Two strategies to infinity: completeness and incompleteness. The completeness of quantum mechanics

The two strategies to infinity originates from the definitive properties of the totality: “to be all”. It is complete being all. However also it is in incomplete because it includes incompleteness within it being all. This seems to be contradictory, but it is very fruitful, in fact, for the definition of the totality rigorously as just that entity determinable by the property to be both complete and incomplete in a consistent way:

The totality contains its externality within it necessarily being all in definition. Thus, any element of the totality has to be doubled by a “twin” both identical and “complementary” (in the generalized sense of Niels Bohr) to the former twin. The “latter twin” as if represents a certain element of the externality of the totality after it has to be inside.

Therefore, the both “twin” dual spaces of the separable complex Hilbert space utilized by quantum mechanics as its basic mathematical formalism are fundamentally necessary as far as quantum mechanics is inherently holistic and thus referred to the whole and its totality.

The totality in mathematics is meant by the concept of infinity. One notices an isomorphism between the opposition “completeness – incompleteness” in both foundations of mathematics and quantum mechanics. The strategy of quantum mechanics to be complete can be interpreted directly onto the problem of how mathematics can be complete. The transition of the strategy is grounded by that isomorphism underlain in turn by the concept of the totality (though being a philosophical one properly).

Once the strategy of completeness is transferred to the foundations of mathematics, it can be visualized by Russell’s paradox of the “barber” as if doubled to himself by a “barber twin”: all elements of the totality are similar to Russell’s “barber”. However this is not contradictory, but productive to be defined any entity as an element of the totality. Quantum mechanics (furthermore be complete) is able to demonstrate how by means of both “twin” spaces of the separable complex Hilbert space.

Another pathway to the same vision to the completeness of mathematics can suggests Henkin’s proposition in virtue of Löb’s proof.

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