ONTOLOGY OF MIRROR SYMMETRY IN LOGIC AND SET …



ONTOLOGY OF MIRROR SYMMETRY IN LOGIC AND SET THEORY

AS A WAY TO SOLVE THE FIRST HILBERT'S PROBLEM.

Alexander ZENKIN

Computing Center of the RAS, Vavilov st. 40, 117967 Moscow GSP-1, Russia.

mailto: alexzen@com2com.ru

Keywords: mirror symmetry, foundations of mathematics, incompleteness of Cantor's set theory, solution of continuum problem, logic and philosophy of science, ontology.

In 1900, at the Second International Congress of Mathematicians in Paris, Hilbert formulated his famous 23 open mathematical problems which were, by his opinion, most important for a progress of Mathematics in the coming century. And the First problem of this famous list was Cantor's Continuum Hypothesis (CH). Cantor first stated the CH in 1878 (Cantor, 1985). The sense of the CH consists in the following. Let X=[0,1] be a set of all real numbers (alias of all proper fractions, alias of all points) of the segment [0,1], N be the set of all finite natural numbers of the infinite series,

1,2,3, ..., n, ... , (*)

and the notation |Y| be a number of elements (alias power or cardinality) of the set Y for any infinite set Y. So that |N| = (0 is, by definition, a minimal infinite cardinal number, and |X| = C is a power of continuum. Using his famous theorem on the cardinality of powers-sets, Cantor constructed the series of transfinite cardinal numbers (powers):

(0 , (1 , (2 ,..., (n , ..., (**)

where (1 is a minimal uncountable cardinal, and formulated the CH in the following form.

CANTOR'S CONTINUUM HYPOTHESIS: C = (1, where С = 2(0 > (0 .

K.Goedel (1939) and P.Cohen (1962) proved the independence of the CH within the framework of the axiomatic set theory of Zermelo-Fraenkel, but the proof of the CH-independence and the CH-solution are obviously quite different things. The situation is described best of all by P.Cohen himself. Concerning a solvability of the CH by means of modern meta-mathematical and set-theoretical methods, Cohen writes in the very beginning of his known book (Cohen, 1969, p. 13): "... Continuum Hypothesis is a rather dramatic example of what can be called (from our today's point of view) an absolutely undecidable assertion ...". And completes the book by the words (Cohen, 1969, p. 282): "Thus, C is greater than (n, ((, ((, where ( = ((, and so on. From this point of view, C is considered as an incredibly large set, which is given to us by some new bold axiom and to approach to which by means of any gradual process of construction is impossible." Taking into account such the quite pessimistic opinion of the leading expert in this area as to a solvability of the CH, we shall use the following more weak, but much more general formulation of the CH or, more precisely, - of the Continuum Problem (CP) as a whole (Zenkin, 2002, 1999, 1997).

CONTINUUM PROBLEM FORMULATION. Is there a set of integers, say D, such that a one-to-one correspondence between the sets, D and X, can be realized so that |D|=|X|=C?

Consider the well-known in mathematics binary tree (Fig.1.) that is a cognitive visual representation (image) (Zenkin, 1991) of the set X alias the continuum: any infinite path,

V a1 a2 a3 ... an ..., where (i(1 [[ai =0] or [ai =1]], (1)

of the tree represents an only real x(X, and any real x(X (2) defines an only path of the tree.

So, there is a 1-1-correspondence between X and a set, say P, of infinite paths of the tree.

Since our Symmetry-Festival is devoted to interdisciplinary and holistic approaches in different areas of a human-being's activity, it's appropriate here to demonstrate the following unexpected connection between the pure mathematical object presented in Fig.1 and the most ancient holistic conceptions of Humankind as to System of Universe.

According to modern mathematical graph theory, the graph shown in Fig.1 is really called a tree, the vertex V of the graph is called a root of the tree, and the main topological property of the graph consists in that any two infinite paths of the tree have no common points except for the root V.

In the far 70s of the last century, when the (elder) author was investigating the CP and all his writing-table was swamped with papers, manuscripts, and pictures like Fig.1, a book about ancient Indian philosophy turned out at the table, absolutely by accident. The book was opened at a page, by accident. The page was saying: "Its roots are at the top, its branches are at the bottom - such is the ancient fig Tree. It is Truth, indeed. It is Brahman. All the worlds are based upon it.... " [Cognition of Brahman. Katha-Upanishada, II, 3].

Whether already the ancient Indians knew well the Tree and were seeing well that the segment [0,1] is a mathematical model of continuity, which is one of the most important, basic concepts of mathematics which, according to Gauss, is "a Queen of all sciences" with the help of which modern humankind tries to comprehend a veritable sense of the Universe?!

Now denote the tree in Fig.1. by TR , rotate it counter-clockwise at 90(, and place a mirror AB at its root V in parallel to its levels. Then, we can see the visual result of such the transformation in Fig. 2, that demonstrates a cognitive visual image of the mirror-symmetrical (MS) 1-1-correspodence, say (, between the original tree TR and its mirror image - the tree TL . The math interpretation of the levels of the trees TR and TL is also shown in Fig. 2.

Consider a real number, x ( [0,1], written in binary system, and its equivalent notation in the form of the corresponding infinite sum:

x = 0.a1 a2 a3 ... an ... ( [pic], ( i(1 [[ai =0] or [ai =1]]. (2)

In connection with the MS-mapping ( there are interesting only two following cases.

CASE 1. Let x be a rational number with a 0-tail, i.e., in (2) (k(i>k [ai =0]. Denote the set of such numbers as Q0. By (, this x is associated with an integer [pic] possessing the property:

[pic], so that [pic] is a finite natural number, (3)

and we have |Q0| = |N| = (0.

CASE 2. Let x be an irrational (or rational with a non-zero tail) number, i.e., in (2) (i(k>i

[ak = 1]. Denote the set of such numbers as D. Then, by virtue of (, this x is associated with an integer [pic] with the following property:

[pic] = S, (4)

By math analysis, the infinite sum S is a limit of the infinite sequence of its partial sums,

s1,s2,s3,. . .,sn,. . ., where sn = [pic] < (, so that sn is a finite natural number. (5)

It is obvious that (5) is an infinite subsequence of (1). The main properties of this new mathematical objects are given by the following statements (Zenkin, 1999, 1997).

MS-THEOREM 1. For all [pic](D the ordinal type of the transfinite integer [pic] is equal to Cantor's minimal transfinite ordinal number (, i.e., Ord([pic]) = (.

MS-COROLLARY 1. Any infinite subsequence of the natural series (*) is an individual mathematical object which is a transfinite ordinal 'integer' of the Cantor's (-type.

MS-COROLLARY 2. For any two real numbers x1, x2 ( X, if x1(x2 then[pic] for (-associated transfinite integers [pic],[pic] ( D.

The MS-mapping gives us a unique opportunity to produce absolutely rigorous ontological, but conditional inferences concerning the existence of such transfinite objects.

'IF-THEN' MS-THEOREM 1. IF a geometrical point x([0,1] is an individual object THEN the corresponding infinite path (1) of the tree TR arrives at its (-level.

COROLLARY 1. All infinite (1)-paths of the tree TR are actual and arrive at the (-level.

'IF-THEN' MS-THEOREM 2. IF an infinite (1)-path of TR arrives at the (-level THEN the (-associated infinite (4)-path [pic] of TL arrives at the (-level of the tree TL as well.

COROLLARY 1. All infinite (4)-paths of the tree TL are actual and arrive at the (-level.

'IF-THEN' MS-THEOREM 3. IF a (4)-path [pic] of the tree TL arrives at the (-level THEN Cantor's ordinal number of the transfinite integer [pic] is equal to (, i.e., Ord{[pic]}=(.

COROLLARY 1 By (, Card{all [pic](TL } = Card{all x(TR }, i.e. |D|=|X|=C.

'IF-THEN' MS-THEOREM 4. IF the geometrical point x of the segment [0,1] exists as an individual object THEN there exists the Cantor least transfinite integer ( and the infinity of all sequences (1) - (5) are thus actual.

The theorem 4 is a strict (unlike Cantor) conditional MS-proof of the Cantor's ( existence.

Thus the new transfinite mathematical objects (4) have the same ontological status as the usual irrational numbers (2). It is quite appropriate here to remind the well-known oracular G.Cantor's words: “Transfinite numbers themselves are, in a certain sense, new irrationalities.[...] It can be certainly said: transfinite numbers stand and fall together with finite irrational numbers.” (Cantor, 1985). All these conditional MS-statements are valid only iff all infinite sets are, according to Cantor, actual. As to a strict algorithmical definition of the concepts of potential and actual infinity see the [FOM]-message at . An artistic dynamical cognitive movie "A colour-musical "black hole" of the continuum problem" will be demonstrated during our report (Zenkin, 2001, 1991).

MAIN CONCLUSION. Let ( be a set of all countable ordinals. According to Cantor, |(| = (1. On the other hand, the set [pic] of all transfinite (-ordinals is a proper subset of (. Consequently, |[pic]| = C ( |(| = (1, and Cantor's CH has thus the following solution: C ( (1. However, the proven fact that there is not the only Cantor's ordinal (, but the uncountable set of different ordinals of (-type means that all Cantor's theory of transfinite ordinals is informally incomplete, is based on a doubtful semantics and must be essentially revised as a whole. It should be emphasized especially once more that all the conditional ontological MS-Theorems above are strictly valid iff the Cantor's very non-naïve axiom "all infinite sets are actual" holds (Zenkin, 2002).

References

Cantor Georg, "Proceedings in Set Theory". - Moscow: NAUKA, 1985 (in Russian).

Cohen, Paul J. Set Theory and the Continuum Hypothesis. - Moscow : MIR, 1969 (in Russian).

Zenkin A.A., Cognitive computer graphics. - Moscow : "Nauka",1991. See Synopsis at: .

Zenkin A.A., Scientific Intuition Of Genii Against Mytho-"Logic" Of Transfinite Cantor's Paradise. International Symposium "Philosophical Insights into Logic and Mathematics ", 2002, NANCY, France.

Zenkin A.A., Goedel's numbering of multi-modal texts. - The Bulletin of Symbolic Logic, Vol.8, No.1, 2002, p. 180.

Zenkin A.A., Cognitive (Semantic) Visualization Of The Continuum Problem. - "Visual Mathematics", Volume 1, No. 2, 1999. at the WEB-Sites:

Zenkins Alexander and Anton, Presentation "The Unity of the Left-Hemispheric, Rational, Abstract Thinking and the Right-Hemispheric, Intuitive, Visual One. Intellectual Aesthetics of Mathematical Abstractions". – The 5th International Congress & Exhibition of the ISIS of Symmetry. Sydney, 2001. Intersections of Art and Science:

Zenkin A.A. Cognitive Visualization of the Continuum Problem and of the Hyper-Real Numbers Theory. - International Conference "Analyse et Logique", UMH, Mons, Belgia, 1997. Abstracts, pp. 93-94 (1997).

A.A.Zenkin, Does Axiomatic Set Theory Need New Axiom? - 12th. International Congress of Logic, Methodology and Philosophy of Science (LMPS-2003). Oviedo (Spain), August 7-13, 2003 Abstract. (Has been accepted for publication. See at:

)

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a)...( ... n ... 4 3 2 1 0 1 2 3 4 ... n ... ( ...

b)...2( ... 2n ... 24 23 22 21 ( 2-1 2-2 2-3 2-4 ... 2-n ... 2-( ...

c)...[pic]( ... [pic]n ... [pic]4 [pic]3 [pic]2 [pic]1 ( a1 a2 a3 a4 ... an ... a( ...

Fig.2. Cognitive Visualization of the Continuum Problem:

a) level's numbers of the trees; b) powers of the base 2 in the binary system;

c) binary representation of the “in-both-side transfinite” hyper-real numbers.

( A.Z., 1975

TL

TR

B

A

x = [pic]

[pic] = [pic]

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