Continuity%20and%20Continuum%20in%20Nonstandard%20Universum

1
Continuity and Continuum in Nonstandard Universum

• Vasil Penchev
• Institute of Philosophical Research
• Bulgarian Academy of Science
• E-mail vasildinev_at_gmail.com
• Publications blog http//www.esnips.com/web/vasil
  penchevsnews

2
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

3
This file is only Part 1 of the entire
presentation and includes

• 1. Motivation
• 2. Infinity and the axiom of choice
• 3. Nonstandard universum

4
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?

5
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

6
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

7
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 Penroses twistors)
• That is the real problem, from which I had started

8
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

9
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

10
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

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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

12
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

13
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

14
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

15
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

16
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

17
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)

18
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?

19
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

20
Infinity and the Axiom of Choice
?

• Some other formulations or corollaries
• Any set can be well ordered (any its subset has a
  least element)
• Zorns lema
• Ultrafilter lema
• Banach-Tarski paradox
• Noncloning theorem in quantum information

21
Infinity and the Axiom of Choice
?

• Zorns 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.

22
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.)

23
Infinity and the Axiom of Choice
?

• BanachTarski 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.

24
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

25
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)

26
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. Thats 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

27
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

28
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 havent 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

29
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.

30
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.

31
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.

32
Infinity and the Axiom of Choice
?

• The Continuum Hypothesis
• The generalized continuum hypothesis (GCH) is
  2Na Na1 . 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

33
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

34
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

35
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

36
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

37
Infinity and the Axiom of Choice
?

• Negation of Continuum Hypothesis
• But as well known, ZFGHC 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
  dont fulfill AF

38
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.

39
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

40
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)

41
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,"

42
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.

43
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

44
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).

45
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

46
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
47
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

48
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

49
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

50
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

51
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 doesnt
  allow any decoherence

52
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

53
Infinity and the Axiom of Choice
are derived from
?

• Classical Quantum
• 1. Copying, Yes No
• 2. Destroying, Yes No
• 3. Non-self-interacting, Yes No
• 4. Energetic medium, Yes No
• 5. Being in space-time Yes No

All these properties
The axiom of chance
No
Yes
54
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)?
The axiom of chance
55
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,
The axiom of chance
56
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)?
The axiom of chance
57
Infinity and the Axiom of Choice
?

• How does the 3. Non-self-interacting (Yes/No)
  descend from
• Self-interacting means
• non-repeating by itself

(No/Yes)?
The axiom of chance
58
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)?
The axiom of chance
59
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)?
The axiom of chance
60
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

61
Nonstandard universum
d
d
Abraham Robinson (October 6, 1918 April 11,
1974)
Leibnitz
62
Nonstandard universum
d
d
Abraham Robinson (October 6, 1918 April 11,
1974)
His Book (1966)
63
Nonstandard universum
d
d
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)
64
Nonstandard universum
d
d

• 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)

65
Nonstandard universum
d
d

• 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?(cgt0, cgt0,
  cgt0) Any finite subset of ? holds for R.
  After that, the finiteness principle (compactness
  theorem) is used

66
Nonstandard universum
d
d

• 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.

67
Nonstandard universum
d
d

• 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.

68
Nonstandard universum
d
d

• 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 Robinsons nonstandard analysis.

69
Nonstandard universum
d
d

• 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.

70
Nonstandard universum
d
d

• 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

71
Nonstandard universum
d
d

• 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.

72
Nonstandard universum
d
d

• Standardisation
• If A is a standard set and P any property,
  classical or otherwise, then there is a unique,
  standard subset B of A whose standard elements
  are precisely the standard elements of A
  satisfying P (but the behaviour of B's
  nonstandard elements is not prescribed).

73
Nonstandard universum
d
d

• Transfer
• If all the parameters
• A, B, C, ..., W
• of a classical formula F have standard values
  then
• F( x, A, B,..., W )
• holds for all x's as soon as it holds for all
  standard xs.

74
Nonstandard universum
d
d

• 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,

75
Nonstandard universum
d
d

• 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.

76
Nonstandard universum
d
d

• The basic idea of both the version of nonstandard
  analysis (as Roninsons as Nelsons) 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)

77
Nonstandard universum
d
d

• 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

78
Nonstandard universum
d
d

• What is ultrafilter?
• Let S be a nonempty set, then an ultrafilter on S
  is a nonempty collection F of subsets of S having
  the following properties
• 1. ? ? F.
• 2. If A, B ? F, then A, B ? F .
• 3. If A,B ? F and A?B?S, then A,B ? F
• 4. For any subset A of S, either A ? F or its
  complement A S A ? F

79
Nonstandard universum
d
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
  ma-ximal filter of nonempty subsets of X.)

80
Nonstandard universum
d
d

• 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

81
Nonstandard universum
d
d

• The philosophical reflection continues Thats
  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

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• 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

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• And at last The generalized so Hilbert space as
  nonstandard universum is delivered again by the
  axiom of choice but this time via Zorns 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

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• 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 operator
• Infinitesimals Compact 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
  transition bet-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
  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

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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 thats 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) x20 doesnt
  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 thats 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 thats 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 Robinsons
  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 numbersThey 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
  from the 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 an
  inde-pendent axiom in a special axiom set

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• 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 ZFGCH, it implies a
  negation of ZF, namely a negation of the axiom
  of foundation AF in ZF. If ZFGCH 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. Thestandard 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

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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 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

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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 ?, ? from the definition of
  qubit can be interpreted as generalized, complex
  possibilities of the coefficients ?, ? from
  relativity

Qubit
Relativity
a (1-b)1/2 bv/c
a2b21 a0?b1? 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 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

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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 c
  restricts 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 point h ???? ???? ??
  ??????????? ?????? ? ??????

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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

129
Forthcoming in 2nd part

• 1. Motivation
• 2. Infinity and the axiom of choice
• 3. Nonstandard universum
• 4. Continuity and continuum
• 5. Nonstandard continuity between two infinitely
  close standard points
• 6. A new axiom of chance
• 7. Two kinds interpretation of quantum mechanics

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That was all of 1st part
CONTINUITY AND CONTINUUM IN NONSTANDARD UNIVERSUM
Vasil Penchev Institute for Philosophical
Research Bulgarian Academy of Science E-mail
vasildinev_at_gmail.com Professional
blog http//www.esnips.com/web/vasilpenchevsnews

• Thank you for your attention!