I imagine someone asking my advice; he says: “I have constructed a proposition (I will use ‘P’ to designate it) in Russell’s symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: ‘P is not provable in Russell’s system.’ Must I not say that this proposition on the one hand is true, and on the other hand is unprovable? For suppose it were false; then it is true that it is provable. And that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable.” — Wittgenstein
Any language of such a set, say Peano Arithmetic PA (or Russell and Whitehead’s Principia Mathematica, or ZFC), expresses – in a finite, unambiguous, and communicable manner – relations between concepts that are external to the language PA (or to Principia, or to ZFC). Each such language is, thus, essentially two-valued, since a relation either holds or does not hold externally (relative to the language).
Further, a selected, finite, number of primitive formal assertions about a finite set of selected primitive relations of, say, PA are defined as axiomatically PA-provable; all other assertions about relations that can be effectively defined in terms of the primitive relations are termed as PA-provable if, and only if, there is a finite sequence of assertions of PA, each of which is either a primitive assertion, or which can effectively be determined in a finite number of steps as an immediate consequence of any two assertions preceding it in the sequence by a finite set of rules of consequence.
The philosophical dimensions of this emerges if we take M as the standard, arithmetical, interpretation of PA, where:
(a) the set of non-negative integers is the domain,
(b) the integer 0 is the interpretation of the symbol “0” of PA,
(c) the successor operation (addition of 1) is the interpretation of the “ ‘ ” function,
(d) ordinary addition and multiplication are the interpretations of “+” and “.“,
(e) the interpretation of the predicate letter “=” is the equality relation.
Now, post-Gödel, the standard interpretation of classical theory seems to be that:
(f) PA can, indeed, be interpreted in M;
(g) assertions in M are decidable by Tarski’s definitions of satisfiability and truth;
(h) Tarskian truth and satisfiability are, however, not effectively verifiable in M.
Tarski made clear his indebtedness to Gödel’s methods,
We owe the method used here to Gödel who employed it for other purposes in his recently published work Gödel. This exceedingly important and interesting article is not directly connected with the theme of our work it deals with strictly methodological problems the consistency and completeness of deductive systems, nevertheless we shall be able to use the methods and in part also the results of Gödel’s investigations for our purpose.
On the other hand Tarski strongly emphasized the fact that his results were obtained independently, even though Tarski’s theorem on the undefinability of truth implies the existence of undecidable sentences, and hence Gödel’s first incompleteness theorem. Shifting gears here, how far was the Wittgensteinian quote really close to Gödel’s? However, the question, implicit in Wittgenstein’s argument regarding the possibility of a semantic contradiction in Gödel’s reasoning, then arises: How can we assert that a PA-assertion (whether such an assertion is PA-provable or not) is true under interpretation in M, so long as such truth remains effectively unverifiable in M? Since the issue is not resolved unambiguously by Gödel in his paper (nor, apparently, by subsequent standard interpretations of his formal reasoning and conclusions), Wittgenstein’s quote can be taken to argue that, although we may validly draw various conclusions from Gödel’s formal reasoning and conclusions, the existence of a true or false assertion of M cannot be amongst them.