Transcript Continuity and Continuum in Nonstandard Universum

Continuity and Continuum in Nonstandard
Universum
Vasil Penchev
Institute of Philosophical Research
Bulgarian Academy of Science
E-mail: [email protected]
Publications blog:
http://www.esnips.com/web/vasilpenchevsnews
1. Motivation
Contents:
2. Infinity and the axiom of choice
3. Nonstandard universum
4. Continuity and continuum
5. Nonstandard continuity between two infinitely
close standard points
6. A new axiom: of chance
7. Two kinds interpretation of quantum mechanics
This file is only Part 1 of the entire presentation and
includes:
1. Motivation
2. Infinity and the axiom of choice
3. Nonstandard universum
:
1. Motivation
:
My problem was:
Given: Two sequences:
: 1, 2, 3, 4, ….a-3, a-2, a-1, a
: a, a-1, a-2, a-3, …, 4, 3, 2, 1
Where a is the power of countable set
The problem:
Do the two sequences  and  coincide or not?
:
1. Motivation
:
At last, my resolution proved out:
That the two sequences:
: 1, 2, 3, 4, ….a-3, a-2, a-1, a
: a, a-1, a-2, a-3, …, 4, 3, 2, 1
coincide or not, is a new axiom (or two different
versions of the choice axiom): the axiom of chance:
whether we can always repeat or not an infinite choice
:
1. Motivation
:
For example, let us be given two Hilbert spaces:
: eit, ei2t, ei3t, ei4t, … ei(a-1)t, eiat
: eiat , ei(a-1)t, … ei4t, ei3t, ei2t, eit
An analogical problem is:
Are those two Hilbert spaces the same or not?
 can be got by Minkowski space  after Legendrelike transformation
:
1. Motivation
:
So that, if:
: eit, ei2t, ei3t, ei4t, … ei(a-1)t, eiat
: eiat , ei(a-1)t, … ei4t, ei3t, ei2t, eit
are the same, then Hilbert space
 is equivalent of the set of all the continuous world
lines in spacetime 
(see also Penrose’s twistors)
That is the real problem, from which I had started
:
1. Motivation
:
About that real problem, from which I had started, my
conclusion was:
There are two different versions about the transition
between the micro-object Hilbert space  and the
apparatus spacetime  in dependence on accepting
or rejecting of “the chance axiom”, but no way to be
chosen between them
:
1. Motivation
:
After that, I noticed that the problem is very easily to
be interpreted by transition within nonstandard
universum between two nonstandard neighborhoods
(ultrafilters) of two infinitely near standard points or
between the standard subset and the properly
nonstandard subset of nonstandard universum
:
1. Motivation
:
And as a result, I decided that only the
highly respected scientists from the honorable and
reverend department “Logic” are that appropriate
public worthy and deserving of being delivered
a report on that most intriguing and even maybe
delicate topic exiting those minds which are more
eminent
:
1. Motivation
:
After that, the very God was so benevolent so that
He allowed me to recognize marvelous
mathematical papers of a great Frenchman, Alain
Connes, recently who has preferred in favor of
sunny California to settle, and who, a long time ago,
had introduced nonstandard infinitesimals by
compact Hilbert operators
1. Motivation
Contents:
2. INFINITY and the AXIOM OF CHOICE
3. Nonstandard universum
4. Continuity and continuum
5. Nonstandard continuity between two infinitely close
standard points
6. A new axiom: of chance
7. Two kinds interpretation of quantum mechanics
Infinity and the Axiom of Choice

A few preliminary notes about how the knowledge of
infinity is possible: The short answer is: as that of
God: in belief and by analogy.The way of
mathematics to be achieved a little knowledge of
infinity transits three stages: 1. From finite perception
to Axioms 2. Negation of some axioms.
3. Mathematics beyond finiteness
Infinity and the Axiom of Choice

The way of mathematics to infinity:
1. From our finite experience and perception to
Axioms: The most famous example is the
axiomatization of geometry accomplished by Euclid
in his “Elements”
Infinity and the Axiom of Choice

The way of mathematics to infinity:
2. Negation of some axioms: the most frequently
cited instance is the fifth Euclid postulate and its
replacing in Lobachevski geometry by one of its
negations. Mathematics only starts from perception,
but its cognition can go beyond it by analogy
Infinity and the Axiom of Choice

The way of mathematics to infinity:
3. Mathematics beyond finiteness: We can postulate
some properties of infinite sets by analogy of finite
ones (e.g. ‘number of elements’ and ‘power’) However
such transfer may produce paradoxes: see as
example: Cantor “naive” set theory
Infinity and the Axiom of Choice

A few inferences about the math full-scale offensive
amongst the infinity:
1. Analogy: well-chosen appropriate properties of
finite mathematical struc-tures are transferred into
infinite ones
2. Belief: the transferred properties are postulated (as
usual their negations can be postulated too)
Infinity and the Axiom of Choice

The most difficult problems of the math offensive
among infinity:
1. Which transfers are allowed by in-finity without
producing paradoxes?
2. Which properties are suitable to be transferred
into infinity?
3. How to dock infinities?
Infinity and the Axiom of Choice

The Axiom of Choice (a formulation):
If given a whatever set A consisting of sets, we always
can choose an element from each set, thereby
constituting a new set B (obviously of the same power as A). So its sense is: we always can transfer the
property of choosing an element of finite set to
infinite one
Infinity and the Axiom of Choice

Some other formulations or corollaries:
1. Any set can be well ordered (any its subset has a
least element)
2. Zorn’s lema
3. Ultrafilter lema
4. Banach-Tarski paradox
5. Noncloning theorem in quantum information
Infinity and the Axiom of Choice

Zorn’s lemma is equivalent to the axiom of choice. Call a
set A a chain if for any two members B and C, either B is
a sub-set of C or C is a subset of B. Now con-sider a set
D with the properties that for every chain E that is a
subset of D, the union of E is a member of D. The lemma states that D contains a member that is maximal,
i.e. which is not a subset of any other set in D.
Infinity and the Axiom of Choice

Ultrafilter lemma: A filter on a set X is a collection
of nonempty subsets of X that is closed under
finite intersection and under superset. An
ultrafilter is a maximal filter. The ultrafilter
lemma states that every filter on a set X is a subset
of some ultrafilter on X (a maximal filter of
nonempty subsets of X.)
Infinity and the Axiom of Choice

Banach–Tarski paradox which says in effect that it is
possible to ‘carve up’ the 3-dimensional solid unit ball
into finitely many pieces and, using only rotation and
translation, reassemble the pieces into two balls each
with the same volume as the original. The proof, like
all proofs involving the axiom of choice, is an
existence proof only.
Infinity and the Axiom of Choice

First stated in 1924, the Banach-Tarski paradox states
that it is possible to dissect a ball into six pieces
which can be reassembled by rigid motions to form
two balls of the same size as the original. The number
of pieces was subsequently reduced to five by
Robinson (1947), although the pieces are extremely
complicated
Infinity and the Axiom of Choice

Five pieces are minimal, although four pieces are
sufficient as long as the single point at the center is
neglected. A generalization of this theorem is that
any two bodies in that do not extend to infinity and
each containing a ball of arbitrary size can be
dissected into each other (i.e., they are
equidecomposable)
Infinity and the Axiom of Choice

Banach-Tarski paradox is very important for quantum
mechanics and information since any qubit is
isomorphic to a 3D sphere. That’s why the paradox
requires for arbitrary qubits (even entire Hilbert
space) to be able to be built by a single qubit from its
parts by translations and rotations iteratively
repeating the procedure
Infinity and the Axiom of Choice

So that the Banach-Tarski paradox implies the
phenomenon of entanglement in quantum
information as two qubits (or two spheres) from one
can be considered as thoroughly entangled. Two
partly entangled qubits could be reckoned as sharing
some subset of an initial qubit (sphere) as if “qubits
(spheres) – Siamese twins”
Infinity and the Axiom of Choice

But the Banach-Tarski paradox is a weaker statement
than the axiom of choice. It is valid only about  3D
sets. But I haven’t meet any other additional
condition. Let us accept that the Banach-Tarski
paradox is equivalent to the axiom of choice for  3D
sets. But entanglement as well 3D space are physical
facts, and then…
Infinity and the Axiom of Choice

But entanglement (= Banach-Tarski paradox) as well
3D space are physical facts, and then consequently,
they are empirical confirmations in favor of the axiom
of choice. This proves that the Banach-Tarski paradox
is just the most decisive confirmation, and not at all, a
refutation of the axiom of choice.
Infinity and the Axiom of Choice

Besides, the axiom of choice occurs in the proofs of:
the Hahn-Banach the-orem in functional analysis, the
theo-rem that every vector space has a ba-sis,
Tychonoff's theorem in topology stating that every
product of compact spaces is compact, and the
theorems in abstract algebra that every ring has a
maximal ideal and that every field has an algebraic
closure.
Infinity and the Axiom of Choice

The Continuum Hypothesis:
The generalized continuum hypothesis (GCH) is not
only independent of ZF, but also independent of ZF
plus the axiom of choice (ZFC). However, ZF plus
GCH implies AC, making GCH a strictly stronger claim
than AC, even though they are both independent of
ZF.
Infinity and the Axiom of Choice

The Continuum Hypothesis:
The generalized continuum hypothesis (GCH) is: 2Na =
Na+1 . Since it can be formulated without AC,
entanglement as an argument in favor of AC is not
expanded to GCH. We may assume the negation of
GHC about cardinalities which are not “alefs”
together with AC about cardinalities which are alefs
Infinity and the Axiom of Choice

Negation of Continuum Hypothesis:
The negation of GHC about cardinali-ties which are
not “alefs” together with AC about cardinalities
which are alefs:
1. There are sets which can not be well ordered. A
physical interpretation of theirs is as physical objects
out of (beyond) space-time. 2. Entanglement about
all the space-time objects
Infinity and the Axiom of Choice

Negation of Continuum Hypothesis:
But the physical sense of 1. and 2.:
1. The non-well-orderable sets consist of well-ordered
subsets (at least, their elements as sets) which are
together in space-time. 2. Any well-ordered set
(because of Banach-Tarski paradox) can be as a set of
entangled objects in space-time
Infinity and the Axiom of Choice

Negation of Continuum Hypothesis:
So that the physical sense of 1. and 2. is ultimately:
The mapping between the set of space-time points
and the set of physical entities is a “many-many”
correspondence: It can be equivalently replaced by
usual mappings but however of a functional space,
namely by Hilbert operators
Infinity and the Axiom of Choice

Negation of Continuum Hypothesis:
Since the physical quantities have interpreted by
Hilbert operators in quantum mechanics and
information (correspondingly, by Hermitian and nonHermitian ones), then that fact is an empirical
confirmation of the negation of continuum
hypothesis
Infinity and the Axiom of Choice

Negation of Continuum Hypothesis:
But as well known, ZF+GHC implies AC. Since we have
already proved both NGHC and AC, the only
possibility remains also the negation of ZF (NZF),
namely the negation the axiom of foundation (AF):
There is a special kind of sets, which will call ‘inseparable sets’ and also don’t fulfill AF
Infinity and the Axiom of Choice

An important example of inseparable set: When
postulating that if a set A is given, then a set B always
exists, such one that A is the set of all the subsets of
B. An instance: let A be a countable set, then B is an
inseparable set, which we can call ‘subcountable set’.
Its power z is bigger than any finite power, but less
than that of a countable set.
Infinity and the Axiom of Choice

The axiom of foundation: “Every nonempty set is
disjoint from one of its elements.“ It can also be
stated as "A set contains no infinitely descending
(membership) sequence," or "A set contains a
(membership) minimal element," i.e., there is an
element of the set that shares no member with the
set
Infinity and the Axiom of Choice

The axiom of foundation
Mendelson (1958) proved that the equivalence of
these two statements necessarily relies on the axiom
of choice. The dual expression is called
º-induction, and is equivalent to the axiom itself (Ito
1986)
Infinity and the Axiom of Choice

The axiom of foundation and its negation: Since we
have accepted both the axiom of choice and the
negation of the axiom of foundation, then we are to
confirm the negation of º-induction, namely “There
are sets containing infinitely descending
(membership) sequence OR without a (membership)
minimal element,"
Infinity and the Axiom of Choice

The axiom of foundation and its negation: So that we
have three kinds of inseparable set: 1.“containing
infinitely descending (membership) sequence” 2.
“without a (membership) minimal element“ 3. Both 1.
and 2.
The alleged “axiom of chance” concerns only 1.
Infinity and the Axiom of Choice

The alleged “axiom of chance” concerning only 1.
claims that there are as inseparable sets “containing
infinitely descending (membership) sequence” as
such ones “containing infinitely ascending
(membership) sequence” and different from the
former ones
Infinity and the Axiom of Choice

The Law of the excluded middle:
The assumption of the axiom of choice is also
sufficient to derive the law of the excluded middle in
some constructive systems (where the law is not
assumed).
Infinity and the Axiom of Choice

A few (maybe redundant) commentaries:
We always can:
1. Choose an element among the elements of a set of
an arbitrary power
2. Choose a set among the sets, which are the
elements of the set A without its repeating
independently of the A power
Infinity and the Axiom of Choice

A (maybe rather useful) commentary:
We always can:
3a. Repeat the choice choosing the same element
according to 1.
3b. Repeat the choice choosing the same set
according to 2.
Infinity and the Axiom of Choice

The sense of the Axiom of Choice:
1. Choice among infinite elements
2. Choice among infinite sets
3. Repetition of the already made choice among
infinite elements
4. Repetition of the already made choice among
infinite sets
Infinity and the Axiom of Choice

The sense of the Axiom of Choice:
If all the 1-4 are fulfilled:
- choice is the same as among finite as among infinite
elements or sets;
- the notion of information being based on choice is
the same as to finite as to infinite sets
Infinity and the Axiom of Choice

At last, the award for your kind patience: The linkages
between my motivation and the choice axiom:
When accepting its negation, we ought to recognize a
special kind of choice and of information in relation
of infinite entities: quantum choice (=measuring) and
quantum information
Infinity and the Axiom of Choice

So that the axiom of choice should be divided into
two parts: The first part concerning quantum choice
claims that the choice between infinite elements or
sets is always possible. The second part concerning
quantum information claims that the made already
choice between infinite elements or sets can be
always repeated
Infinity and the Axiom of Choice

My exposition is devoted to the nega-tion only of the
“second part” of the choice axiom. But not more than
a couple of words about the sense for the first part to
be replaced or canceled: When doing that, we accept
a new kind of entities: whole without parts in principle, or in other words, such kind of superposition
which doesn’t allow any decoherence
Infinity and the Axiom of Choice

Negating the choice axiom second part is the
suggested “axiom of chance” properly speaking. Its
sense is: quantum information exists, and it is
different than “classical” one. The former differs from
the latter in five basic properties as following:
copying, destroying, non-self-interacting, energetic
medium, being in space-time: “Yes” about classical
and “No” about quantum information
Infinity and the Axiom of Choice

Classical Quantum
1. Copying,
Yes
No
2. Destroying,
Yes
No
3. Non-self-interacting, Yes
No
4. Energetic medium, Yes
No
5. Being in space-timeYes
No
Infinity and the Axiom of Choice

How does the “1. Copying” (Yes/No) descend from
(No/Yes)?
It is obviously: “Copying” means that a set of choices
is repeated, and
consequently, it has been able to be repeated
Infinity and the Axiom of Choice

If the case is: “1. Copying – No” from
- Yes,
then that case is the non-cloning theorem in
quantum information: No qubit can be copied
(Wootters, Zurek, 1982)
Infinity and the Axiom of Choice

How does the “2. Destroying” (Yes/No) descend from
(No/Yes)?
“Destroying” is similar to copying:
As if negative copying
Infinity and the Axiom of Choice

How does the “3. Non-self-interacting” (Yes/No)
descend from
(No/Yes)?
Self-interacting means
non-repeating by itself
Infinity and the Axiom of Choice

How does the “4. Energetic medium” (Yes/No)
descend from
(No/Yes)?
Energetic medium means for repeating to be turned
into substance, or in other words, to be carried by
medium obeyed energy conservation
Infinity and the Axiom of Choice

How does the “5. Being in space-time” (Yes/No)
descend from
(No/Yes)?
‘Being of a set in space-time’ means that the set is
well-ordered which fol-lows from the axiom of choice.
‘No axiom of chance’ means
that the wellordering in space-time is conserved
1. Motivation
Contents:
2. Infinity and the axiom of choice
3. NONSTANDARD UNIVERSUM
4. Continuity and continuum
5. Nonstandard continuity between two infinitely close
standard points
6. A new axiom: of chance
7. Two kinds interpretation of quantum mechanics
Nonstandard universum
Leibnitz
Abraham Robinson
(October 6, 1918
– April 11, 1974)
Nonstandard universum
His Book (1966)
Abraham Robinson
(October 6, 1918
– April 11, 1974)
Nonstandard universum
“It is shown in this book that
Leibniz ideas can be fully
vindicated and that they lead
to a novel and fruitful approach
to classical Analysis and many
other branches of
mathematics” (p. 2)
His Book (1966)
Nonstandard universum
“…G.W.Leibniz argued that the theory of
infinitesimals implies the introduction of ideal
numbers which might be infinitely small or
infinitely large compared with the real numbers
but which were to possess the same
properties as the latter.” (p. 2)
Nonstandard universum
The original approach of A. Robinson:
1. Construction of a nonstandard model of R (the real
continuum): Nonstan-dard model (Skolem 1934): Let
A be the set of all the true statements about R, then:
 = A(c>0, c>0`, c>0``…): Any finite subset of  holds
for R. After that, the finiteness principle
(compactness theorem) is used:
Nonstandard universum
2. The finiteness principle: If any fi-nite subset of a
(infinite) set  posses-ses a model, then the set 
possesses a model too. The model of  is not
isomorphic to R & A and it is a nonstandard
universum over R & A. Its sense is as follow: there is a
nonstandard neighborhood x about any standard
point x of R.
Nonstandard universum
The properties of nonstandard neighborhood x
about any standard point x of R: 1) The “length” of x
in R or of any its measurable subset is 0. 2) Any x in
R is isomorphic to (R & A) itself. Our main problem is
about continuity and continuum of two
neighborhoods x and y between two neighbor
well ordered standard points x and y of R.
Nonstandard universum
Indeed, the word of G.W.Leibniz “that the theory of
infinitesimals implies the introduction of ideal
numbers which might be infinitely small or infinitely
large compared with the real numbers but which
were to possess the same properties as the latter”
(Robinson, p. 2) are really accomplished by
Robinson’s nonstandard analysis.
Nonstandard universum
Another possible approach was developed by was
developed in the mid-1970s by the mathematician
Edward Nelson. Nelson introduced an entirely
axiomatic formulation of non-standard analysis that
he called Internal Set Theory or IST. IST is an
extension of Zermelo-Fraenkel set theory or it is a
conservative extension of ZFC.
Nonstandard universum
In IST alongside the basic binary membership
relation , it introduces a new unary predicate
standard which can be applied to elements of the
mathematical universe together with three axioms
for reasoning with this new predicate (again IST): the
axioms of Idealization, Standardization, Transfer
Nonstandard universum
Idealization:
For every classical relation R, and for arbit-rary values for all
other free variables, we have that if for each standard, finite
set F, there exists a g such that R(g, f ) holds for all f in F,
then there is a particular G such that for any standard f we
have R (G, f ), and conversely, if there exists G such that for
any standard f, we have R(G, f ), then for each finite set F,
there exists a g such that R(g, f ) holds for all f in F.
Nonstandard universum
Standardisation
If A is a standard set and P any property, classical or
otherwise, then there is a unique, standard subset B
of A whose standard elements are precisely the
standard elements of A satisfying P (but the
behaviour of B's nonstandard elements is not
prescribed).
Nonstandard universum
Transfer
If all the parameters
A, B, C, ..., W
of a classical formula F have standard values
then
F( x, A, B,..., W )
holds for all x's as soon as it holds for all
standard xs.
Nonstandard universum
The sense of the unary predicate standard:
If any formula holds for any finite standard
set of standard elements, it holds for all the universum. So
that standard elements are only those which establish, set
the standards, with which all the elements must be in
conformity: In other words, the standard elements, which
are always as finite as finite number, establish, set the
standards about infinity. Next, …
Nonstandard universum
So that the suggested by Nelson IST is a constructivist
version of nonstandard analysis. If ZFC is consistent, then
ZFC + IST is consistent. In fact, a stronger statement can be
made: ZFC + IST is a conservative extension of ZFC: any
classical formula (correct or incorrect!) that can be proven
within internal set theory can be proven in the ZermeloFraenkel axioms with the Axiom of Choice alone.
Nonstandard universum
The basic idea of both the version of nonstandard
analysis (as Roninson’s as Nelson’s) is repetition of all
the real continuum R at, or better, within any its point
as nonstandard neighborhoods about any of them.
The consistency of that repetition is achieved by the
notion of internal set (i.e. as if within any standard
element)
Nonstandard universum
That collapse and repetition of all infinity into any its
point is accomp-lished by the notion of ultrafilter in
nonstandard analysis. Ultrafilter is way to be
transferred and thereby repeated the topological
properties of all the real continuum into any its point,
and after that, all the properties of real conti-nuum to
be recovered from the trans-ferred topological
properties
Nonstandard universum
What is ‘ultrafilter’?
Let S be a nonempty set, then an ultrafilter on S is a
nonempty collection F of subsets of S having the following
properties:
1.   F.
2. If A, B  F, then A, B  F .
3. If A,B  F and ABS, then A,B  F
4. For any subset A of S, either A  F or its complement A`=
S AF
Nonstandard universum
Ultrafilter lemma: A filter on a set X is a collection
of nonempty subsets of X that is closed under finite
intersection and under superset. An ultrafilter is a
maximal filter. The ultrafilter lemma states that every
filter on a set X is a subset of some ultrafilter on X (a
ma-ximal filter of nonempty subsets of X.)
Nonstandard universum
A philosophical reflection: Let us remember the BanachTarski paradox: entire Hilbert space can be delivered only
by repetition ad infinitum of a single qubit (since it is
isomorphic to 3D sphere)as well the paradox follows from
the axiom of choice. However nonstandard analysis
carries out the same idea as the BanachTarski paradox about 1D sphere, i.e. a point:
all the nonstandard universum can be
recovered from a point, since the universum
is within it
Nonstandard universum
The philosophical reflection continues: That’s why
nonstandard analysis is a good tool for quantum
mechanics: Nonstandard universum (NU) possesses as if
fractal structure just as Hilbert space. It allows all quantum
objects to be described as internal sets absolutely similar to
macro-objects being described as external or standard sets.
The best advantage is that NU can describe the transition
between internal and external set, which is our main
problem
Nonstandard universum
Something still a little more: If Hilbert spa-ce is isomorphic
to a well ordered sequence of 3D spheres delivered by the
axiom of choice via the Banach-Tarski paradox, then 1. It is
at least comparable unless even iso-morphic to Minkowski
space; 2. It is getting generalized into nonstandard
universum as to arbitrary number dimensions, and even as
to fractional number dimensions as we will see. So that
qubit is getting generalized into internal set with ultrafilter
structure
Nonstandard universum
And at last: The generalized so Hilbert space as
nonstandard universum is delivered again by the axiom of
choice but this time via Zorn’s lemma (an equivalent to the
axiom of choice) via ultrafilter lemma (a weaker statement
than the axiom of choice). Nonstandard universum admits
to be in its turn generalized as in the gauge theories, when
internal and external set differ in structure, as in varying
the nonstandard connection between two points as we will
do
Nonstandard universum
Thus we have already pioneered to Alain Connes’
introducing of infinitesimals as compact Hilbert operators
unlike the rest Hilbert operators representing transformations of standard sets. He has suggested the following
“dictionary”:
Complex variable
Real variable
Infinitesimals
Hilbert operator
Self-adjoint operator
Compact operator
Nonstandard universum
The sense of compact operator: if it is ap-plied to
nonstandard universum, it trans-forms a nonstandard
neighborhood into a nonstandard neighborhood, so that it
keeps division between standard and nonstandard
elements. If the nonstandard universum is built on Hilbert
space instead of on real continuum, then Connes defined
infinite-simals on the Cartesian product of Hilbert spaces.
So that it requires the axiom of choice for the existence of
Cartesian product
Nonstandard universum
I would like to display that Connes’ infinitesimals possesses
an exceptionally important property: they are
infinitesimals both in Hilbert and in Minkowski space: so
that they describe very well transformations of Minkowski
space into Hilbert space and vice versa: Math speaking,
Minkowski operator is compact if and only if it is compact
Hilbert operator. You might kindly remember that
transformations between those spaces was my initial
motivation
Nonstandard universum
Minkowski operator is compact if and only if it is compact
Hilbert operator. Before a sketch of proof, its sense and
motivation: If we describe the transformations of Minkowski space into Hilbert space and vice versa, we will be able
to speak of the transition between the apparatus and the
microobject and vice versa as well of the transition between the coherent and collapsed state of the wave function
Y and its inverse transition, i.e. of the collapse and decollapse of Y.
Nonstandard universum
Before a sketch of proof, its sense and motivation: Our
strategic purpose is to be built a united, common language
for us to be able to speak both of the apparatus and of the
microobject as well, and the most impor-tant, of the
transition and its converse bet-ween them. The creating of
such a language requires a different set-theory foundation
including: 1. The axiom of choice. 2. The foundation axiom
negation. 3. The generalized continuum hypothesis
negation
Nonstandard universum
Before a sketch of proof, its sense and motivation: The
axiom of foundation is available in quantum mechanics by
the collapse of wave function. Let us represent the coherent
state as infinity since, if the Hilbert space is separable, then
any its point is a coherent superposition of a countable set
of components. The “collapse” represents as if a descending
avalanche from the infinity to some finite value observed
with various probability.
Nonstandard universum
Before a sketch of proof, its sense and motivation: If that’s
the case, the axiom of foundation AF is available just as the
requirement for the wave function to collapse from the
infinity as an avalanche since AF forbids a smooth,
continuous, infinite lowering, sinking. It would be an
equivalent of the AF negation. A smooth, continuous,
infinite process of lowering admits and even suggests the
possibility of its reversibility
Nonstandard universum
A note: Let us accept now the AF negation, and
consequently , a smooth reversibility between coherent and
“collapsed” state. Then: P = Ps Pr, where Ps is the
probability from the coherent superposition to a given
value, and Pr is the probability of reversible process. So that
the quantum mechanical probability attached to any
observable state could be interpreted as a finite relation
between two infinities
Nonstandard universum
A Minkowski operator is compact if and only if it is a
compact Hilbert operator. A sketch of proof:
Wave function Y: RR  RR
Hilbert space: {RR}  {RR}
Hilbert operators:
{RR}  {RR}  {RR}  {RR}
Using the isomorphism of Möbius and Lorentz group as
follows:
Nonstandard universum
{RR}  {RR}  {RR}  {RR}
 (the isomorphism)
{RR  R}R  {RR  R}R:
i.e. Minkowski space operators.
The sense of introducing of nonstandard infinitesimals by
compact Hilbert operators is for them to be invariant
towards (straight and inverse) transformations between
Hilbert space and Minkowski space
Nonstandard universum
A little comment on the theorem:
A Minkowski operator is compact if and only if it is a
compact Hilbert operator
Defining nonstandard infinitesimals as compact Hilbert
operators we are introducing infinitesimals being able to
serve both such ones of the transition between Minkowski
and Hilbert space (the apparatus and the microobject) and
such ones of both spaces
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A little more comment on the theorem:
Let us imagine those infinitesimals, being operators, as sells
of phase space: they are smoothly decreasing from the
minimal cell of the apparatus phase space via and beyond
the axiom of foundation to zero, what is the phase space
sell of the microobject. That decreasing is to be described
rather by Jacobian than Hamiltonian or Lagrangian
Nonstandard universum
A little more comment on the theorem:
Hamiltonian describes a system by two independent linear
systems of equalities [as if towards the reference frame
both of the apparatus (infinity) and of microobject
(finiteness)]
Lagrangian does the same by a nonlinear system of
equalities [the current curvature is relation between the
two reference frames above]
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A little more comment on the theorem:
Jacobian describes the bifurcation, two-forked direction(s)
from a nonlinear system to two linear systems when the
one united, common description is already impossible and
it is disintegrating to two independent each of other
descriptions
Jacobian describes as well entanglement as bifurcations
and such process.
Nonstandard universum
A few slides are devoted to alternative ways for
nonstandard infinitesimals to be introduced:
- smooth infinitesimal analysis
- surreal numbers.
Both the cases are inappropriate to our purpose or
can be interpreted too close-ly or even identical to
the nonstandard infinitesimal of A. Robinson
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“Intuitively, smooth infinitesimal analysis can be
interpreted as describing a world in which lines are made
out of infinitesimally small segments, not out of points.
These seg-ments can be thought of as being long enough to
have a definite direction, but not long enough to be curved.
The construction of discontinuous functions fails because a
function is identified with a curve, and the curve cannot be
constructed pointwise” (Wikipedia, “Smooth …”)
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“We can imagine the intermediate value theorem's
failure as resulting from the ability of an infinitesimal
segment to straddle a line. Similarly, the BanachTarski paradox fails because a volume cannot be
taken apart into points” (Wikipedia, “Smooth
infinitesimal analysis”) “. Consequently, the axiom of
choice fails too.
Nonstandard universum
The infinitesimals x in smooth infinitesimal analysis
are nilpotent (nilsquare): x2=0 doesn’t mean and
require that x is necessarily zero. The law of the
excluded middle is denied: the infinitesimals are such
a middle, which is between zero and nonzero. If that’s
the case all the functions are continuous and
differentiable infinitely.
Nonstandard universum
The smooth infinitesimal analysis does not satisfy our
requirements even only because of denying the
axiom of choice or the Banach - Tarski paradox. But I
think that another version of nilpotent infinitesimals
is possible, when they are an orthogonal basis of
Hilbert space and the latter is being transformed by
compact operator. If that’s the case, it is too similar to
Alain Connes’ ones.
Nonstandard universum
By introducing as zero divisors, the infinitesimals are
interested because of possibility for the phase space
sell to be zero still satisfying uncertainty. It means
that the bifurcation of the initial nonlinear reference
frame to two linear frames correspondingly of the
apparatus and of the object is being represented by
an angle decreasing from p/2 to 0.
Nonstandard universum
The infinitesimals introduced as surreal numbers
unlike hyperreal numbers (equal to Robinson’s
infinitesimals):
Definition: “If L and R are two sets of surreal numbers
and no member of R is less than or equal to any
member of L then { L | R } is a surreal number”
(Wikipedia, “Surreal numbers).
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About the surreal numbers:They are a proper class
(i.e. are not a set), ant the biggest ordered field (i.e.
include any other field). Comparison rule: “For a
surreal number x = { XL | XR } and y = { YL | YR } it holds
that x ≤ y if and only if y is less than or equal to no
member of XL, and no member of YR is less than or
equal to x.” (Wikipedia)
Nonstandard universum
Since the comparison rule is recursive, it requires
finite or transfinite induction . Let us now consider
the following subset N of surreal numbers: All the
surreal numbers S  0. 2N has to contain all the well
ordered falling sequences from the bottom of 0. The
numbers of N from the kind
{N/ 0  N} are especially important for our purpose
Nonstandard universum
For example, we can easily to define our initial
problem in their terms:
Let  and  be:
 = {q: q  {N | 0}}
 = {w: w  {0 | 0  N}}
Our problem is whether  and  co-incide or not? If
not, what is power of   ? Our hypothesis is: the
ans-wer of the former question is an inde-pendent
axiom in a special axiom set
Nonstandard universum
That special axiom set includes: the axiom of choice
and a negation of the generalized continuum
hypothesis (GCH). Since the axiom of choice is a
corollary from ZF+GCH, it implies a negation of ZF,
namely: a negation of the axiom of foundation AF in
ZF. If ZF+GCH is the case, our problem does not arise
since the infinite degres-sive sequences  are
forbidden by AF
Nonstandard universum
However a permission and introducing of the infinite
degressive sequences , and consequently, a AF
negation is required by quantum information, or
more particularly, by a discussing whether Hilbert
and Minkowski space are equivalent or not, or more
generally, by a considering whether any common
language about the apparatus & the microobject is
possible
Nonstandard universum
Comparison between “standard” and nonstandard
infinitesimals. The“standard” infinitesimals exist
only in boundary transition. Their sense represents
velocity for a point-focused sequence to converge to
that point. That velocity is the ratio between the two
neighbor intervals between three discrete successive
points of the sequence in question
Nonstandard universum
More about the sense of “standard” infinitesimals: By
virtue of the axiom of choice any set can be well ordered as
a sequence and thereby the ratio between the two neighbor
intervals between three discrete successive points of the
sequence in question is to exist just as before: in the proper
case of series. However now, the “neighbor” points of an
arbitrary set are not discrete and consequently the intervals
between them are zero
Nonstandard universum
Although the “neighbor” points of an arbit-rary set are not
discrete, and consequently, the intervals between them are
zero, we can recover as if “intervals” between the well-
ordered as if “discrete” neighbor points by means of
nonstandard infini-tesimals. The nonstandard
infinitesimals are such intervals. The representation of
velocity for a sequence to converge remains in force by the
nonstandard infinitesimals
Nonstandard universum
But the ratio of the neighbor intervals can be also
considered as probability, thereby the velocity itself
can be inter-preted as such probability as above. Two
opposite senses of a similar inter-pretation are
possible: 1) about a point belonging to the sequence:
as much the velocity of convergence is higher as
the probability of a point of the series in question to
be there is bigger;
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2) about a point not belonging to the sequence: as
much the velocity of convergence is higher as the
probability of a point out of the series in question to
be there is less; i.e. the sequence thought as a process
is steeper, and the process is more nonequilibrium,
off-balance, dissipative while a balance, equilibrium,
non-dissipative state is much more likely in time
Nonstandard universum
The same about a cell of phase space:
The same can be said of a cell of phase space: as much
a process is steeper, and the process is more
nonequilibrium, off-balance, dissipative as the
probability of a cell belonging to it is higher
while a balance, equilibrium, non-dissipative state
out of that cell is much more likely in time
Nonstandard universum
Our question is how the probability in quantum
mechanics should be interpre-ted? A possible
hypothesis is: the pro-babilities of non-commutative,
comple-mentary quantities are both the kinds
correspondingly and interchangeably.
For example, the coordinate probability corresponds
to state, and the momentum probability to process.
But that is rather an analogy
Nonstandard universum
The physical interpretation of the velo-city for a
series to converge is just as velocity of some physical
process. If the case is spatial motion, then the connection between velocity and probability is fixed by
the fundamental constant c:
Where: v is velocity, p is probability
Nonstandard universum
The coefficients ,  from the definition of qubit can
be interpreted as generalized, complex possibilities of
the coefficients ,  from relativity:
Qubit:
2
2
 + =1
|0+|1 = q
Relativity:
1/2
 = (1-)
=v/c
Nonstandard universum
The interpretation of the ratio between nonstandard
infinitesimals both as velocity and as probability. The
ratio between “stanadard” infinitesimals which exist
only in boundary transit
Nonstandard universum
But we need some interpretation of complex
probabilities, or, which is equi-valent, of complex
nonstandard neigh-borhoods. If we reject AF, then we
can introduce the falling, descending from the
infinity, but also infinite series as purely, properly
imaginary nonstandard neighborhoods: The real
components go up to infinity. The imaginary ones go
down to finiteness
Nonstandard universum
After that, all the complex probabilities are ushered
in varying the ties, “hyste-reses” “up” or “down”
between two well ordered neighbor standard points.
Wave function being or not in separable Hilbert space
(i.e. with countable or non-countable power of its
components) is well interpreted as nonstandard
straight line (or its rational subset). Operators
transform such lines
Nonstandard universum
Consequently, there exists one more bridge of
interpretation connecting Hilbert and 3D or
Minkowski space.
What do the constants c and h inter-pret from the
relations and ratios bet-ween two neighbor
nonstandard inter-vals? It turns out that c restricts
the ra-tio between two neighbor nonstandard
intervals both either “up” or “down”
Nonstandard universum
And what about the constant h? It guarantees on
existing of: both the sequences, both the
nonstandard neighborhoods “up” and “down”. It is
the unit of the central symmetry transforming
between the nonstandard neighborhoods “up” and
“down” of any standard point h като площ на
хистерезиса надолу и нагоре
Nonstandard universum
And what about the constant h? It gua-rantees on
existing of: both the sequen-ces, both the
nonstandard neighbor-hoods “up” and “down”. It is
the unit of the central symmetry transforming
between the nonstandard neighborhoods “up” and
“down” of any stan-dard point. However another
interpretation is possible about the constant h …
Nonstandard universum
One more interpretation of h: as the square of the
hysteresis between the “up” and the “down”
neighborhood between two standard points. Unlike
standard continuity a parametric set of nonstandard
continuities is available. The parameter g = Dp/Dx =
Dm/Dt =
= (DE)2/c2h displays the hysteresis “rectangularity”
degree
Nonstandard universum
One more interpretation of h: The sense of g is
intuitively very clear: As more points “up” and
“down” are common as both the hysteresis branches
are closer. So the standard continuity turns out an
extreme peculiar case of nonstan-dard continuity,
namely all the points “up” and “down” are common
and both the hysteresis branches coincide: The
hysteresis is canceled
Nonstandard universum
By means of the latter interpretation we can interpret
also phase space as non-standard 3D space. Any cell
of phase space represents the hysteresis between 3D
points well ordered in each of the three dimensions.
The connection bet-ween phase space and Hilbert
space as different interpretation of a basic space,
nonstandard 3D space, is obvious
Nonstandard universum
What do the constants c and h interpret as limits of a
phase space cell deformation?
c.1.dx  dy  h.dx
Here 1 is the unit of curving [distance x mass]
Forthcoming in 2nd part:
1. Motivation
2. Infinity and the axiom of choice
3. Nonstandard universum
4. Continuity and continuum
5. Nonstandard continuity between two
infinitely close standard points
6. A new axiom: of chance
7. Two kinds interpretation of quantum
mechanics
CONTINUITY AND CONTINUUM
IN NONSTANDARD UNIVERSUM
Vasil Penchev
Institute for Philosophical Research
Bulgarian Academy of Science
E-mail: [email protected]
Professional blog:
http://www.esnips.com/web/vasilpenchevsnews
st
That was all of 1 part
Thank you for your attention!