Continuity and Continuum in Nonstandard Universum Vasil Penchev Institute of Philosophical Research Bulgarian Academy of Science E-mail: email@example.com Publications blog:http://www.esnips.com/web/vasilpenchevsnews
Contents: 1.Motivation 2. Infinity and the axiom of choice 3.Nonstandard universum 4.Continuity and continuum 5.Nonstandard continuity between two infinitely close standard points 6.A new axiom: of chance 7.Two kinds interpretation of quantum mechanics
This file is only Part 1 of the entire presentation and includes: 1.Motivation 2. Infinity and the axiom of choice 3.Nonstandard universum
: : 1.Motivation My problem was: Given: Two sequences: : 1, 2, 3, 4, ….a-3, a-2, a-1, a : a, a-1, a-2, a-3, …, 4, 3, 2, 1 Where a is the power of countable set The problem: Do the two sequences and coincide or not?
: : 1.Motivation At last, my resolution proved out: That the two sequences: : 1, 2, 3, 4, ….a-3, a-2, a-1, a : a, a-1, a-2, a-3, …, 4, 3, 2, 1 coincide or not, is a new axiom (or two different versions of the choice axiom): the axiom of chance: whether we can always repeat or not an infinite choice
: : 1.Motivation For example, let us be given two Hilbert spaces: : eit, ei2t, ei3t, ei4t, … ei(a-1)t, eiat : eiat , ei(a-1)t, … ei4t, ei3t, ei2t, eit An analogical problem is: Are those two Hilbert spaces the same or not? can be got by Minkowski space after Legendre-like transformation
: : 1.Motivation So that, if: : eit, ei2t, ei3t, ei4t, … ei(a-1)t, eiat : eiat , ei(a-1)t, … ei4t, ei3t, ei2t, eit are the same, then Hilbert space is equivalent of the set of all the continuous world lines in spacetime (see also Penrose’s twistors) That is the real problem, from which I had started
: : 1.Motivation About that real problem, from which I had started, my conclusion was: Thereare two different versions about the transition between the micro-object Hilbert space and the apparatus spacetime in dependence on accepting or rejecting of “the chance axiom”, but no way to be chosen between them
: : 1.Motivation After that, I noticed that the problem is very easily to be interpreted by transition within nonstandard universum between two nonstandardneighborhoods (ultrafilters) of two infinitely near standard points or between the standard subset and the properly nonstandard subset of nonstandard universum
: : 1.Motivation And as a result, I decided that only the highly respected scientists from the honorable and reverend department “Logic” are that appropriate public worthy and deserving of being delivered a report on that most intriguing and even maybe delicate topicexiting those minds which are more eminent
: : 1.Motivation After that, the very God was so benevolent so that He allowed me to recognize marvelous mathematical papers of a great Frenchman, Alain Connes, recently who has preferred in favor of sunny California to settle, and who, a long time ago, had introduced nonstandard infinitesimals by compact Hilbert operators
Contents: 1.Motivation 2. INFINITY and the AXIOM OF CHOICE 3.Nonstandard universum 4.Continuity and continuum 5.Nonstandard continuity between two infinitely close standard points 6.A new axiom: of chance 7.Two kinds interpretation of quantum mechanics
Infinity and the Axiom of Choice A few preliminary notes about how the knowledge of infinity is possible: The short answer is: as that of God: in belief and by analogy.The way of mathematics to be achieved a little knowledge of infinity transits three stages: 1. From finite perception to Axioms 2. Negation of some axioms. 3. Mathematics beyond finiteness
Infinity and the Axiom of Choice The way of mathematics to infinity: 1. From our finite experience and perception to Axioms: The most famous example is the axiomatization of geometry accomplished by Euclid in his “Elements”
Infinity and the Axiom of Choice The way of mathematics to infinity: 2. Negation of some axioms: the most frequently cited instance is the fifth Euclid postulateand its replacing in Lobachevski geometry by one of its negations. Mathematics only starts from perception, but its cognition can go beyond it by analogy
Infinity and the Axiom of Choice The way of mathematics to infinity: 3. Mathematics beyond finiteness: We can postulate some properties of infinite sets by analogy of finite ones (e.g. ‘number of elements’ and ‘power’) However such transfer may produce paradoxes: see as example: Cantor “naive” set theory
Infinity and the Axiom of Choice A few inferences about the math full-scale offensive amongst the infinity: 1. Analogy: well-chosen appropriate properties of finite mathematical struc-tures are transferred into infinite ones 2. Belief: the transferred properties are postulated (as usual their negations can be postulated too)
Infinity and the Axiom of Choice The most difficult problems of the math offensive among infinity: Which transfers are allowed by in-finity without producing paradoxes? Which properties are suitable to be transferred into infinity? How to dock infinities?
Infinity and the Axiom of Choice The Axiom of Choice (a formulation): If given a whatever set A consisting of sets, we always can choose an element from each set, thereby constituting a new set B (obviously of the same po-wer as A). So its sense is: we always can transfer the property of choosing an element of finite set to infinite one
Infinity and the Axiom of Choice Some other formulations or corollaries: Any set can be well ordered (any its subset has a least element) Zorn’s lema Ultrafilter lema Banach-Tarski paradox Noncloning theorem in quantum information
Infinity and the Axiom of Choice Zorn’s lemma is equivalent to the axiom of choice. Call a set A a chain if for any two members B and C, either B is a sub-set of C or C is a subset of B. Now con-sider a set D with the properties that for every chain E that is a subset of D, the union of E is a member of D. The lem-ma states that D contains a member that is maximal, i.e. which is not a subset of any other set in D.
Infinity and the Axiom of Choice Ultrafilter lemma: A filter on a set X is a collection of nonempty subsets of X that is closed under finite intersection and under superset. An ultrafilter is a maximal filter. The ultrafilter lemma states that every filter on a set X is a subset of some ultrafilter on X (a maximal filter of nonempty subsets of X.)
Infinity and the Axiom of Choice Banach–Tarski paradox which says in effect that it is possible to ‘carve up’ the 3-dimensional solid unit ball into finitely many pieces and, using only rotation and translation, reassemble the pieces into two balls each with the same volume as the original. The proof, like all proofs involving the axiom of choice, is an existence proof only.
Infinity and the Axiom of Choice First stated in 1924, the Banach-Tarski paradox states that it is possible to dissect a ball into six pieces which can be reassembled by rigid motions to form two balls of the same size as the original. The number of pieces was subsequently reduced to five by Robinson (1947), although the pieces are extremely complicated
Infinity and the Axiom of Choice Five pieces are minimal, although four pieces are sufficient as long as the single point at the center is neglected. A generalization of this theorem is that any two bodies in that do not extend to infinity and each containing a ball of arbitrary size can be dissected into each other (i.e., they are equidecomposable)
Infinity and the Axiom of Choice Banach-Tarski paradox is very important for quantum mechanics and information since any qubit is isomorphic to a 3D sphere. That’s why the paradox requires for arbitrary qubits (even entire Hilbert space) to be able to be built by a single qubit from its parts by translations and rotations iteratively repeating the procedure
Infinity and the Axiom of Choice So that theBanach-Tarski paradox implies the phenomenon of entanglement in quantum information as two qubits (or two spheres) from one can be considered as thoroughly entangled. Two partly entangled qubits could be reckoned as sharing some subset of an initial qubit (sphere) as if “qubits (spheres) – Siamese twins”
Infinity and the Axiom of Choice But theBanach-Tarski paradox is a weaker statement than the axiom of choice. It is valid only about 3D sets. But I haven’t meet any other additional condition. Let us accept that the Banach-Tarski paradox is equivalent to the axiom of choice for 3D sets. But entanglement as well 3D space are physical facts, and then…
Infinity and the Axiom of Choice But entanglement (= Banach-Tarski paradox) as well 3D space are physical facts, and then consequently, they are empirical confirmations in favor of the axiom of choice. This proves that the Banach-Tarski paradox is just the most decisive confirmation, and not at all, a refutation of the axiom of choice.
Infinity and the Axiom of Choice Besides, the axiom of choice occurs in the proofs of: the Hahn-Banach the-orem in functional analysis, the theo-rem that every vector space has a ba-sis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that every ring has a maximal ideal and that every field has an algebraic closure.
Infinity and the Axiom of Choice The Continuum Hypothesis: The generalized continuum hypothesis (GCH) is not only independent of ZF, but also independent of ZF plus the axiom of choice (ZFC). However, ZF plus GCH implies AC, making GCH a strictly stronger claim than AC, even though they are both independent of ZF.
Infinity and the Axiom of Choice The Continuum Hypothesis: The generalized continuum hypothesis (GCH) is: 2Na = Na+1. Since it can be formulated without AC, entanglement as an argument in favor of AC is not expanded to GCH. We may assume the negation of GHC about cardinalities which are not “alefs” together with AC about cardinalities which are alefs
Infinity and the Axiom of Choice Negation of Continuum Hypothesis: The negation of GHC about cardinali-ties which are not “alefs” together with AC about cardinalities which are alefs: 1. There are sets which can not be well ordered. A physical interpretation of theirs is as physical objects out of (beyond) space-time. 2. Entanglement about all the space-time objects
Infinity and the Axiom of Choice Negation of Continuum Hypothesis: But the physical sense of 1. and 2.: 1. The non-well-orderable sets consist of well-ordered subsets (at least, their elements as sets) which are together in space-time. 2. Any well-ordered set (because of Banach-Tarski paradox) can be as a set of entangled objects in space-time
Infinity and the Axiom of Choice Negation of Continuum Hypothesis: So that the physical sense of 1. and 2. is ultimately: The mapping between the set of space-time points and the set of physical entities is a “many-many” correspondence: It can be equivalently replaced by usual mappings but however of a functional space, namely by Hilbert operators
Infinity and the Axiom of Choice Negation of Continuum Hypothesis: Since the physical quantities have interpreted by Hilbert operators in quantum mechanics and information (correspondingly, by Hermitian and non-Hermitian ones), then that fact is an empirical confirmation of the negation of continuum hypothesis
Infinity and the Axiom of Choice Negation of Continuum Hypothesis: But as well known, ZF+GHC implies AC. Since we have already proved both NGHC and AC, the only possibility remains also the negation of ZF (NZF), namely the negation the axiom of foundation (AF): There is a special kind of sets, which will call ‘insepa-rable sets’ and also don’t fulfill AF
Infinity and the Axiom of Choice An important example of inseparable set: When postulating that if a set A is given, then a set B always exists, such one that A is the set of all the subsets of B. An instance: let A be a countable set, then B is an inseparable set, which we can call ‘subcountable set’. Its power z is bigger than any finite power, but less than that of a countable set.
Infinity and the Axiom of Choice The axiom of foundation: “Every nonempty set is disjoint from one of its elements.“ It can also be stated as "A set contains no infinitely descending (membership) sequence," or "A set contains a (membership) minimal element," i.e., there is an element of the set that shares no member with the set
Infinity and the Axiom of Choice The axiom of foundation Mendelson (1958) proved that the equivalence of these two statements necessarily relies on the axiom of choice. The dual expression is called º-induction, and is equivalent to the axiom itself (Ito 1986)
Infinity and the Axiom of Choice The axiom of foundation and its negation: Since we have accepted both the axiom of choice and the negation of the axiom of foundation, then we are to confirm the negation ofº-induction, namely “There are sets containing infinitely descending (membership) sequence OR without a (membership) minimal element,"
Infinity and the Axiom of Choice The axiom of foundation and its negation: So that we have three kinds of inseparable set: 1.“containing infinitely descending (membership) sequence”2.“without a (membership) minimal element“ 3. Both 1. and 2. The alleged “axiom of chance” concerns only 1.
Infinity and the Axiom of Choice The alleged “axiom of chance” concerning only 1. claims that there are as inseparable sets “containing infinitely descending (membership) sequence”as such ones “containing infinitely ascending (membership) sequence” and different from the former ones
Infinity and the Axiom of Choice The Law of the excluded middle: The assumption of the axiom of choice is also sufficient to derive the law of the excluded middle in some constructive systems (where the law is not assumed).
Infinity and the Axiom of Choice A few (maybe redundant) commentaries: We always can: 1. Choose an element among the elements of a set of an arbitrary power 2. Choose a set among the sets, which are the elements of the set A without its repeating independently of the A power
Infinity and the Axiom of Choice A (maybe rather useful) commentary: We always can: 3a. Repeat the choice choosing the same element according to 1. 3b. Repeat the choice choosing the same set according to 2. Not (3a & 3b) is the new axiom of chance
Infinity and the Axiom of Choice The sense of the Axiom of Choice: Choice among infinite elements Choice among infinite sets Repetition of the already made choice among infinite elements Repetition of the already made choice among infinite sets
Infinity and the Axiom of Choice The sense of the Axiom of Choice: If all the 1-4 are fulfilled: - choice is the same as among finite as among infinite elements or sets; - the notion of information being based on choice is the same as to finite as to infinite sets
Infinity and the Axiom of Choice At last, the award for your kind patience: The linkages between my motivation and the choice axiom: When accepting its negation, we ought to recognize a special kind of choice and of information in relation of infinite entities: quantum choice (=measuring) and quantum information
Infinity and the Axiom of Choice So that the axiom of choice should be divided into two parts: The first part concerning quantum choice claims that the choice between infinite elements or sets is always possible.The second part concerning quantum information claims that the made already choice between infinite elements or sets can be always repeated