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 elaborated 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 point 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 exists (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.
|