The document discusses the relationship between Gödel's incompleteness theorems and the completeness of quantum mechanics, proposing that both concepts may be interconnected through mathematical structures. It highlights the importance of Hilbert space in unifying arithmetic and geometry, suggesting that mathematics can self-found if it embraces multiple arithmetics and quantum theory. Ultimately, it posits that the foundation of mathematics is linked to the concept of quantum information, thereby establishing a connection between infinite and finite structures.

2. • Bulgarian Academy of Science: Institute for
the Study of Societies and Knowledge:
Dept. of Logical Systems and Models
vasildinev@gmail.com
10:15 - 10:45, June 29th , University of
Istanbul, Room “C”
5th Congress in Universal Logic,
University of Istanbul, Turkey, 25-30 June 2015

3. The Gödel incompleteness can be modeled on
the alleged incompleteness of quantum
mechanics
Then the proved and even experimentally
confirmed completeness of quantum mechanics
can be reversely interpreted as a strategy of
completeness as to the foundation of
mathematics

4. The one supposes that the Gödel incompleteness
originates from the deficiency of the
mathematical structure, on which mathematics
should be grounded
However that deficiency can imply two alternative
and maybe equivalent ways for the cherished
completeness:
 Qualitative deficiency: some other
mathematical structure rather than arithmetic
(e.g. geometry)
 Quantitative deficiency: arithmetic but more
than one (e.g. two ones)

5. Which is the mathematical structure, on which
completeness can be proved?
In tradition originating from Hilbert and Gödel, that
should be arithmetic, but what are the reasons for
that choice?
Indeed arithmetic seems to be the simplest one, but
whether not too simple in order to be able to be
sufficient for grounding completeness?
In fact, the Gödel incompleteness theorems means
only that it is insufficient, but nothing about some
other one eventually ...

6. Set theory and arithmetic were what was put as
the base of mathematics
However set theory seemed to be controversial
allowing of paradoxes such as that of Russell (1902)
unlike arithmetic
So, the Hilbert idea (1928) for grounding set theory
on arithmetic appeared
That idea has not ever been more than one
hypothesis and still less its refusing can mean
anything about the foundation of mathematics at all

7. In fact, there is a well-known result, that of Gentzen
(1936)
It claims the self-foundation of arithmetic and thus
of mathematics at all merely substituting induction
with transfinite induction (and even only to 𝜔 𝜔
is
what is necessary)
One can distinguish the Peano arithmetic from the
newly Gentzen one only by the axiom of induction
Then the Gentzen arithmetic would be sufficient
for the self-foundation of mathematics

8. However the transfinite induction seems to involve
implicitly and in advance infinity , that
controversial concept of set theory just which is
what should be to be grounded
Thus (along with his real or alleged complicity in
Nazism unlike Gödel who was a refugee from it)
Gentzen’s result has tended to be neglected in
favor of Gödel’s
In fact the real problem should be: What is
transfinite induction in comparison with the
standard, “finite” one?

9. Induction is the only “interesting” axiom among the
Peano ones in turn abstracted from Dedekind’s
(1888), which grounded arithmetic on set theory, and
therefore breaking the vicious circle
Transfinite induction has used to be thought as a kind
of super-induction in infinity rather than to (or until)
infinity and thus containing the usual one as a true
subset
However it can be not less well defined as a second
induction therefore a second and independent Peano
arithmetic along with the “first”, standard one

10. Transfinite induction can be (e.g.) defined as a second
and independent induction thus:
 Merely postulating it as such: After that the first
and second induction can be ordered
(not idempotent) or not (i.e. idempotent)
 By distinguishing the successor function as
follows:
 No one-to-one mapping of sets of 𝑛 and 𝑛 + 1
elements for the first induction (always 𝑛 ≠ 𝑛 + 1)
 There is at least a one-to-one mapping of sets of
𝜔 and 𝜔 + 1 elements for the second induction (not

11. Arithmetic and furthermore mathematics can be
self-founded consistently
This is able not to involve infinity either explicitly
or implicitly (which is an interpretation of
Gentzen’s finitism)
Infinity can be equally well defined as both
continuation of finiteness (continuity) and a leap
to a new dimension of finiteness (discreteness)

12. The concept of (quantum) information as the quantity
of choices underlies the foundation of mathematics
in fact:
Indeed the unit of information (a bit) is the choice
between two equally probable alternatives and thus
describes the mapping between a single arithmetic
(finiteness) and two ones (infinity)
The unit of quantum information (a qubit) is the
choice among an infinite set of alternatives and
describes the mapping between the “finite”
arithmetic and the “infinite” set theory

13. Quantum mechanics being a physical and
thus experimental science can be
nevertheless thoroughly reformulated in
terms of (quantum) information
Then quantum mechanics should be
interpreted as an empirical doctrine about
infinity after (quantum) information can
describe the relation between infinity and
finiteness quantitatively

14. Quantum mechanics is inherently dualistic theory for it
rests on the system of two fundamentally different
elements:
o The studied quantum entity, and
o The macroscopic apparatus measuring it
Of course both are finite, but two too different kinds of
finiteness: microscopic and macroscopic
If quantum mechanics studies eventually infinity in an
experimental way, this turns out to be possible just by
reducing infinity to a second and independent finiteness

15. If the first lesson repeated Gentzen’s, the
second one is unique and furthermore allows of
building a link between it and Gentzen’s
It consists in involving Hilbert space, a properly
geometric structure in its foundation and thus in
the foundation of mathematics
Indeed mathematics turns out to be able to
found itself as both two arithmetics and
geometry implicitly including arithmetic

16. Anyway why the arithmetic?
This turns out to be a random historical fact
appealing to intuition or to intellectual authorities
such as Cantor, Frege, Russell, Hilbert, “Nicolas
Bourbaki”, etc. rather than to any mathematical
proof
However arithmetic keeps its place in the
foundation of mathematics but forced to share it
whether with still one and independent arithmetic
or with geometry generalizing it in a sense

17. The so-called Gödel incompleteness theorems (1931)
demonstrated that set theory reducible to a single
arithmetic is irrelevant as the ground of mathematics
However they said nothing about some other
mathematical structures relevant for self-grounding
of mathematics
The quantum strategy allows of at least two direction
for researching those structures relevant to
completeness and still one corresponding to their
unification in terms of information as well

18. One can utilize an analogy to the so-called
fundamental theorem of algebra:
It needs a more general structure than the real
numbers, within which it can be proved
Analogically, the self-foundation of mathematics
needs some more general structure than the
positive integers in order to be provable

19. Still one key is Einstein’s failure (however
nevertheless exceptionally fruitful) to show that
quantum mechanics is incomplete
The triple article (1935) designated merely “EPR” as
well Schrödinger’s study (also 1935) forecast the
phenomena of entanglement on the base of
Hilbert space
The incompleteness of set theory and arithmetic
and the alleged incompleteness of quantum
mechanics can be linked to each other inherently ...

20. The close friendship of the Princeton refugees Gödel
and Einstein (Yourgrau 2006) might address that fact
However, Gödel came to Princeton in 1940
The famous triple article of Einstein, Podolsky, and
Rosen “Can Quantum-Mechanical Description of
Physical Reality Be Considered Complete?” was
published in 1935
So, there should exist a common mathematical
structure underlying both “completeness and
incompleteness”

21. The mathematical formalism of quantum
mechanics is based on the complex Hilbert
space featuring by a few important properties
relevant to that structure capable to underlie
mathematics:
It is a generalization of positive integers
It is both discrete and continuous (even smooth)
It is invariant to the axiom of choice

22. Hilbert space is a generalization of positive integers:
Thus it involves countable infinity
Indeed it can be considered as a countable series of
“empty” qubits equivalent to 3D unit balls
If one “shrinks” these unit balls to 3D points (balls
with zero radius), Hilbert space will degenerate to
Peano arithmetic

23. Hilbert space is both discrete and continuous
(even smooth) in a sense:
It is that mathematical structure, in which the
main problem of quantum mechanics about
uniformly describing both discrete and smooth
(continuous) motion can be resolved
Furthermore, it is discrete between any two
qubits but smooth (continuous) within each of
them
Thus it can unify arithmetic and geometry

24. Hilbert space is invariant to the axiom of choice in a
sense:
Indeed any point in it (a wave function in quantum
mechanics) can be interpreted both as:
o The characteristic function of a certain probability
distribution of a single coherent state before
measurement, i.e. before choice (the Born
interpretation of quantum mechanics)
o The smooth space-time trajectory of a “world”
after measurement, i.e. after choice (the many-
worlds interpretation of quantum mechanics)

25. This would mean the unification of:
• The externality and internality of any infinite set
• Model and reality in principle
• The probabilistic and deterministic consideration
of the modeled reality
• Along with that property of it to allow of uniformly
describing both discrete and smooth motion for
resolving the main problem of quantum mechanics

26. One can say that the crucial concept of all those
unifications is that of choice and thus (quantum)
information as the quantity of choices
Indeed it allows of reducing
o Two arithmetics to only one single (as bits of
information)
o Geometry to arithmetic (as qubits of quantum
information)
o And even much, much more: qubits of quantum
information to bits of information

27. The essence of set theory is the concept of
infinity and its link to arithmetic
Even more, that essence of set theory allows of
it to ground all mathematics though in a way yet
not consistent enough
Right the concept of information is what can
capture that core consistently

28. The Schrödinger equation is the most fundamental
equation in quantum mechanics
By the concept of (quantum) information, it can be
interpreted in terms of the foundation of mathematics
Then its sense would merely be that both ways for
infinity to be represented are equivalent two each
other. That is:
oA bit and a qubit can be equated energetically,
i.e. per a unit of time
oInfinity is quantitatively equivalent to a second
finiteness

29. One can describe that simple way for the Gödel
undecidable statements to be resolvable in two
arithmetics (besides Gentzen’s proof by transfinite
induction):
Any statement of that kind can be interpreted as if
its Gödel number coincide with that of its negation
The second dimension (for the second arithmetic)
allows the Gödel numbers of the statement and its
negation to be different always, i.e. for any
statement

30. Then once the Gödel incompleteness can be anyway
sidestepped, mathematics can found itself
consistently at a certain and rather surprising cost:
Mathematics turns out to be equivalent to the being
itself rather than to some true and thus limited part
of it: Of course, this might be called quantum
Pythagoreanism
Mathematics can self-ground only at the cost of
identifying with the world

31. Infinity is equivalent to a second and
independent finiteness
Two independent Peano arithmetics as well as
one single Hilbert space as an unification of
geometry and arithmetic are sufficient to the
self-foundation of mathematics
Quantum mechanics is inseparable from the
foundation of mathematics and thus from set
theory particularly

32. Dedekind, R. (1888) “Was sind und was sollen die Zahlen?“
Einstein, A., B. Podolsky, N. Rosen (1935) “Can Quantum-Mechanical
Description of Physical Reality Be Considered Complete?”
Gentzen, G. (1936) “Die Widerspruchfreiheit der reinen Zahlentheorie“
Gödel, K. (1931) “Über formal unentscheidbare Sätze der Principia
mathematica und verwandter Systeme I”
Hilbert, D. (1928) “Die Grundlagen Der Elementaren Zahlentheorie“
Russell, B. (1902) “Letter to Frege”
Schrödinger, E. (1935) “Die gegenwärtige Situation in der
Quantenmechanik”
Yourgrau, P. (2006) A World Without Time: The Forgotten Legacy of
Gödel and Einstein. New York: Perseus Books Group

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