Undecidability and Modern Physics

    Let us construct a symbolic formal system with the following elements:

1. An arbitrary axiomatic system which contains Gödel’s axiomatic system together with its rules of inference, say G_a.

2. The functions and relations of the system are recursively defined and free from contradiction.

3. We construct an isomorphic representation of the subsystem of non-numerical symbols by a system of positive integers, ascribing natural numbers to the symbols. Therefore, we can express any formula in numerical terms (particularly as a sequence of primes, as Gödel did, but the fundamental theorem of arithmetic is irrelevant for our argument), and also express proofs as sequences of positive integers.

4. We construct a set of formulas F which are directly provable within the system and which represent common expressions of our calculus. 

Then, for every formula f_i ∈ F, there is a numerical formula p_i ∈ P, for P⊆ F, such that G_a⊢ p_i.

Construct now a f_j which expresses “this formula is not provable”, a valid and meaningful expression of our calculus. Therefore, there is a p_j numerical formula which corresponds to f_j.

The undecidability theorem says that p_j is undecidable (unentscheidbare).

Proof:

Suppose p_j is true. Then p_j is not provable, but p_j ∈ F, set of directly provable formulas, so there is a contradiction.

Suppose p_j is false. Then ¬ p_j is true, i.e. p_j is provable, but p_j says that it is not provable, so there is a contradiction.

Therefore, we cannot decide about p_j

   We say that a system S is complete when for a given formula A∈ S, we can prove either A or ¬A. In our case, G_a is not complete.

   Physics considers itself immune to Gödel’s theorems. Gordon Kane, with the blessings of Edward Witten, has publically rejected the validity of the completeness theorem for physics, to name but a single and significant case of the irrelevant effects which Gödel theorems had in physics. But I fail to see how.

   The laws of a conceptual system of physics are equivalent, at best, to theorems of a calculus which express necessary connections. Then a true formula of physics would be provable from the conceptual physical frame (made by both theoretical and observational concepts) where it belongs, i.e. formulas are proven by experiment but experiments are only meaningful within a particular conceptual frame. We can construct for such system an isomorphic image of its semantic formulas (those that say if a formula of the physical system is either true or false) in the system of the positive integers, and make a calculus of formulas that operates like G_a, establishing functions and relations recursively defined all the way to statements of our more basic experience (Lebenswelt). Nonetheless, by the undecidability theorem, such calculus is not complete. In turn, this implies that conceptual physical systems are incomplete.

   What is then physics talking about when it says that is unveiling the ultimate laws of nature? It is building a contemporary narrative of the universe, repeating the traditional unveiling of Isis, though the image constructed has not more logical foundations than the old theosophical one. Inasmuch as physics insists in giving ontological proofs of a particular constitution of the universe based in arithmetic, it will only give an incomplete image. Of course, if it renounces arithmetic, it would have to return to the old shaman visions, and not even that, for arithmetic is just a generalization of the individuation processes and time experience in high organisms. However, this does not mean that there is a complete conceptual construction, or could ever be one. In fact, when considering that for any statement of a calculus either A or ¬A is provable, we are declaring a principle of epistemological omnipotence very much related to a belief in universal laws. Undecidability is the logical declaration of the mirage of the  physical universal law, too heavy a blow for the traditional epistemological aspirations of the Queen of Sciences, modern physics.