Continuity and Continuum in Nonstandard Universum
Vasil PenchevInstitute of Philosophical Research
Bulgarian Academy of ScienceE-mail: [email protected]
Publications blog: http://www.esnips.com/web/vasilpenchevsnews
Contents:1. Motivation
2. Infinity and the axiom of choice
3. Nonstandard universum4. Continuity and continuum5. Nonstandard continuity
between two infinitely close standard points
6. A new axiom: of chance7. Two kinds interpretation of
quantum mechanics
1. Motivation2. Infinity and the axiom of
choice3. Nonstandard universum
This file is only Part 1 of the entire presentation and
includes:
1. MotivationMy 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 setThe 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. MotivationFor 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 Legendre-like
transformation
::
1. MotivationSo 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. MotivationAbout 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
::
Contents:1. Motivation
2. INFINITY and the AXIOM OF CHOICE3. Nonstandard universum4. Continuity and continuum5. Nonstandard continuity
between two infinitely close standard points
6. A new axiom: of chance7. 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 ones2. 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 po-wer 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 lema4.Banach-Tarski paradox5.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 lem-ma 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 non-Hermitian 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 ‘insepa-rable 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 foundationMendelson (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 power2. 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 sets3.Repetition of the already made choice among infinite
elements4.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 prin-ciple, 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 Quantum1. Copying, YesNo2. Destroying, Yes
No3. Non-self-interacting, Yes
No4. Energetic medium, Yes
No5. Being in space-timeYes
No
Infinity and the Axiom of Choice
How does the “1. Copying” (Yes/No) descend from
It is obviously: “Copying” means that a set of choices is repeated, and consequently, it has been able to be repeated
(No/Yes)?
Infinity and the Axiom of Choice
If the case is: “1. Copying – No” from
then that case is the non-cloning theorem in quantum information: No qubit can be copied (Wootters, Zurek, 1982)
- Yes,
Infinity and the Axiom of Choice
How does the “2. Destroying” (Yes/No) descend from
“Destroying” is similar to copying: As if negative copying
(No/Yes)?
Infinity and the Axiom of Choice
How does the “3. Non-self-interacting” (Yes/No) descend from
Self-interacting meansnon-repeating by itself
(No/Yes)?
Infinity and the Axiom of Choice
How does the “4. Energetic medium” (Yes/No) descend
from
Energetic medium means for repeating to be turned into substance, or in other words, to be carried by medium obeyed energy conservation
(No/Yes)?
Infinity and the Axiom of Choice
How does the “5. Being in space-time” (Yes/No) descend from
‘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 well-ordering in space-time is
conserved
(No/Yes)?
Contents:1. Motivation
2. Infinity and the axiom of choice3. NONSTANDARD UNIVERSUM4. Continuity and continuum5. Nonstandard continuity
between two infinitely close standard points
6. A new axiom: of chance7. Two kinds interpretation of
quantum mechanics
Nonstandard universum
Abraham Robinson (October 6, 1918 – April 11, 1974)Leibnitz
Nonstandard universum
Abraham Robinson (October 6, 1918 – April 11, 1974)His Book (1966)
Nonstandard universum
His Book (1966)
“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)
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 universumIdealization:
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
StandardisationIf 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 universumTransfer
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 Zermelo-Fraenkel 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 universumThat 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 ABS, 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 Banach-Tarski
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 Banach-Tarski 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 transfor-mations of standard sets. He has
suggested the following “dictionary”:
Complex variable Hilbert operator Real variable Self-adjoint operatorInfinitesimals 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 Minkow-ski 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 bet-ween the
coherent and collapsed state of the wave function and its inverse transition, i.e. of the collapse and de-collapse of .
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 : RR RR
Hilbert space: {RR} {RR}Hilbert operators:
{RR} {RR} {RR} {RR}Using the isomorphism of Möbius and Lorentz group as follows:
Nonstandard universum
{RR} {RR} {RR} {RR} (the isomorphism)
{RR R}R {RR 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
Nonstandard universum
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]
Nonstandard universum
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
Nonstandard universum“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
…”)
Nonstandard universum
“We can imagine the intermediate value theorem's failure as resulting from the
ability of an infinitesimal segment to straddle a line. Similarly, the Banach-Tarski
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 universumThe 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 /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).
Nonstandard universum
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 asthe probability of a point of the series in question to be
there is bigger;
Nonstandard universum
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 con-nection 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: Relativity:2+2=1|0+|1 = q
= (1-)1/2
=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 = p/x = m/t == (E)2/c2h displays the
hysteresis “rectangularity” degree
Nonstandard universum
One more interpretation of h: The sense of 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.dxHere 1 is the unit of curving
[distance x mass]
Forthcoming in 2nd part:1. Motivation2. Infinity and the axiom of choice3. Nonstandard universum
4. Continuity and continuum5. Nonstandard continuity between two infinitely close standard points6. A new axiom: of chance7. Two kinds interpretation of quantum mechanics
That was all of 1st part
Thank you for your attention!
CONTINUITY AND CONTINUUM IN NONSTANDARD UNIVERSUM
Vasil PenchevInstitute for Philosophical Research
Bulgarian Academy of ScienceE-mail: [email protected]
Professional blog:http://www.esnips.com/web/vasilpenchevsnews
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