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”.
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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.
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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