the “usual method of proving consistency”. One should notice here that the notion of truth in a structure, central to the very definition of satisfiability or validity, was nowhere analysed in either Gödel’s dissertation or his published revision of it. There was in fact a long tradition of using the informal notion of satisfiability (compare the work of Löwenheim, Skolem and others).

Some months later, in 1930, Gödel solved three other problems posed by Hilbert in Bologna by showing that arithmetic of natural numbers and all richer ←27 | 28→theories are essentially incomplete (provided they are consistent) (cf. Gödel 1931). It is interesting to see how Gödel arrived at this result.

Gödel himself wrote on his discovery in a draft reply to a letter dated 27th May 1970 from Yossef Balas, then a student at the University of Northern Iowa (cf.Wang 1987, pp. 84–85). Gödel indicated there that it was precisely his recognition of the contrast between the formal definability of provability and the formal undefinability of truth that led him to his discovery of incompleteness. One finds also there the following statement:

[...] long before, I had found the correct solution of the semantic paradoxes in the fact that truth in a language cannot be defined in itself.

On the base of this quotation, one can argue that Gödel obtained the result on the undefinability of truth independently of A. Tarski (cf. Tarski 1933).17

Note also that Gödel was convinced of the objectivity of the concept of mathematical truth. In a letter to Hao Wang (cf.Wang 1974, p. 9) he wrote:

[...] it should be noted that the heuristic principle of my construction of undecidable number-theoretical propositions in the formal systems of mathematics is the highly transfinite concept of ‘objectivemathematical truth’ as opposed to that of ‘demonstrability’.

In this situation, one should ask why Gödel did not mention the undefinability of truth, in his writings. In fact, Gödel even avoided the terms “true” and “truth” as well as the very concept of being true (he used the term “richtige Formel” and not the term “wahre Formel”). In the paper “Über formal unentscheidbare Sätze” (1931) the concept of a true formula occurs only at the end of Section 1 where Gödel explains the main idea of the proof of the first incompleteness theorem (but again the term “inhaltlich richtige Formel” and not the term “wahre Formel” appears here). Indeed, talking about the construction of a formula which should express its own unprovability invokes the interpretation of the formal system.

On the other hand, the term “truth” occurred in Gödel’s lectures on the incompleteness theorems at the Institute for Advanced Study in Princeton in the spring of 1934.He discussed there, among other things, the relation between the existence of undecidable propositions and the possibility of defining the concept “true (false) sentence” of a given language in the language itself. Considering the relation of his arguments to the paradoxes, in particular to the paradox of “The Liar”, Gödel indicates that the paradox disappears when one notes that the notion “false statement in a language B” cannot be expressed in B. Even more, “the paradox can be considered as a proof that ‘false statement in B’ cannot be expressed in B”.

←28 | 29→

What were the reasons of avoiding the concept of truth by Gödel? An answer can be found in a crossed-out passage of a draft of Gödel’s reply to the letter of the student Yossef Balas (mentioned already above). Gödel wrote there:

However in consequence of the philosophical prejudices of our times 1. nobody was looking for a relative consistency proof because [it]was considered axiomatic that a consistency proof must be finitary in order to make sense, 2. a concept of objective mathematical truth as opposed to demonstrability was viewed with greatest suspicion and widely rejected as meaningless.

Hence, it leads us to the conclusion formulated by S. Feferman in 1984 in the following way:

[...] Gödel feared that work assuming such a concept [i.e., the concept of mathematical truth –my remark, R.M.] would be rejected by the foundational establishment, dominated as it was by Hilbert’s ideas. Thus he sought to extract results from it which would make perfectly good sense even to those who eschewed all non-finitarymethods in mathematics.

Though Gödel tried to avoid concepts not accepted by the foundational establishment, his own philosophy of mathematics was in fact Platonist. He was convinced that (cf.Wang 1996, p. 83):

It was the anti-Platonic prejudice which prevented people from getting my results. This fact is a clear proof that the prejudice is a mistake.

Gödel’s theorem on the completeness of first-order logic and his discovery of the incompleteness phenomenon together with the undefinability of truth vs. definability of formal demonstrability showed that formal provability cannot be treated as an analysis of truth, that the former is in fact weaker than the latter. It was also shown in this way that Hilbert’s dreams to justify classical mathematics by means of finitistic methods cannot be fully realized. Those results together with Tarski’s definition of truth (in the structure) and Carnap’s work on the syntax of a language led also to the establishing of syntax and semantics in the 1930s.

On the other hand, it should be added that Gödel shared Hilbert’s “rationalistic optimism” (to use Hao Wang’s term) insofar as informal proofs were concerned. In fact, Gödel retained the idea of mathematics as a system of truth, which is complete in the sense that “every precisely formulated yes-or-no question in mathematics must have a clear-cut answer” (cf. Gödel 1970). He rejected however – in the light of his incompleteness theorem – the idea that the basis of these truths is their derivability from axioms. In his Gibbs lecture of 1951, Gödel distinguishes between the system of all true mathematical propositions from that of all demonstrable mathematical propositions, calling them, respectively, mathematics in the objective and subjective sense. He claimed also that it is objective mathematics that no axiom system can fully comprise.

←29 | 30→

Gödel’s incompleteness theorems and in particular his recognition of the undefinability of the concept of truth indicated a certain gap in Hilbert’s program and showed in particular, roughly speaking, that (full) truth cannot be comprised by provability and, generally, by syntactic means. The former can be only approximated by the latter. Hence there arose a problem: How should Hilbert’s finitistic point of view be extended?

Hilbert in his lecture in Hamburg in December 1930 (cf. Hilbert 1931) proposed to admit a new rule of inference. This rule was similar to the ω-rule, but it had rather informal character (a system obtained by admitting it would be semiformal). In fact, Hilbert proposed that whenever A(z) is a quantifier-free formula for which it can be shown (finitarily) that A(z) is a correct (richtig) numerical formula for each particular numerical instance z, then its universal generalization ∀xA(x) may be taken as a new premise (Ausgangsformel) in all further proofs.

Gödel pointed in many places that new axioms are needed to settle both undecidable arithmetical and set-theoretic propositions. In 1931 (p. 35), he stated that “[...] there are number-theoretic problems that cannot be solved with number-theoretic, but only with analytic or, respectively, set-theoreticmethods”.Andin1933 (p. 48)he wrote: “there are arithmetic propositions which cannot be proved even by analysis but only by methods involving extremely large infinite cardinals and similar things”. In (1970) Gödel proposed “cultivating (deepening) knowledge of the abstract concepts themselves which lead to the setting up of these mechanical systems”. In (1972) (this paper was a revised and expanded English version of 1958), Gödel claimed that concrete finitary methods are insufficient to prove the consistency of elementary number theory and some abstract concepts must be used in addition. In the paper (1946), Gödel explicitly called for an effort to use progressively more powerful transfinite theories to derive new arithmetical theorems.

Also Zermelo proposed to allow infinitary methods to overcome restrictions revealed by Gödel. According to Zermelo, the existence of undecidable propositions was a consequence of the restriction of the notion of proof to finitistic methods (he said here about “finitistic prejudice”). This situation could be changed if one used a more general “scheme” of proof. Zermelo had here in mind an infinitary logic, in which there were infinitely long sentences and rules of inference with infinitely many premises. In such a logic, he insisted, “all propositions are decidable!” He thought of quantifiers as infinitary conjunctions or disjunctions of unrestricted cardinality and conceived of proofs not as formal deductions from given axioms but as metamathematical determinations of the truth or falsity of a proposition. Thus syntactic considerations played no role in his thinking.

To give a rough account of how those suggestions and proposals to extend the finitistic point of view do in fact work, let us quote some technical results. We ←30 | 31→restrict ourselves to the case of the arithmetic of natural numbers, more exactly to Peano arithmetic PA.

Generally speaking, one can obtain completions of PA by:

 admitting the ω-rule,

 adding new axioms (in particular reflection principles) and

 adding (partial) notion(s) of truth.

Let us start by considering the case of the ω-rule, i.e., of the following rule:

Denote by (PA)ω Peano arithmetic PA with the ω-rule. One can easily see that (PA)ω is complete – it follows from the fact that its unique model up to isomorphism is the standard model Hence

One can ask: How many times must the ω-rule be applied to obtain a complete extension of PA? To give an answer, let us define the following hierarchy of theories where T is any first-order theory in the language L(PA) of Peano arithmetic:

To	=	T,
	=	Tα ∪{φ ∶ φ is of the form ∀xψ(x) and for every n ∈N},
Tα+1	=	the smallest set of formulas containing and closed under the rules of inference of PA,
Tλ	=	for λ limit.

One can now prove that

Theorem 1

Recall the hierarchy of formulas of the language L(PA). Let be the set of all quantifier free formulas and all formulas with bounded quantifiers. Define to be the set of all formulas of the form ∃xψ for and to be the set of all formulas of the form ∀xψ for We also define Δas the set of all n formulas equivalent (in PA) to a formula and to a formula. One can prove that

Theorem 2 For every n ∈ N the theory PAn is complete with respect to sentences.

In PA one can define partial notions of truth, i.e., one can define satisfaction and truth for formulas of a given class of the arithmetical hierarchy. Denote by ←31 | 32→the definition of satisfaction for SatΠformulas; similarly let denote the definition of satisfaction for formulas. Note that and formulas are and that and (for n ≥ 1) are and respectively. Let further and denote truth predicates for and sentences.18 In the sequel, we shall identify formulas defining satisfaction and truth and their extensions in the standard model No.

The previous theorem can now be formulated as:

In the definition of the hierarchy Tα no restriction was put on formulas to which the ω-rule was applied. Consider now a hierarchy in which such a restriction is put. So let T be any theory in the language L(PA). Define the following hierarchy of theories (cf. Niebergall 1996):

T(o)	=	T,
	=	is of the form ∀xψ(x) and and
		for every n ∈N},
T(α+1)	=	the smallest set of formulas containing and closed
		under the rules of inference of PA,
T(λ)	=	for λ limit.

Hence the ω-rule is now applied at stage n to formulas only.

One has the following

Theorem 3 (Niebergall 1996) For any n ∈N,

The above theorems19 indicate interconnections between Peano arithmetic augmented with the ω-rule and the partial truths. Other connections between them can be formulated in the language of interpretability. So let S ≼ T denote that a theory S is relatively interpretable in the theory T (in the sense of Tarski 1953). We have now the following facts (cf. Niebergall 1996):

←32 | 33→

Theorem 4 Let ConS (for an appropriate theory S) denote a statement of L(PA) stating that S is consistent. Then

The above theorems indicate that the arithmetical truth, i.e., the set Th(No) of all arithmetical sentences true in the standard model No, can be approximated by syntactical methods, i.e. by demonstrability – though not by finitary means (one uses here the ω-rule).

So far, we have considered Peano arithmetic and partial truths. Ask now: What about PA and the full truth? Gödel’s and Tarski’s theoremshows that the truth predicate for L(PA) cannot be defined in PA. But one can extend the language L(PA) by adding a new binary predicate S called satisfaction class and characterizing it axiomatically by adding to Peano arithmetic PA axioms being an appropriate modification of Tarski’s definition of satisfaction (cf. Krajewski 1976, where this notion was introduced, or Murawski, 1997). Note that since those axioms form a finite set of axioms, one can write them as a single formula of the language L(PA) ∪ S. Denote this theory as PA + “S is a satisfaction class”. One can extend this theory by adding new axioms stating special properties of S. In particular, one can demand that S is full, i.e., S decides any formula of L(PA) on any valuation or that S is Γinductive for Γ being a given class of formulas of the language L(PA) ∪ S, i.e., that the induction axiom holds for all formulas of the class Γ (if Γ is the class of all formulas of L(PA) ∪ S then one says that the satisfaction class S is inductive).

Since theories T of the indicated type are extensions of PA one can ask what about natural numbers can be proved in T, i.e., one can consider theories of the type

Theorems of PAT are those sentences of the language L(PA) of Peano arithmetic (hence sentences about natural numbers) which can be proved in the stronger theory T. A natural problem of finding an axiomatization of the theory PAT arises.

One can easily see that the following theories are conservative extensions of PA:

(a)	PA + “S is a satisfaction class”,
(b)	PA + “S is a full satisfaction class” and
(c)	PA + “S is an inductive satisfaction class”.

←33 | 34→

This means that one can prove in those theories exactly the same theorems about natural numbers (i.e., formulas of the language L(PA)) as in Peano arithmetic PA. Hence the addition of a new notion, i.e., of a notion of a satisfaction class (and consequently a notion of truth), with properties indicated in (a)–(c) does not increase the proof-theoretical power of a theory with respect to sentences of the language L(PA). On the other hand, the assumption that a satisfaction class is full and inductive gives a nonconservative extension of PA. In fact one can prove in this theory the consistency of PA.

The theories PAT for T being PA+ “S is a full inductive satisfaction class” can be characterized by transfinite induction or the consistency of appropriate ω-logics. Denote by Γ − PA(S) the theory PA + “S is a full Γ-inductive satisfaction class” and by PA(S) the theory PA+ “S is a full inductive satisfaction class”.

Consider the following sequence of formulas of the language L(PA) (one uses here arithmetization):

Γn+1(φ) = “there exists a proof of the formula φ

based on

Observe that in this system of ω-logic only the application of the ω-rule increases the degree of complexity of a proof.

Theorem 5 (Kotlarski 1986)

It can also be proved (cf. Kotlarski 1986) that the theory PA(S) is equal to the theory

The last sentence can be read as: “S makes all theorems of PA true”. It is equivalent to the -inductiveness of the satisfaction class S.

The system of ω-logic described above can be iterated in the transfinite and one can axiomatize theories and PAPA(S) by consistency statements of appropriate systems of this logic (cf. Kotlarski and Ratajczyk 1990a).

Define for an ordinal α a sequence in the following way: and put ωn = ωn(ω). Let now TI(ρ), where ρ is an ordinal, denote the scheme of transfinite induction up to ρ. Then the following theorem holds.

←34 | 35→

Theorem 6 (Kotlarski and Ratajczyk 1990b) Let m be a natural number. Then

The above theorems show how strong is Peano arithmetic augmented with an appropriate notion of satisfaction (and truth). One can see that only by assuming that the added notion of satisfaction (truth) is full and at least inductive one obtains a proper extension of PA. It is interesting that such extensions are equivalent to PA extended by appropriate forms of transfinite induction or by the statements of the consistency of appropriate systems of ω-logic. In other words, the above theorems show in particular that what can be proved about natural numbers using Peano axioms and the notion of satisfaction (truth) that is assumed to be full and -inductive is exactly the same as what can be proved in PA plus transfinite induction for ordinals (for all k ∈N) or in PA plus appropriate consistency statements. Similarly for PA plus full inductive satisfaction (truth) on the one hand and PA plus transfinite induction for ordinals (for all k ∈N) or PA plus appropriate consistency statements on the other. They show also that by adding to PA the notion of satisfaction (truth) and assuming that it is full and makes all theorems of PA true, one obtains a theory with exactly the same theorems about natural numbers as by taking PA augmented with a concept of a full and inductive satisfaction (truth) or PA plus appropriate consistency statements. In the above considerations, we restricted ourselves to formal proofs and to the semantical notion of truth in mathematics. We tried to show how the awareness of differences between them has been developed – from the hopes that formal proofs provide sufficient means to exhaust the mathematical truth to the discovery of various limitations of them. Let us finish with some general remarks.

Concepts of proof and truth are (even in mathematics) ambiguous. One should distinguish between working proofs of everyday mathematics and idealized formal proofs used by logicians. On the other hand, a proof in mathematics has various aspects and can be studied from various points of view. One can distinguish psychological, social, cultural and logical aspects of proofs. a proof can be studied as a mathematical or as an epistemological object. The former is precisely defined on the basis of mathematical logic, and the latter is a vague concept. The former is an idealization of proofs occurring in a research practice of mathematicians, is a reconstruction of them. Recently, one can observe in the philosophy of mathematics a tendency to concentrate on the actual research practice of mathematicians rather than on idealized foundational reconstructions of it and consequently to study the methods actually used by mathematicians.

←35 | 36→

Similar distinctions can be made with respect to the concept of truth. The semantical concept of truth precisely defined by Tarski is in fact a mathematical notion. It provides a definition of truth in mathematics;20 it is the concept of truth for a model in a formal language (its essential feature is to define truth in terms of reference or satisfaction on the basis of a particular kind of syntactico-semantical analysis of the language). But one can also speak about epistemic truth – cf., e.g. Isaacson (1987, 1992) where it is argued that Peano arithmetic is complete with respect to an epistemic notion of arithmetical truth.

The distinction between proof and truth in mathematics presupposes of course some philosophical assumptions. In fact for pure formalists and for intuitionists there exists no truth/proof problem. For them a mathematical statement is true just in case it is provable, and proofs are syntactic or mental constructions of our own making. In the case of a Platonist (realist) philosophy of mathematics, the situation is different. One can say that Platonist approach to mathematics enabled Gödel to state the problem and to be able to distinguish between proof and truth, between syntax and semantics.21

14 ,,Wenn wir die Tatsachen eines bestimmtenmehr oder minder umfassenden Wissensgebiete zusammenstellen, so bemerken wir bald, daß diese Tatsachen einer Ordnung fähig sind. Diese Ordnung erfolgt jedesmal mit Hilfe eines gewissen Fachwerkes von Begriffen in der Weise, daß dem einzelnen Gegenstande des Wissensgebietes ein Begriff dieses Fachwerkes und jeder Tatsache innerhalb des Wissensgebietes eine logische Beziehung zwischen den Begriffen entspricht. Das Fachwerk der Begriffe ist nicht Anderes als die Theorie des Wissensgebietes”.

15 ,,Ja, es ist doch selbsverständlich eine jede Theorie nur ein Fachwerk oder Schema von Begriffen nebst ihren nothwendigen Beziehungen zu einander, und die Grundelemente können in beliebiger Weise gedacht werden. Wenn ich unter meinen Punkten irgendwelche Systeme von Dingen, z.B. das System: Liebe, Gesetz, Schornsteinfeger [...] denke und dann nur meine sämtlichen Axiome als Beziehungen zwischen diesen Dingen annehme, so gelten meine Sätze, z.B. der Pythagoras auch von diesen Dingen.Mit anderen Worten: eine jede Theorie kann stets auf unendliche viele Systeme von Grundelementen angewandt werden”.

16 Cf. Gödel’s letter to Hao Wang dated 7th December 1967 – see Wang (1974), p. 8.

17 For the problem of the priority of proving the undefinability of truth, see Woleński (1991) and Murawski (1998).

18 Construction of SatΣn and SatΠn can be found in Kaye (1991) and Murawski (1999).

19 Note that many of those theorems hold not only for Peano arithmetic PA but also for a broad class of theories – cf.Niebergall (1996).

20 One should distinguish the truth in mathematics and the truth of mathematics.

21 Note that, as indicated above, Hilbert was not interested in philosophical questions and did not consider them.