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

1. Incompleteness:
Gödel and Einstein

2. Vasil Penchev
vasildinev@gmail.com, vaspench@abv.bg
http://www.scribd.com/vasil7penchev
http://www.wprdpress.com/vasil7penchev
CV: http://old-philosophy.issk-bas.org/CV/cvpdf/V.Penchev-CV-eng.pdf

3. 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

4. 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

5. 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

6. 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

8. 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

9. 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

10. 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

11. 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.

12. 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

13. 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

14. 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

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

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

17. 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

18. 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”

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. The Gödel
incompleteness
The EPR
incompleteness
“Spooky” action
at any distance

32. Quantum mechanics and infinity

33. 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

34. 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

35. Incompleteness:

36. 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

37. 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)

38. 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

39. 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

40. 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

41. 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.)