Continuity and Continuum in    Nonstandard Universum               Vasil Penchev    Institute of Philosophical Research   ...
Content1. Motivation                             s:2. Infinity and the axiom of choice3. Nonstandard universum4. Continuit...
This file is only Part 1 of the  entire presentation and            includes: 1. Motivation 2. Infinity and the axiom of c...
:       1. Motivation            :         My problem was:     Given: Two sequences:   : 1, 2, 3, 4, ….a-3, a-2, a-1, a...
:        1. Motivation          :  At last, my resolution proved                    out:      That the two sequences:   ...
:         1. Motivation                :For example, let us be given two            Hilbert spaces:  : eit, ei2t, ei3t,...
:         1. Motivation                :               So that, if:  : eit, ei2t, ei3t, ei4t, … ei(a-1)t, eiat  : eiat...
:      1. Motivation       : About that real problem, from     which I had started, my         conclusion was:There are ...
:      1. Motivation        :   After that, I noticed that the   problem is very easily to beinterpreted by transition w...
:       1. Motivation        :  And as a result, I decided that             only thehighly respected scientists from   t...
:      1. Motivation        :After that, the very God was sobenevolent so that He allowed  me to recognize marvelousmath...
Content1. Motivation                                   s:2. INFINITY and the AXIOM OF CHOICE3. Nonstandard universum4. Con...
Infinity and the Axiom of            Choice                A few preliminary notes abouthow the knowledge of infinity isp...
Infinity and the Axiom of            Choice                    The way of mathematics to              infinity:1. From ou...
Infinity and the Axiom of             Choice                    The way of mathematics to              infinity: 2. Negat...
Infinity and the Axiom of             Choice                   The way of mathematics to              infinity:3. Mathema...
Infinity and the Axiom of             Choice                  A few inferences about the math full-scale offensive amongs...
Infinity and the Axiom of             Choice                  The most difficult problems of     the math offensive among...
Infinity and the Axiom of             Choice                     The Axiom of Choice (a          formulation):    If give...
Infinity and the Axiom of            Choice                 Some other formulations or             corollaries:1. Any set...
Infinity and the Axiom of            Choice                 Zorn’s lemma is equivalent to the    axiom of choice. Call a ...
Infinity and the Axiom of            Choice               Ultrafilter lemma: A filter on    a set X is a collection of  n...
Infinity and the Axiom of             Choice               Banach–Tarski paradox whichsays in effect that it is possiblet...
Infinity and the Axiom of             Choice                First stated in 1924, the Banach- Tarski paradox states that ...
Infinity and the Axiom of             Choice               Five pieces are minimal, althoughfour pieces are sufficient as...
Infinity and the Axiom of             Choice                 Banach-Tarski paradox is very     important for quantummecha...
Infinity and the Axiom of             Choice                   So that the Banach-Tarskiparadox implies the phenomenon   ...
Infinity and the Axiom of             Choice               But the Banach-Tarski paradox is   a weaker statement than the...
Infinity and the Axiom of             Choice                 But entanglement (= Banach-  Tarski paradox) as well 3D  spa...
Infinity and the Axiom of             Choice                  Besides, the axiom of choiceoccurs in the proofs of: the Ha...
Infinity and the Axiom of             Choice                   The Continuum Hypothesis:    The generalized continuum   h...
Infinity and the Axiom of             Choice                    The Continuum Hypothesis:    The generalized continuumhyp...
Infinity and the Axiom of             Choice               Negation of Continuum Hypothesis:     The negation of GHC abou...
Infinity and the Axiom of             Choice               Negation of Continuum Hypothesis:But the physical sense of 1. ...
Infinity and the Axiom of             Choice               Negation of Continuum Hypothesis:  So that the physical sense ...
Infinity and the Axiom of             Choice               Negation of Continuum Hypothesis:Since the physical quantities...
Infinity and the Axiom of             Choice               Negation of Continuum Hypothesis:    But as well known, ZF+GHC...
Infinity and the Axiom of            Choice                   An important example of     inseparable set: When  postulat...
Infinity and the Axiom of             Choice                The axiom of foundation: ‚Every nonempty set is disjoint from...
Infinity and the Axiom of             Choice                     The axiom of foundationMendelson (1958) proved that the ...
Infinity and the Axiom of             Choice                The axiom of foundation and its negation: Since we have accep...
Infinity and the Axiom of             Choice                 The axiom of foundation and its  negation: So that we have t...
Infinity and the Axiom of            Choice               The alleged ‚axiom of chance‛ concerning only 1. claims that th...
Infinity and the Axiom of             Choice               The Law of the excluded middle:  The assumption of the axiom o...
Infinity and the Axiom of             Choice                     A few (maybe redundant)          commentaries:          ...
Infinity and the Axiom of             Choice                      A (maybe rather useful)           commentary:          ...
Infinity and the Axiom of             Choice                 The sense of the Axiom of Choice:1. Choice among infinite el...
Infinity and the Axiom of             Choice                The sense of the Axiom of Choice:    If all the 1-4 are fulfi...
Infinity and the Axiom of            Choice               At last, the award for your kind patience: The linkages between...
Infinity and the Axiom of             Choice                   So that the axiom of choiceshould be divided into two part...
Infinity and the Axiom of                            Choice My exposition is devoted to the  nega-tion only of the ‚secon...
Infinity and the Axiom of                            ChoiceNegating the choice axiom second part is the suggested ‚axiom ...
Infinity and the Axiom of                            Choice                  ClassicalQuantum1. Copying,        Yes     N...
Infinity and the Axiom of             Choice                How does the ‚1. Copying‛(Yes/No) descend from               ...
Infinity and the Axiom of             Choice                  If the case is: ‚1. Copying – No‛from                      ...
Infinity and the Axiom of             Choice                How does the ‚2. Destroying‛(Yes/No) descend from            ...
Infinity and the Axiom of             Choice               How does the ‚3. Non-self-interacting‛ (Yes/No) descendfrom   ...
Infinity and the Axiom of             Choice                  How does the ‚4. Energetic   medium‛ (Yes/No) descendfrom  ...
Infinity and the Axiom of             Choice               How does the ‚5. Being in space-time‛ (Yes/No) descend from   ...
Content1. Motivation                                  s:2. Infinity and the axiom of choice3. NONSTANDARD UNIVERSUM4. Cont...
Nonstandard universum            Abraham Robinson            (October 6, 1918 Leibnitz   – April 11, 1974)
Nonstandard universum                  Abraham Robinson                  (October 6, 1918His Book (1966)   – April 11, 1974)
Nonstandard universum                  ‚It is shown in this                  book that Leibniz                  ideas can ...
Nonstandard universum  ‚…G.W.Leibniz argued thatthe theory of infinitesimalsimplies the introduction of ideal numbers whic...
Nonstandard universum    The original approach of A.              Robinson:1. Construction of a nonstandardmodel of R (the...
Nonstandard universum 2. The finiteness principle: If any fi-nite subset of a (infinite)  set  posses-ses a model, thenth...
Nonstandard universum  The properties of nonstandard    neighborhood x about any    standard point x of R: 1) The  ‚lengt...
Nonstandard universum Indeed, the word of G.W.Leibniz‚that the theory of infinitesimalsimplies the introduction of ideal  ...
Nonstandard universum  Another possible approach was developed by was developed in        the mid-1970s by the  mathematic...
Nonstandard universum  In IST alongside the basic binary      membership relation , itintroduces a new unary predicatesta...
Nonstandard universum               Idealization: For every classical relation R, andfor arbit-rary values for all other f...
Nonstandard universum          Standardisation If A is a standard set and P any      property, classical or    otherwise, ...
Nonstandard universum            Transfer    If all the parameters           A, B, C, ..., Wof a classical formula F have ...
Nonstandard universum  The sense of the unary predicate             standard:If any formula holds for any finite          ...
Nonstandard universumSo that the suggested by Nelson IST is       a constructivist version of     nonstandard analysis. If...
Nonstandard universumThe basic idea of both the version   of nonstandard analysis (as    Roninson’s as Nelson’s) is    rep...
Nonstandard universum That collapse and repetition of  all infinity into any its point is  accomp-lished by the notion of ...
Nonstandard universum           What is ‘ultrafilter’?    Let S be a nonempty set, then an     ultrafilter on S is a nonem...
Nonstandard universumUltrafilter lemma: A filter on   a set X is a collection of nonempty subsets of X that isclosed under...
Nonstandard universum  A philosophical reflection: Let us     remember the Banach-Tarskiparadox: entire Hilbert space can ...
Nonstandard universum     The philosophical reflection  continues: That’s why nonstandard analysis is a good tool for quan...
Nonstandard universum   Something still a little more: If   Hilbert spa-ce is isomorphic to a     well ordered sequence of...
Nonstandard universum    And at last: The generalized so     Hilbert space as nonstandard universum is delivered again by ...
Nonstandard universumThus we have already pioneered to    Alain Connes’ introducing of infinitesimals as compact Hilbert o...
Nonstandard universumThe sense of compact operator: if it      is ap-plied to nonstandard      universum, it trans-forms a...
Nonstandard universum I would like to display that Connes’       infinitesimals possesses an exceptionally important prope...
Nonstandard universum  Minkowski operator is compact if   and only if it is compact Hilbert     operator. Before a sketch ...
Nonstandard universum Before a sketch of proof, its sense    and motivation: Our strategic       purpose is to be built a ...
Nonstandard universum  Before a sketch of proof, its sense     and motivation: The axiom of foundation is available in qua...
Nonstandard universum  Before a sketch of proof, its senseand motivation: If that’s the case, the axiom of foundation AF i...
Nonstandard universum  A note: Let us accept now the AF   negation, and consequently , a   smooth reversibility between co...
Nonstandard universumA Minkowski operator is compact if and only if it is a compact Hilbert    operator. A sketch of proof...
Nonstandard universum {RR}  {RR}  {RR}  {RR}    (the isomorphism)    {RR  R}R  {RR  R}R: i.e. Minkowski spa...
Nonstandard universum  A little comment on the theorem:A Minkowski operator is compact if  and only if it is a compact Hil...
Nonstandard universum   A little more comment on the              theorem: Let us imagine those infinitesimals, being oper...
Nonstandard universum   A little more comment on the              theorem: Hamiltonian describes a system bytwo independen...
Nonstandard universum    A little more comment on the              theorem: Jacobian describes the bifurcation,  two-forke...
Nonstandard universum    A few slides are devoted to        alternative ways for nonstandard infinitesimals to be         ...
Nonstandard universum  ‚Intuitively, smooth infinitesimal    analysis can be interpreted asdescribing a world in which lin...
Nonstandard universum‚We can imagine the intermediate    value theorems failure as resulting from the ability of an     in...
Nonstandard universum  The infinitesimals x in smooth    infinitesimal analysis are    nilpotent (nilsquare): x2=0doesn’t ...
Nonstandard universumThe smooth infinitesimal analysis       does not satisfy our requirements even only becauseof denying...
Nonstandard universum By introducing as zero divisors,the infinitesimals are interested  because of possibility for thepha...
Nonstandard universumThe infinitesimals introduced as    surreal numbers unlike hyperreal numbers (equal to   Robinson’s i...
Nonstandard universum About the surreal numbers:Theyare a proper class (i.e. are not a   set), ant the biggest ordered    ...
Nonstandard universum  Since the comparison rule is recursive, it requires finite ortransfinite induction . Let us nowcons...
Nonstandard universum  For example, we can easily to   define our initial problem in             their terms:           Le...
Nonstandard universum That special axiom set includes:    the axiom of choice and a   negation of the generalizedcontinuum...
Nonstandard universum    However a permission and    introducing of the infinite   degressive sequences , and  consequent...
Nonstandard universum Comparison between ‚standard‛ and nonstandard infinitesimals.   The‚standard‛ infinitesimals      ex...
Nonstandard universum More about the sense of ‚standard‛infinitesimals: By virtue of the axiom     of choice any set can b...
Nonstandard universumAlthough the ‚neighbor‛ points of anarbit-rary set are not discrete, andconsequently, the intervals b...
Nonstandard universum  But the ratio of the neighborintervals can be also considered   as probability, thereby the   veloc...
Nonstandard universum2) about a point not belonging to    the sequence: as much thevelocity of convergence is higheras the...
Nonstandard universum The same about a cell of phase              space:The same can be said of a cell ofphase space: as m...
Nonstandard universum     Our question is how the     probability in quantum mechanics should be interpre-ted? A possible ...
Nonstandard universum The physical interpretation of  the velo-city for a series to converge is just as velocity of  some ...
Nonstandard universum  The coefficients ,  from the     definition of qubit can be   interpreted as generalized,   compl...
Nonstandard universum The interpretation of the ratio       between nonstandard  infinitesimals both as velocity   and as ...
Nonstandard universumBut we need some interpretation  of complex probabilities, or, which is equi-valent, of complex nonst...
Nonstandard universum   After that, all the complex  probabilities are ushered in varying the ties, ‚hyste-reses‛  ‚up‛ or...
Nonstandard universumConsequently, there exists onemore bridge of interpretation connecting Hilbert and 3D or      Minkows...
Nonstandard universum And what about the constant h?It guarantees on existing of: both     the sequences, both thenonstand...
Nonstandard universumAnd what about the constant h? It gua-rantees on existing of: both     the sequen-ces, both the  nons...
Nonstandard universumOne more interpretation of h: as   the square of the hysteresisbetween the ‚up‛ and the ‚down‛   neig...
Nonstandard universum  One more interpretation of h: The sense of g is intuitively very  clear: As more points ‚up‛ and‚do...
Nonstandard universum      By means of the latter interpretation we can interpret     also phase space as non-  standard 3...
Nonstandard universumWhat do the constants c and hinterpret as limits of a phase   space cell deformation?    c.1.dx  dy ...
Forthcoming in 2nd part:1. Motivation2. Infinity and the axiom of choice3. Nonstandard universum4. Continuity and continuu...
CONTINUITY AND CONTINUUM      IN NONSTANDARD UNIVERSUM                  Vasil Penchev      Institute for Philosophical Res...
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Vasil Penchev. Continuity and Continuum in Nonstandard Universum

  1. 1. Continuity and Continuum in Nonstandard Universum Vasil Penchev Institute of Philosophical Research Bulgarian Academy of Science E-mail: vasildinev@gmail.com Publications blog:http://www.esnips.com/web/vasilpenchevsnews
  2. 2. Content1. Motivation s:2. Infinity and the axiom of choice3. Nonstandard universum4. Continuity and continuum5. Nonstandard continuity between two infinitely close standard points6. A new axiom: of chance7. Two kinds interpretation of
  3. 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. 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 
  5. 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, 1coincide or not, is a new axiom (or two different versions of the choice axiom): the axiom of
  6. 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
  7. 7. : 1. Motivation : So that, if: : eit, ei2t, ei3t, ei4t, … ei(a-1)t, eiat : eiat , ei(a-1)t, … ei4t, ei3t, ei2t, eitare 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
  8. 8. : 1. Motivation : About that real problem, from which I had started, my conclusion was:There are two different versionsabout the transition between themicro-object Hilbert space  and the apparatus spacetime  in dependence on accepting or rejecting of ‚the chance
  9. 9. : 1. Motivation : After that, I noticed that the problem is very easily to beinterpreted by transition withinnonstandard universum betweentwo nonstandard neighborhoods (ultrafilters) of two infinitelynear standard points or between the standard subset and theproperly nonstandard subset of
  10. 10. : 1. Motivation : And as a result, I decided that only thehighly respected scientists from the honorable and reverend department ‚Logic‛ are that appropriate public worthy and deserving of being delivereda report on that most intriguing
  11. 11. : 1. Motivation :After that, the very God was sobenevolent so that He allowed me to recognize marvelousmathematical papers of a great Frenchman, Alain Connes, recently who has preferred in favor of sunnyCalifornia to settle, and who, a long time ago, had introduced
  12. 12. Content1. Motivation s:2. INFINITY and the AXIOM OF CHOICE3. Nonstandard universum4. Continuity and continuum5. Nonstandard continuity between two infinitely close standard points6. A new axiom: of chance7. Two kinds interpretation of
  13. 13. Infinity and the Axiom of Choice  A few preliminary notes abouthow the knowledge of infinity ispossible: The short answer is: as that of God: in belief and byanalogy.The way of mathematics to be achieved a little knowledge of infinity transits three stages: 1. From finiteperception to Axioms 2. Negation
  14. 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. 15. Infinity and the Axiom of Choice  The way of mathematics to infinity: 2. Negation of some axioms: themost frequently cited instance isthe fifth Euclid postulate and its replacing in Lobachevskigeometry by one of its negations. Mathematics only starts from
  16. 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
  17. 17. Infinity and the Axiom of Choice  A few inferences about the math full-scale offensive amongst the infinity:1. Analogy: well-chosenappropriate properties of finitemathematical struc-tures aretransferred into infinite ones2. Belief: the transferred
  18. 18. Infinity and the Axiom of Choice  The most difficult problems of the math offensive among infinity:1. Which transfers are allowed by in-finity without producing paradoxes?2. Which properties are suitable
  19. 19. Infinity and the Axiom of Choice  The Axiom of Choice (a formulation): If given a whatever set Aconsisting of sets, we always can choose an element from each set, thereby constituting a newset B (obviously of the same po- wer as A). So its sense is: we
  20. 20. Infinity and the Axiom of Choice  Some other formulations or corollaries:1. Any set can be well ordered (any its subset has a least element) 2. Zorn’s lema 3. Ultrafilter lema 4. Banach-Tarski paradox
  21. 21. 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.
  22. 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
  23. 23. Infinity and the Axiom of Choice Banach–Tarski paradox whichsays in effect that it is possibleto ‘carve up’ the 3-dimensionalsolid unit ball into finitely manypieces and, using only rotationand translation, reassemble thepieces into two balls each withthe same volume as the original.The proof, like all proofs
  24. 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 bereassembled by rigid motions toform two balls of the same size as the original. The number ofpieces was subsequently reduced to five by Robinson
  25. 25. Infinity and the Axiom of Choice Five pieces are minimal, althoughfour pieces are sufficient as longas the single point at the centeris neglected. A generalization ofthis theorem is that any twobodies in that do not extend toinfinity and each containing aball of arbitrary size can bedissected into each other (i.e.,
  26. 26. Infinity and the Axiom of Choice  Banach-Tarski paradox is very important for quantummechanics 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 beable to be built by a single qubit from its parts by translations
  27. 27. Infinity and the Axiom of Choice  So that the Banach-Tarskiparadox implies the phenomenon of entanglement in quantum information as two qubits (or two spheres) from one can be considered as thoroughlyentangled. Two partly entangled qubits could be reckoned assharing some subset of an initial
  28. 28. Infinity and the Axiom of Choice But the Banach-Tarski paradox is a weaker statement than the axiom of choice. It is valid onlyabout  3D sets. But I haven’t meet any other additional condition. Let us accept that the Banach- Tarski paradox is equivalent tothe axiom of choice for  3D sets. But entanglement as well 3D
  29. 29. Infinity and the Axiom of Choice  But entanglement (= Banach- Tarski paradox) as well 3D space are physical facts, and then consequently, they areempirical confirmations in favor of the axiom of choice. This proves that the Banach-Tarskiparadox is just the most decisive confirmation, and not at all, a
  30. 30. Infinity and the Axiom of Choice  Besides, the axiom of choiceoccurs in the proofs of: the Hahn- Banach the-orem in functional analysis, the theo-rem that every vector space has a ba- sis, Tychonoffs theorem in topology stating that every product of compact spaces is compact, and the theorems inabstract algebra that every ring
  31. 31. Infinity and the Axiom of Choice  The Continuum Hypothesis: The generalized continuum hypothesis (GCH) is not only independent of ZF, but alsoindependent 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
  32. 32. Infinity and the Axiom of Choice  The Continuum Hypothesis: The generalized continuumhypothesis (GCH) is: 2Na = Na+1 . Since it can be formulated withoutAC, entanglement as an argumentin favor of AC is not expanded toGCH. We may assume the negationof GHC about cardinalities whichare not ‚alefs‛ together with AC
  33. 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 asphysical objects out of (beyond)
  34. 34. Infinity and the Axiom of Choice Negation of Continuum Hypothesis:But the physical sense of 1. and 2.: 1. The non-well-orderable setsconsist of well-ordered subsets(at least, their elements as sets)which are together in space-time.2. Any well-ordered set (becauseof Banach-Tarski paradox) can be as a set of entangled objects in
  35. 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
  36. 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
  37. 37. Infinity and the Axiom of Choice Negation of Continuum Hypothesis: But as well known, ZF+GHCimplies AC. Since we have already proved both NGHC and AC, theonly possibility remains also the negation of ZF (NZF), namely thenegation the axiom of foundation (AF): There is a special kind of sets, which will call ‘insepa-
  38. 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 alwaysexists, 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
  39. 39. Infinity and the Axiom of Choice The axiom of foundation: ‚Every nonempty set is disjoint fromone 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
  40. 40. Infinity and the Axiom of Choice  The axiom of foundationMendelson (1958) proved that the equivalence of these twostatements necessarily relies on the axiom of choice. The dual expression is called º-induction, and is equivalent to
  41. 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
  42. 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‛
  43. 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
  44. 44. Infinity and the Axiom of Choice The Law of the excluded middle: The assumption of the axiom ofchoice is also sufficient to derive the law of the excluded middle in some constructive systems (where the law is not assumed).
  45. 45. Infinity and the Axiom of Choice  A few (maybe redundant) commentaries: We always can:1. Choose an element among theelements of a set of anarbitrary power2. Choose a set among the
  46. 46. Infinity and the Axiom of Choice  A (maybe rather useful) commentary: We always can:3a. Repeat the choice choosing thesame element according to 1.3b. Repeat the choice choosing thesame set according to 2.
  47. 47. Infinity and the Axiom of Choice  The sense of the Axiom of Choice:1. Choice among infinite elements 2. Choice among infinite sets 3. Repetition of the already made choice among infinite elements 4. Repetition of the already
  48. 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
  49. 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, weought to recognize a special kind of choice and of information in relation of infinite entities:
  50. 50. Infinity and the Axiom of Choice  So that the axiom of choiceshould be divided into two parts: The first part concerning quantum choice claims that thechoice between infinite elements or sets is always possible. The second part concerning quantum information claims that the made already choice between
  51. 51. Infinity and the Axiom of  Choice My exposition is devoted to the nega-tion only of the ‚second part‛ of the choice axiom. Butnot more than a couple of words about the sense for the firstpart 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
  52. 52. Infinity and the Axiom of  ChoiceNegating 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
  53. 53. Infinity and the Axiom of  Choice ClassicalQuantum1. Copying, Yes No2. Destroying, Yes No3. Non-self-interacting, Yes No4. Energetic medium, Yes No5. Being in space-time Yes No
  54. 54. Infinity and the Axiom of Choice How does the ‚1. Copying‛(Yes/No) descend from (No/Yes)?It is obviously: ‚Copying‛ meansthat a set of choices isrepeated, andconsequently, it has been able to
  55. 55. Infinity and the Axiom of Choice If the case is: ‚1. Copying – No‛from - Yes,then that case is the non-cloningtheorem in quantum information:No qubit can be copied
  56. 56. Infinity and the Axiom of Choice How does the ‚2. Destroying‛(Yes/No) descend from (No/Yes)?‚Destroying‛ is similar tocopying:As if negative copying
  57. 57. Infinity and the Axiom of Choice How does the ‚3. Non-self-interacting‛ (Yes/No) descendfrom (No/Yes)? Self-interacting means non-repeating by itself
  58. 58. Infinity and the Axiom of Choice  How does the ‚4. Energetic medium‛ (Yes/No) descendfrom (No/Yes)?Energetic medium means forrepeating to be turned intosubstance, or in other words, to
  59. 59. Infinity and the Axiom of Choice How does the ‚5. Being in space-time‛ (Yes/No) descend from (No/Yes)? ‘Being of a set in space-time’ means that the set is well- ordered which fol-lows fromthe axiom of choice. ‘No axiom of chance’ means that the well-
  60. 60. Content1. Motivation s:2. Infinity and the axiom of choice3. NONSTANDARD UNIVERSUM4. Continuity and continuum5. Nonstandard continuity between two infinitely close standard points6. A new axiom: of chance7. Two kinds interpretation of
  61. 61. Nonstandard universum Abraham Robinson (October 6, 1918 Leibnitz – April 11, 1974)
  62. 62. Nonstandard universum Abraham Robinson (October 6, 1918His Book (1966) – April 11, 1974)
  63. 63. 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 AnalysisHis Book (1966) and many other branches of
  64. 64. Nonstandard universum ‚…G.W.Leibniz argued thatthe theory of infinitesimalsimplies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the
  65. 65. Nonstandard universum The original approach of A. Robinson:1. Construction of a nonstandardmodel of R (the real continuum): Nonstan-dard model (Skolem 1934): Let A be the set of all thetrue statements about R, then:  = A(c>0, c>0`, c>0``…): Any finite subset of  holds for R. After
  66. 66. Nonstandard universum 2. The finiteness principle: If any fi-nite subset of a (infinite) set  posses-ses a model, thenthe set  possesses a model too.The model of  is not isomorphic to R & A and it is a nonstandarduniversum over R & A. Its sense is as follow: there is a nonstandard neighborhood x
  67. 67. Nonstandard universum The properties of nonstandard neighborhood x about any standard point x of R: 1) The ‚length‛ of x in R or of any its measurable subset is 0. 2) Any x in R is isomorphic to (R & A)itself. Our main problem is about continuity and continuum of two neighborhoods x and y between two neighbor well
  68. 68. Nonstandard universum Indeed, the word of G.W.Leibniz‚that the theory of infinitesimalsimplies 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
  69. 69. 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-
  70. 70. Nonstandard universum In IST alongside the basic binary membership relation , itintroduces a new unary predicatestandard 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
  71. 71. Nonstandard universum Idealization: For every classical relation R, andfor arbit-rary values for all other free variables, we have that if for each standard, finite set F, thereexists a g such that R(g, f ) holds forall f in F, then there is a particular Gsuch 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
  72. 72. Nonstandard universum Standardisation If A is a standard set and P any property, classical or otherwise, then there is a unique, standard subset B of A whose standard elements areprecisely the standard elements of A satisfying P (but the behaviour of Bs nonstandard
  73. 73. Nonstandard universum Transfer If all the parameters A, B, C, ..., Wof a classical formula F have standard values then F( x, A, B,..., W ) holds for all xs as soon asit holds for all standard xs.
  74. 74. Nonstandard universum The sense of the unary predicate standard:If any formula holds for any finite standardset 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
  75. 75. Nonstandard universumSo that the suggested by Nelson IST is a constructivist version of nonstandard analysis. If ZFC isconsistent, then ZFC + IST is consistent. In fact, a stronger statement can be made: ZFC + IST is a conservative extension of ZFC: any classicalformula (correct or incorrect!) that can be proven within internal set theory can be proven in the
  76. 76. Nonstandard universumThe basic idea of both the version of nonstandard analysis (as Roninson’s as Nelson’s) is repetition of all the realcontinuum R at, or better, within any its point as nonstandard neighborhoods about any of them. The consistency of that repetition is achieved by the
  77. 77. Nonstandard universum That collapse and repetition of all infinity into any its point is accomp-lished by the notion of ultrafilter in nonstandardanalysis. Ultrafilter is way to be transferred and thereby repeated the topological properties of all the real continuum into any its point, andafter that, all the properties of
  78. 78. 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  F4. For any subset A of S, either A  F
  79. 79. Nonstandard universumUltrafilter lemma: A filter on a set X is a collection of nonempty subsets of X that isclosed under finite intersection and under superset. Anultrafilter is a maximal filter. The ultrafilter lemma statesthat every filter on a set X is asubset of some ultrafilter on X
  80. 80. Nonstandard universum A philosophical reflection: Let us remember the Banach-Tarskiparadox: entire Hilbert space can be delivered only by repetition adinfinitum of a single qubit (since it isisomorphic to 3D sphere)as well theparadox follows from the axiom of choice. However nonstandard analysis carries out the same idea as theBanach-Tarski paradox about 1D sphere, i.e. a point: all the nonstandard universum can
  81. 81. 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 bestadvantage is that NU can describe the
  82. 82. 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 leastcomparable unless even iso-morphic to Minkowski space; 2. It is getting generalized into nonstandard universum as to arbitrary numberdimensions, and even as to fractional
  83. 83. 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 ultrafilterlemma (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
  84. 84. Nonstandard universumThus we have already pioneered to Alain Connes’ introducing of infinitesimals as compact Hilbert operators unlike the rest Hilbertoperators representing transfor- mations of standard sets. He has suggested the following ‚dictionary‛:Complex variable Hilbert
  85. 85. Nonstandard universumThe sense of compact operator: if it is ap-plied to nonstandard universum, it trans-forms a nonstandard neighborhood into anonstandard 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
  86. 86. Nonstandard universum I would like to display that Connes’ infinitesimals possesses an exceptionally important property: they are infinitesimals both in Hilbert and in Minkowski space: so that they describe very welltransformations of Minkowski space into Hilbert space and vice versa: Math speaking, Minkowski operator is compact if and only if it iscompact Hilbert operator. You might
  87. 87. Nonstandard universum Minkowski operator is compact if and only if it is compact Hilbert operator. Before a sketch ofproof, 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
  88. 88. Nonstandard universum Before a sketch of proof, its sense and motivation: Our strategic purpose is to be built a united, common language for us to be able to speak both of theapparatus 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
  89. 89. Nonstandard universum Before a sketch of proof, its sense and motivation: The axiom of foundation is available in quantum mechanics by the collapse of wave function. Let us represent thecoherent state as infinity since, if the Hilbert space is separable, then anyits point is a coherent superposition of a countable set of components. The ‚collapse‛ represents as if a descending avalanche from the
  90. 90. Nonstandard universum Before a sketch of proof, its senseand 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 AFforbids 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
  91. 91. Nonstandard universum A note: Let us accept now the AF negation, and consequently , a smooth reversibility between coherent and ‚collapsed‛ state. Then: P = Ps - Pr, where Ps is the probability from the coherentsuperposition to a given value, andPr is the probability of reversible process. So that the quantummechanical probability attached to
  92. 92. Nonstandard universumA 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:
  93. 93. Nonstandard universum {RR}  {RR}  {RR}  {RR}  (the isomorphism) {RR  R}R  {RR  R}R: i.e. Minkowski space operators. The sense of introducing of nonstandard infinitesimals bycompact Hilbert operators is for them to be invariant towards (straight and inverse)
  94. 94. Nonstandard universum A little comment on the theorem:A Minkowski operator is compact if and only if it is a compact Hilbert operator Defining nonstandard infinitesimals as compact Hilbert operators we are introducing infinitesimals being able to serve both such ones of the transition between Minkowski andHilbert space (the apparatus and the
  95. 95. Nonstandard universum A little more comment on the theorem: Let us imagine those infinitesimals, being operators, as sells of phasespace: they are smoothly decreasing from the minimal cell of the apparatus phase space via and beyond the axiom of foundation tozero, what is the phase space sell of the microobject. That decreasing is
  96. 96. Nonstandard universum A little more comment on the theorem: Hamiltonian describes a system bytwo 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 anonlinear system of equalities [the
  97. 97. Nonstandard universum A little more comment on the theorem: Jacobian describes the bifurcation, two-forked direction(s) from a nonlinear system to two linear systems when the one united, common description is alreadyimpossible and it is disintegrating to two independent each of other descriptions
  98. 98. Nonstandard universum A few slides are devoted to alternative ways for nonstandard infinitesimals to be introduced: - smooth infinitesimal analysis - surreal numbers.Both the cases are inappropriate to our purpose or can beinterpreted too close-ly or even
  99. 99. Nonstandard universum ‚Intuitively, smooth infinitesimal analysis can be interpreted asdescribing a world in which lines are made out of infinitesimally small segments, not out of points. Theseseg-ments can be thought of as being long enough to have a definite direction, but not long enough to be curved. The construction ofdiscontinuous functions fails because a function is identified with a curve,
  100. 100. Nonstandard universum‚We can imagine the intermediate value theorems 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
  101. 101. Nonstandard universum The infinitesimals x in smooth infinitesimal analysis are nilpotent (nilsquare): x2=0doesn’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
  102. 102. Nonstandard universumThe smooth infinitesimal analysis does not satisfy our requirements even only becauseof denying the axiom of choice orthe 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
  103. 103. Nonstandard universum By introducing as zero divisors,the infinitesimals are interested because of possibility for thephase space sell to be zero still satisfying uncertainty. It meansthat the bifurcation of the initial nonlinear reference frame to two linear frames correspondingly of the
  104. 104. Nonstandard universumThe infinitesimals introduced as surreal numbers unlike hyperreal numbers (equal to Robinson’s infinitesimals): Definition: ‚If L and R are twosets of surreal numbers and no member of R is less than orequal to any member of L then {
  105. 105. Nonstandard universum About the surreal numbers:Theyare 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 orequal to no member of XL, and no
  106. 106. Nonstandard universum Since the comparison rule is recursive, it requires finite ortransfinite induction . Let us nowconsider the following subset N of surreal numbers: All thesurreal 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
  107. 107. Nonstandard universum For example, we can easily to define our initial problem in their terms: Let  and  be:  = {q: q  {N | 0}}  = {w: w  {0 | 0  N}}Our problem is whether  and  co-incide or not? If not, what ispower of   ? Our hypothesis is: the ans-wer of the former
  108. 108. Nonstandard universum That special axiom set includes: the axiom of choice and a negation of the generalizedcontinuum hypothesis (GCH). Since the axiom of choice is acorollary from ZF+GCH, it implies a negation of ZF, namely: a negation of the axiom offoundation AF in ZF. If ZF+GCH is thecase, our problem does not arise
  109. 109. Nonstandard universum However a permission and introducing of the infinite degressive sequences , and consequently, a AF negation is required by quantum information, or more particularly, by a discussingwhether Hilbert and Minkowski space are equivalent or not, ormore generally, by a considering
  110. 110. Nonstandard universum Comparison between ‚standard‛ and nonstandard infinitesimals. The‚standard‛ infinitesimals exist only in boundary transition. Their sense represents velocity for a point- focused sequence to converge to that point. That velocity is the ratio between the two neighborintervals between three discrete
  111. 111. Nonstandard universum More about the sense of ‚standard‛infinitesimals: By virtue of the axiom of choice any set can be well ordered as a sequence and thereby the ratio between the two neighbor intervals between three discrete successive points of the sequence inquestion 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
  112. 112. Nonstandard universumAlthough the ‚neighbor‛ points of anarbit-rary set are not discrete, andconsequently, the intervals between them are zero, we can recover asif ‚intervals‛ between the well- ordered as if ‚discrete‛ neighbor points by means ofnonstandard infini-tesimals. The nonstandard infinitesimals aresuch intervals. The representation
  113. 113. Nonstandard universum But the ratio of the neighborintervals 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: asmuch the velocity of convergence
  114. 114. Nonstandard universum2) about a point not belonging to the sequence: as much thevelocity of convergence is higheras 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,
  115. 115. Nonstandard universum The same about a cell of phase space:The same can be said of a cell ofphase space: as much a process issteeper, and the process is more nonequilibrium, off-balance,dissipative as the probability of a cell belonging to it is higher while a balance, equilibrium,
  116. 116. Nonstandard universum Our question is how the probability in quantum mechanics should be interpre-ted? A possible hypothesis is: the pro-babilities of non- commutative, comple-mentary quantities are both the kinds correspondingly and interchangeably. For example, the coordinate
  117. 117. Nonstandard universum The physical interpretation of the velo-city for a series to converge is just as velocity of some physical process. If the case is spatial motion, then thecon-nection between velocity and probability is fixed by the fundamental constant c:
  118. 118. Nonstandard universum The coefficients ,  from the definition of qubit can be interpreted as generalized, complex possibilities of thecoefficients ,  from relativity: Qubit: Relativity:  2+2=1  = (1-) 1/2 |0+|1 = q =v/c
  119. 119. Nonstandard universum The interpretation of the ratio between nonstandard infinitesimals both as velocity and as probability. The ratio between ‚stanadard‛infinitesimals which exist only in boundary transit
  120. 120. Nonstandard universumBut we need some interpretation of complex probabilities, or, which is equi-valent, of complex nonstandard neigh-borhoods. If we reject AF, then we canintroduce the falling, descending from the infinity, but also infinite series as purely,properly imaginary nonstandard neighborhoods: The real
  121. 121. Nonstandard universum After that, all the complex probabilities are ushered in varying the ties, ‚hyste-reses‛ ‚up‛ or ‚down‛ between twowell ordered neighbor standard points. Wave function being or not in separable Hilbert space (i.e. with countable or non- countable power of itscomponents) is well interpreted
  122. 122. Nonstandard universumConsequently, there exists onemore bridge of interpretation connecting Hilbert and 3D or Minkowski space.What do the constants c and hinter-pret from the relations and ratios bet-ween two neighbor nonstandard inter- vals? It turns out that c
  123. 123. Nonstandard universum And what about the constant h?It guarantees on existing of: both the sequences, both thenonstandard neighborhoods ‚up‛ and ‚down‛. It is the unit of thecentral symmetry transforming between the nonstandard neighborhoods ‚up‛ and ‚down‛of any standard point h като площ
  124. 124. Nonstandard universumAnd 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 thenonstandard neighborhoods ‚up‛ and ‚down‛ of any stan-dard
  125. 125. Nonstandard universumOne more interpretation of h: as the square of the hysteresisbetween the ‚up‛ and the ‚down‛ neighborhood between twostandard points. Unlike standard continuity a parametric set of nonstandard continuities is available. The parameter g = Dp/Dx = Dm/Dt =
  126. 126. Nonstandard universum One more interpretation of h: The sense of g is intuitively very clear: As more points ‚up‛ and‚down‛ are common as both thehysteresis branches are closer.So the standard continuity turnsout an extreme peculiar case ofnonstan-dard continuity, namely all the points ‚up‛ and ‚down‛ are common and both the
  127. 127. Nonstandard universum By means of the latter interpretation we can interpret also phase space as non- standard 3D space. Any cell of phase space represents the hysteresis between 3D points well ordered in each of thethree dimensions. The connectionbet-ween phase space and Hilbert
  128. 128. Nonstandard universumWhat do the constants c and hinterpret 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. 129. Forthcoming in 2nd part:1. Motivation2. Infinity and the axiom of choice3. Nonstandard universum4. Continuity and continuum5. Nonstandard continuitybetween two infinitely closestandard points6. A new axiom: of chance7. Two kinds interpretation of
  130. 130. CONTINUITY AND CONTINUUM IN NONSTANDARD UNIVERSUM Vasil Penchev Institute for Philosophical Research Bulgarian Academy of Science E-mail: vasildinev@gmail.com Professional blog:http://www.esnips.com/web/vasilpenchevsnews That was all of 1 st part Thank you for your

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