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Incompleteness: Gรถdel and Einstein

Vasil Penchev, CV:

Two incompletenesses: • The thesis is: Einstein, Podolsky and Rosen’s argument (1935, Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? ) is another interpretation of the famous Gödel incompleteness argument (1931, Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I ) in terms of quantum mechanics

The incompleteness of quantum mechanics • Quantum mechanics needs the half of variables necessary to be exhaustively described in comparison with a system in classical mechanics. The other half is both equivalent and complementary to the former and thus redundant • Another viewpoint to the same fact, shared by Einstein, is the theory of quantum mechanics is incomplete and should be completed in a future theory • Accordingly, he wasted much time to prove that imperfectness of quantum mechanics

Incompleteness in Gödel • After Gödel had demonstrated (1930) in a non-constructive way that a finite axiomatics can be consistent and complete, he showed (1931) in a constructive way that an infinite axiomatics (as including Peano’s axioms about the natural numbers) can be consistent if and only if it is incomplete • Thus he managed to investigate the link between infinity and incompleteness in a formal and logical way as to the foundation of mathematics

The link between the two incompleteness • The close friendship between the Princeton refugees Einstein and Gödel might address that link • However Kurt Gödel came to Princeton in 1940, while Einstein, Podolsky, and Rosen had already published their famous article “Can quantummechanical description of physical reality be considered complete?” five years ago (1935) • Consequently no one of both could influence the other but they shared rather a common philosophical viewpoint, which is expressed differently in the cited works

The underlying structure: • However the outlines of a common set-theory structure interpretable in both ways are much more essential concerning the incompleteness of infinity: • If two so great thinkers and scientists shared a common philosophical viewpoint to the link of infinity and incompleteness, it is much worth to determine a formal structure underlying their treating of incompleteness correspondingly in quantum mechanics and the foundation of mathematics

Infinity as a bridge between the two incompleteness • Gödel’s two papers (1930 and 1931) addresses clearly infinity as a possible condition of incompleteness in mathematics in the sense exacted by them • In fact, quantum mechanics is the first experimental science, which has involved infinity by its mathematical formalism, that of Hilbert space • Infinity is the pathway necessity to link the incompleteness in mathematics to that in quantum mechanics

A model of the openness (incompleteness) of infinity • An arbitrary infinite countable set “A” and another set “B” so that their intersection is empty are given • The general model of incompleteness, which is going to be constructed, is general enough as it is based on set theory underlying all mathematics • Only the most fundamental and thus simplest properties of an infinite set will be necessary for that purpose

Infinity and finiteness compared in relation to openness and incompleteness • Both completeness and incompleteness are well distinguishable as to finiteness: • Completeness supposes that any operations defined over any finite sets do not transcend them while incompleteness displays that they can do it sometimes • This legible boundary turns out to be unclear and even inconsistent jumping into infinity.

The construction: • One constitutes their union “C”, which will be an infinite set whatever B is: • The idea is to demonstrate that infinity generates a similar internal image of any external entity, just being necessary universal after infinity is truly infinite (sorry for the tautology) • Even more, one can distinguish the external entity from its internal image

Openness and universality as to infinity • One may say that there are two strategies or “philosophies” after that leap into infinity has been just made and any orientation in the unknown infinity is necessary for the thought to survive: • One should keep either to completeness or to incompleteness for the infinity seems both complete and incomplete being as universal as open

The mapping • Utilizing the axiom of choice, a one-to-one mapping “f ” exists: • To be the deduction rigorous, the language of set theory is used. However the underlying ideas are fundamentally philosophical • That mapping should equate in a sense the external entity and internal image in a common whole

The role of the axiom of choice in the construction The axiom of choice can be interpreted in two ways in the case: ďƒ˜As the set of all constructive ways, in which a mapping between the two sets at issue can be built ďƒ˜As all ways that mapping to exist independent of whether it can be constructed in any way somehow or not

The complement and its image • One designates the image of B into A through f by “B(f)â€? so that B(f) is a true subset of A

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The necessity of an image of openness into universality • The set-theory construction only makes visible a more general and philosophical idea: • Infinity should reconcile two properties seeming contradictory and inconsistent to each other: universality and openness • Indeed universality means completeness as any entity should be within the universality in a sense • However openness means incompleteness as some entities should be outside it to be able to be open to them

Image and Simile If the axiom of choice holds, there is always an internal and equivalent image as B(f) for any external set as B However the relation between “B” and “B(f)” is ambiguous in a sense: According to “f”, or the universality of infinity, “B” and “B(f)” should be identical, indistinguishable. Then “B(f)” is an exact image of “B” However according to the openness of infinity, they should be only similar, distinguishable, or “B(f)” is a simile of “B”

Incompleteness in the completeness • So the B(f) constructed thus is both identical (a copy) and only similar (non-identical) to B • One can say that B(F) allows of representing incompleteness within the completeness of an universality • That construction elucidates that both infinity and universality as well as the totality as a philosophical generalization of them are necessarily ambiguous in relation to the property of completeness/ incompleteness

Undecidability • That equivocality implies undecidability in a logical sense for any interpretation of the construction: • Indeed if one accepts that B(f) coincides with B, whether an element b of B belongs or not to A is an undecidable problem as far as b(f) coincides with b • The logical undecidability can be thought more generally in a philosophical sense as the equivocality of “image and simile” as to infinity or the totality

Undecidability and Infinity • Both infinity and the totality imply that equivocality and thus the corresponding undecidability if they are formalized in a rigorous way by means of a logical axiomatics e.g. as what Gödel utilized or that of a mathematical structure e.g. as Hilbert space used by quantum mechanics • However the latters share both a common set theory structure described above and a fundamental philosophical property of the totality only visualized in particular by that Gödel insolubility

The necessity of the axiom of choice • However if the axiom of choice is not valid, one cannot guarantee that f exists and should display how a constructive analog of “f” can be built • Consequently the axiom of choice supplies a more general consideration both as to the constructive and as to non-constructive case • It frees us from the inconvenience of a too lengthy, awkward and intricate construction for its result is directly postulated without being expressed explicitly

The invariance to the axiom of choice • However the relation of that construction to the axiom of choice is more sophisticated: • It serves as “stairs”, which can be removed after the construction is accomplished so that it can be reached both in the “stairs” of the axiom of choice and by a “jump” leaping them (or it) • Indeed: If A and a subset B’ of it are given, B’ can be interpreted as the “image and simile” of some unknown B and therefore implying only the pure existence of B removing the “stairs” of “f” • This the construction both needs the axiom of choice and is invariant to it

Constructiveness vs. the axiom of choice • If one shows how “f” to be constructed at least in one case, this will be a constructive proof of undecidablity as what Gödel’s is • In fact, the almost entire volume of Gödel’s paper (1931) addresses how the difficulties for a constructive proof can be overcome • He constructed a concrete procedure, by which to show explicitly one case of an insoluble statement and thus to demonstrate just in a constructive way the existence of those propositions under the conditions of the theorem

About the Gödel number of Godel’s theorem

However the equivocality discussed above can be referred to Gödel’s proof by the following question: What is the Gödel number of the so-called first incompleteness theorem? It contains the set of all natural numbers by Peano’s axioms. Then: If that set is considered as a singularity, the Gödel number of it is finite, but the formulation of the theorem is not constructive as it refers to an infinite set as actually infinite If that set is considered as constructively infinite, the Gödel number of the theorem should be infinite and thus the same as that of its negation

From the mathematical to the physical incompleteness • In fact, the paper of Einstein – Podolsky – Rosen interprets the same structure discussed above: • Indeed quantum mechanics is the first experimental theory, which introduces infinity to describe theoretically the investigated phenomena • It was forced and decided to do this too difficultly after dramatic discussions during decades • However Einstein never accepted this step for the paradoxical corollaries as if blaming quantum mechanics

The EPR argument and quantum information

• The genius of Einstein becomes obvious even in his mistakes: • The EPR argument did not manage to demonstrate the incompleteness of quantum mechanics • However it did much more opening the universe of quantum correlations and the phenomena of entanglement and thus the new physical discipline of quantum information • In final analysis, quantum information can be deduced of that extraordinary step for infinity to be involved in an empirical science like quantum mechanics

The essence of the EPR argument • There is an initial quantum system Q, which is divided into two other systems P and S moving with some relative speed to each other in space-time • The key word is “quantum”: being “classical” the EPR argument could not be reproduced • It is just the quantum consideration of a mechanical system, which necessarily involves infinity and just this is the essence of EPR

Infinity as the essence of the EPR argument • In the context of Einstein, quantum mechanics can be thought as a kind of a further generalization of his famous principle of general relativity that the laws of nature should be invariant to any smooth motion • The generalization implicitly involved by quantum mechanics should be that the laws of nature should be invariant to any motion including quantum rather than to a smooth one • Just the latter involves infinity necessarily

The set-theory core of the EPR argument • For Q, P, and S are quantum systems and they are represented by three infinite-dimensional Hilbert spaces, the EPR argument can be bared to a set-theory core: • Indeed the fact that infinity is embedded in some physical entities like quantum “particles” moving to each other in space-time is accidental to the essence of EPR once quantum mechanics is forced to use infinity in the mathematical model

The GĂśdel incompleteness

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The EPR incompleteness “Spooky� action at any distance

Quantum mechanics and infinity • Indeed infinity in quantum mechanics can be directly deduced by the generalized Einstein principle above: • That principle requires some common or joint measure both for a quantum and for a smooth motion • The only possible measure of that kind requires Hilbert space as the space of the measured: It is not only infinitely dimensional in general, but even any finitely dimensional subspace of it requires infinity. For any unit of its basis (đ?‘’ đ?‘–đ?‘›đ?œ” , đ?‘› is a natural number) requires implicitly infinity

Dividing an infinity into two infinite parts ... • Consequently, the set-theory core can be reduced to the following after one has replaced the moving quantum particles by three Hilbert spaces corresponding to them and the Hilbert spaces are reduced to infinite sets in turn: • There is an initial infinity Q, which is divided into two infinities P and S, each of which suggests an external viewpoint to the other • This is not more than the set-theory structure extracted above by the first incompleteness theorem of Gödel

The definition of infinity in thus: • In turn, infinity can be defined as what can be divided into parts, which are equivalent to it in some sense • That definition of infinity is a kind of philosophical generalization of Dedekind’s one • Involving that Dedekind definition, at least a weaker form of the axiom of choice is necessary • Thus after one has introduced the axiom of choice, itself, that definition of infinity is acceptable

Incompleteness: • So each of the two pairs (P, S) and (S, P) models the structure extracted by GĂśdel’s paper (1931) and can serve to demonstrate the “incompleteness of quantum mechanicsâ€?, which produces that description of reality • Indeed e.g. for (P, S), P corresponds to A, and S to B. Thus S generates an “image and simileâ€? in P in the way described above. It will be designated by “S(f) ⊆ đ?‘ˇâ€? • Furthermore, Einstein, Podolsky, and Rosen interpreted the existence of S(f) as the proof of the incompleteness of quantum mechanics in their paper (1935)

The incompleteness of infinity • However, that “S(f)” cannot exclude the completeness of quantum mechanics as completeness and incompleteness do not contradict to each other as to infinity • Infinity can be interpreted by a suitable discrete topology therefore implying the wellordering theorem and the axiom of choice • Indeed, any discrete topology is “clopen”, both closed and open, therefore implying similarly both completeness and incompleteness of infinity

The contemporary physical interpretation • Indeed only the pure existence of “S(f)” can be stated on the set-theory ground. However, the pair [S(f),S] implicates some mapping “f”, which can depict “S” into “S(f)” by the mediation of the axiom of choice • Furthermore, a non-empty Q(f) implies some restriction of the degrees of freedom (DOF) of P and S as well as of the corresponding physical systems, from which they are extracted as their core • That restriction of DOF is experimentally observable and designated as “entanglement” (of the quantum systems “P” and “S” in the case)

The interpretation of “entanglement” as a generalization of ‘physical force’ The action of any physical force onto any physical entity results in some restriction of DOF Consequently, entanglement can be interpreted as a generalization of ‘physical force’ or ‘force field’, where the restriction of DOF includes an arbitrary change of the probability for a physical event to occur Even more, infinity underlying entanglement (as this is discussed above) is what grounds ‘physical force’ or ‘force field’ by its extraordinary property to be both complete and incomplete

Conclusions: • However the cause of the alleged incompleteness in EPR is the paradoxical property of infinity rather than the description of quantum mechanics once forced to introduce infinity in itself • Even much more, that involvement of infinity in an empirical and experimental science such as quantum mechanics turns out to be exceptionally fruitful by the concept and phenomena of entanglement

The totality both universal and open • One can try to continue and generalize that course of thought leading from infinity to physical reality to reality at all:

• The totality just being both universal and open is what is able to generate reality

References: • Einstein, A., B. Podolsky and N. Rosen. 1935. Can QuantumMechanical Description of Physical Reality Be Considered Complete? ‒ Physical Review, 1935, 47, 777-780. • Gödel, K. 1930. Die Vollständigkeit der Axiome des logischen Funktionenkalküls. – Monatshefte der Mathematik und Physik. Bd. 37, No 1 (December, 1930), 349-360 (Bilingual German ‒ English edition: K. Gödel. The completeness of the axioms of the functional calculus of logic. ‒ In: K. Gödel. Collected Works. Vol. I. Publications 1929 – 1936. Oxford: University Press, New York: Clarendon Press ‒ Oxford, 1986, 103-123.) • Gödel, K. 1931. Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I. ‒ Monatshefte der Mathematik und Physik. Bd. 38, No 1 (December, 1931), 173-198. (Bilingual German ‒ English edition: K. Gödel. The formally undecidable propositions of Principia mathematica and related systems I. ‒ In: K. Gödel. Collected Works. Vol. I. Publications 1929 – 1936. Oxford: University Press, New York: Clarendon Press ‒ Oxford, 1986, 144-195.)

Incompleteness: Gödel and Einstein