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6. SOME MIRROR-LIKE PROOFS
OF MIRROR-LIKE THEOREMS

Probably, the first use of mirror-arguments in metamathematical proofs belong to S.Kleene (see his proof of the Cantor-Bernstein Theorem in [12], p. 18 ). Modern and much more advanced and systematic investigations in this very promising area of logic are elaborating in Visual Inference Lab of the Indiana University by J.Barwise, J. Etchemendy, E.Hammer [13].

Using their ideas on symmetry and multimedia argumentation (the visual proof, almost by L.E.J.Brouwer) and results obtained above, we formulate some mirror-statements (Mirror-Theorems) based on the cognitive visual image of the 1-1-correspondence between binary trees TR and TL shown in Fig. 2.

M-THEOREM 1. IF the geometrical point x of the segment [0,1] is an individual object THEN the corresponding infinite path x of the tree TR attaines its w -level.

COROLLARY 1. All infinite paths x of the tree TR attaine its w -level.

COROLLARY 2. The path x = 0.000...1w (the geometric poit in usual sense) is the maximal transfinite small number (maximal infinitesimal): since |x| = Card { x } = 2 - w.

M-THEOREM 2. IF an infinite path x of the tree TR attaines the w -level THEN the corresponding infinite path x- of the tree TL attaines the w -level of the tree TL.

COROLLARY 1. All infinite paths x- of the tree TL attaine its w -level.

M-THEOREM 3. IF a path x- of the tree TL attaines the w -level THEN Ord{x-}= w.

COROLLARY 1. The path x- = 1w ... 000. is the minimal transfinite large number: since |x-| = Card {x-} = 2 w = À0 .

COROLLARY 2 By virtue of Y , Card{all x-Î TL } = Card{all xÎ TR }.

M-THEOREM 4. IF the geometrical point x of the segment [0,1] exists as an individual object THEN there exists the Cantor least trans-finite integer w .

M-THEOREM 5. IF a path x- of the tree TL attaines the w -level THEN there exist (by Peano!) (w+1)-th level in the tree TL and an infinte path x- which attaines the (w+1)-level.

COROLLARY 1. All infinite paths x- of the tree TL attaine its (w+1)-level.

M-THEOREM 6. IF the transfinite (w+1)-level in the tree TL exists THEN there exists the corresponding transfinite (w+1)-level of the tree TR.

COROLLARY 1. All transfinite paths x of the tree TR attaine its (w+1)-level.

COROLLARY 2. There is the infinitesimal x = 0.000...0w .1w+1. of the order (w+1) and Card{x}=2-(w +1)

COROLLARY 3. There exist infinitesimals x of any transfinite-small order a , so that

Ord{ x } = a and Card { x } = 2-a.

Remark here that all the Mirror-Theorems are condition statements. Therefore all objections and doubts concerning the tree TL (particularly, the existence and rather unusual properties of the transfinite integers x-ÎX -) are mirror-likely reflected into the same objections and doubts concerning the tree TR (particularly, the existence and properties of the usual real numbers xÎ D).

Some of obvious methodological consequences of the consideration above are presented in Fig. 3 and Fig. 4.



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