Gödel's Vision

Gödel's Vision and Semantic Unfolding in Mathematical Logic: Gödel's First Incompleteness Theorem

The conceptual essence of Gödel's first incompleteness theorem may be summarized in the assertion that if a formal system containing arithmetic, i.e. any arithmetic structure endowed with the operations of addition and multiplication, is consistent, then it contains undecidable propositions, namely statements whose truth or falsity cannot be expressed within the language of this formal system.

Strong Self-Referentiality and Unfolding

Gödel's proof of the first incompleteness theorem is based on the explicit construction of an arithmetical formula that asserts its own non-provability, and thus, it is undecidable within the language of its formal system. From our viewpoint, the particular interest in Gödel's proof stems from three factors:

1. According to a remark of Gödel himself, there exists an obvious analogy between his undecidable proposition within a formal system containing arithmetic and classical semantic paradoxes, like the Liar paradox. The analogy is based on the existence of strong self-referentiality;

2. The correct solution of these semantic paradoxes, derives from the method of proof that Gödel devised in order to evade the logical obstruction of direct strong self-referentiality within the language of his formal system. Concisely put, this method of proof involved an argument requiring a stratification into two levels, one of which is called the mathematical level and the other the metamathematical level. In other words, the evasion of strong self-referentiality required a process of unfolding into another level of hypostasis, such that the direct obstruction is avoided via ascending to another level and then descending back. In this way, indirect self-reference via unfolding to another level, leads to a well-defined legitimate statement and not to a paradox.

3. The process of ascending from the mathematical to the metamathematical level and then descending back, or the other way round symmetrically, effecting indirect self-reference, and thus eventually, producing a legitimate statement asserting its own unprovability, required the conception of a "metaphor organon'' furnishing bidirectional bridges for translating between these two levels. This is precisely the role of "Gödel's numbering'' or "Gödel's ordering'' idea.

Gödel's Logical Conjugation

For reasons of historical continuity in the development of mathematical ideas, we refer to "Gödel's numbering'' as "Gödel's gnomon'', that effects the bidirectional process of ascending and descending between the mathematical and the metamathematical level.

This abstraction consists in thinking of a gnomon as a means to indicate or label propositions at both the mathematical and the metamathematical level in such a way that a certain type of translation equivalence can be established between these two levels. Therefore, "Gödel's gnomon'' is actually an encoding/decoding device between the mathematical and the metamathematical level based on a criterion of symmetry, in the sense of a common measure of ordering propositions appropriately at both levels.

In this train of thought, the deep role of ``Gödel's gnomon'' is to "logically conjugate'' the intractable problem of direct, strong self-referentiality at one level of hypostasis by a definite tractable process at the other level of hypostasis. The latter tractable process is the process of "Cantor's diagonalization''.

Conclusively, Gödel's theorem referring to a true proposition at the metamathematical level, is proved by descending at the mathematical level, such that a particular argument can be formulated by means of an infinite closure operator at this level ("Cantor's diagonalization'' process), which is then transferred back to the metamathematical level by means of ascending the inverse bridge to prove the theorem.

Gödel's Phraselogy

However (and this is the decisive point) it follows from the correct solution of the semantic paradoxes that the "truth'' of the propositions of a language cannot be expressed in the same language, while provability (being an arithmetical relation) can. Hence true is different from provable.

Such a theorem has in spite of contrary appearance nothing circular in itself since it asserts, to begin with, the unprovability of a quite definite formula, and only later it develops (so to speak, by accident) that this formula is just the one expressing the theorem itself.

Temporality of Unfolding

According to our knowledge, no sufficient attention has been paid to the subtle point implicated by the phrase "only later it develops" , in the above quotation of Gödel, although it bears a fundamental significance from the perspective of our analysis of Gödel's first incompleteness theorem.

The reason is that Gödel's argument requires a process of semantic unfolding from the mathematical to the metamathematical level of hypostasis, which is of an implicated temporal character, according to Gödel's conception in the presented wording. In this way, indirect self-reference taking place in the temporal dimensionality (called dia-stasis in ancient Greek) opened up via unfolding to another level, leads to a well-defined legitimate statement and not to a paradox.