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):
