Continuity and continuum in nonstandard universum
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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. Contents:. 1. M otivation 2. I nfinity and the axiom of choice

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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


Contents

Contents:

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


This file is only part 1 of the entire presentation and includes

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

:

:

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 motivation1

:

:

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 motivation2

:

:

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 Legendre-like transformation


1 motivation3

:

:

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 motivation4

:

:

1.Motivation

About that real problem, from which I had started, my conclusion was:

Thereare 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 motivation5

:

:

1.Motivation

After that, I noticed that the problem is very easily to be interpreted by transition within nonstandard universum between two nonstandardneighborhoods (ultrafilters) of two infinitely near standard points or between the standard subset and the properly nonstandard subset of nonstandard universum


1 motivation6

:

:

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 topicexiting those minds which are more eminent


1 motivation7

:

:

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


Contents1

Contents:

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


Infinity and the axiom of choice

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 choice1

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 choice2

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 postulateand 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 choice3

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 choice4

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 choice5

Infinity and the Axiom of Choice

The most difficult problems of the math offensive among infinity:

Which transfers are allowed by in-finity without producing paradoxes?

Which properties are suitable to be transferred into infinity?

How to dock infinities?


Infinity and the axiom of choice6

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 choice7

Infinity and the Axiom of Choice

Some other formulations or corollaries:

Any set can be well ordered (any its subset has a least element)

Zorn’s lema

Ultrafilter lema

Banach-Tarski paradox

Noncloning theorem in quantum information


Infinity and the axiom of choice8

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 choice9

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 choice10

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 choice11

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 choice12

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 choice13

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 choice14

Infinity and the Axiom of Choice

So that theBanach-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 choice15

Infinity and the Axiom of Choice

But theBanach-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 choice16

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 choice17

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 choice18

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 choice19

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 choice20

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 choice21

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 choice22

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 choice23

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 choice24

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 choice25

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 choice26

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 choice27

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 choice28

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 ofº-induction, namely “There are sets containing infinitely descending (membership) sequence OR without a (membership) minimal element,"


Infinity and the axiom of choice29

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 choice30

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 choice31

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 choice32

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 choice33

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.

Not (3a & 3b) is the new axiom of chance


Infinity and the axiom of choice34

Infinity and the Axiom of Choice

The sense of the Axiom of Choice:

Choice among infinite elements

Choice among infinite sets

Repetition of the already made choice among infinite elements

Repetition of the already made choice among infinite sets


Infinity and the axiom of choice35

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 choice36

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 choice37

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 choice38

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 choice39

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 choice40

are derived from

All these properties

Infinity and the Axiom of Choice

Classical Quantum

1. Copying, YesNo

2. Destroying,YesNo

3. Non-self-interacting, YesNo

4. Energetic medium, YesNo

5. Being in space-timeYesNo

The axiom of chance

No

Yes


Infinity and the axiom of choice41

Infinity and the Axiom of Choice

(No/Yes)?

The axiom of chance

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


Infinity and the axiom of choice42

Infinity and the Axiom of Choice

- Yes,

The axiom of chance

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)


Infinity and the axiom of choice43

Infinity and the Axiom of Choice

(No/Yes)?

The axiom of chance

How does the “2. Destroying” (Yes/No) descend from

“Destroying” is similar to copying:

As if negative copying


Infinity and the axiom of choice44

Infinity and the Axiom of Choice

(No/Yes)?

The axiom of chance

How does the “3. Non-self-interacting” (Yes/No) descend from

Self-interacting means

non-repeating by itself


Infinity and the axiom of choice45

Infinity and the Axiom of Choice

(No/Yes)?

The axiom of chance

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


Infinity and the axiom of choice46

Infinity and the Axiom of Choice

(No/Yes)?

The axiom of chance

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’ meansthat the well-ordering in space-time is conserved


Contents2

Contents:

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


Nonstandard universum

d

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Nonstandard universum

Abraham Robinson

(October 6, 1918

– April 11, 1974)

Leibnitz


Nonstandard universum1

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Nonstandard universum

Abraham Robinson

(October 6, 1918

– April 11, 1974)

His Book (1966)


Nonstandard universum2

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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 universum3

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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 universum4

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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 universum5

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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 xabout any standard point x of R.


Nonstandard universum6

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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xandybetween two neighbor well ordered standard pointsxandyofR.


Nonstandard universum7

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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 universum8

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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 universum9

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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 universum10

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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 universum11

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Nonstandard universum

Standardisation

IfAis a standard set and P any property, classical or otherwise, then there is a unique, standard subsetBofAwhose standard elements are precisely the standard elements ofAsatisfyingP(but the behaviour ofB's nonstandard elements is not prescribed).


Nonstandard universum12

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Nonstandard universum

Transfer

If all the parameters

A, B, C, ..., W

of a classical formulaFhave standard values then

F( x, A, B,..., W )

holds for all x's as soon as it holds for all standardxs.


Nonstandard universum13

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Nonstandard universum

The sense of the unary predicate standard:

If any formula holds for any finitestandard

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 universum14

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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 aconservative extensionof 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 universum15

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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 universum16

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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 universum17

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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 complementA`= S ­ AF


Nonstandard universum18

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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 universum19

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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 universum20

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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 thatNUcan describe the transition between internal and external set, which is our main problem


Nonstandard universum21

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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 universum22

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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 universum23

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Nonstandard universum

Thus we have already pioneered to Alain Connes’ introducing of infinitesimalsas compact Hilbert operators unlike the rest Hilbert operators representing transfor-mations of standard sets. He has suggested the following “dictionary”:

Complex variableHilbert operator

Real variableSelf-adjoint operator

InfinitesimalsCompact operator


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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


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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


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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 transitionbet-ween the coherent and collapsed state of the wave function Y and its inverse transition, i.e. of the collapse and de-collapse of Y.


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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 thetransition 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


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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.


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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


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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


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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:


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{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


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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


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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.


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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 Banach-Tarski paradox fails because a volume cannot be taken apart into points” (Wikipedia, “Smooth infinitesimal analysis”) “.Consequently, the axiom of choice fails too.


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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.


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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.


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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.


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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)


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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 fromthe bottom of 0. The numbers of N from the kind

{N/ 0  N} are especially important for our purpose


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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 aninde-pendent axiom in a special axiomset


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That special axiomset 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


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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


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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 betweenthe two neighbor intervals between three discrete successive points of the sequence in question


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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 betweenthe 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


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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


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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


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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


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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


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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

V = CP


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The coefficients ,  fromthe definition of qubit can be interpreted as generalized, complex possibilities of the coefficients ,  from relativity:

Qubit:

Relativity:

a = (1-b)1/2

b=v/c

a2+b2=1

a|0+b|1 = q


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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


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But we need some interpretation of complex probabilities, or, which is equi-valent, of complex nonstandard neigh-borhoods. If we rejectAF, 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


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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


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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 crestricts the ra-tio between two neighbor nonstandard intervals both either “up” or “down”


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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 pointh като площ на хистерезиса надолу и нагоре


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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 …


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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


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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


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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


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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 2 nd part

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.Anew axiom: of chance

7.Two kinds interpretation of quantum mechanics


That was all of 1 st part

CONTINUITY AND CONTINUUM

IN NONSTANDARD UNIVERSUM

That was all of 1st part

Vasil Penchev

Institute for Philosophical Research

Bulgarian Academy of Science

E-mail: [email protected]

Professional blog:

http://www.esnips.com/web/vasilpenchevsnews

Thank you for your attention!


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