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: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

1. Motivation



This file is only Part 1 of the entire presentation and

includes:

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 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. 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. Motivation

So that, if:: eit, ei2t, ei3t, ei4t, … ei(a-1)t, eiat : eiat , ei(a-1)t, … ei4t, ei3t, ei2t, eit

are the same, then Hilbert space

is equivalent of the set of all the continuous world lines in

spacetime (see also Penrose’s twistors)

That is the real problem, from which I had started

::

1. Motivation

About that real problem, from which I had started, my

conclusion was:There are two different

versions about the transition between the micro-object

Hilbert space and the apparatus spacetime in

dependence on accepting or rejecting of “the chance axiom”, but no way to be

chosen between them

::

1. Motivation

After that, I noticed that the problem is very easily to be interpreted by transition

within nonstandard universum between two nonstandard

neighborhoods (ultrafilters) of two infinitely near standard

points or between the standard subset and the

properly nonstandard subset of nonstandard universum

::

1. Motivation

And as a result, I decided that only the

highly respected scientists from the honorable and

reverend department “Logic” are that appropriate public

worthy and deserving of being delivered

a report on that most intriguing and even maybe delicate topic exiting those

minds which are more eminent

::

1. Motivation

After that, the very God was so benevolent so that He allowed me to recognize marvelous mathematical

papers of a great Frenchman, Alain Connes, recently who

has preferred in favor of sunny California to settle, and

who, a long time ago, had introduced nonstandard

infinitesimals by compact Hilbert operators

::

Contents:1. Motivation

2. 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


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”



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



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


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)


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?


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


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


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.


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.)


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.


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


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)


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


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”


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…


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.


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.


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.


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


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



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



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



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



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


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.


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


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)


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,"


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.


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


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).


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


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.


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


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


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


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


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


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


Classical Quantum1. Copying, YesNo2. Destroying, Yes

No3. Non-self-interacting, Yes

No4. Energetic medium, Yes

No5. Being in space-timeYes

No


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)?


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,


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

“Destroying” is similar to copying: As if negative copying

(No/Yes)?


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

Self-interacting meansnon-repeating by itself

(No/Yes)?


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)?


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


3. NONSTANDARD UNIVERSUM





Nonstandard universum

Abraham Robinson (October 6, 1918 – April 11, 1974)Leibnitz


Abraham Robinson (October 6, 1918 – April 11, 1974)His Book (1966)


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)


“…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)


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:


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.


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.


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.


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.


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.


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.


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, …


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.


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)


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


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


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.)


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


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


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


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


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


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


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


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 .


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


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.


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


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


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:


{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


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


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



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]



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.


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


“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

…”)


“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.


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.


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.


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.


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).


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)


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


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


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


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


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


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


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


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;


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


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


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


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


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


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


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


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


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”


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


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 …


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


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


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


What do the constants c and

h interpret as limits of a phase space cell deformation?

c.1.dx dy h.dx

Here 1 is the unit of curving [distance x mass]

Forthcoming in 2nd part:1. 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

continuity and continuum in nonstandard universum

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