6. SOME MIRRORLIKE PROOFS
OF MIRRORLIKE THEOREMS
Probably, the first use of mirrorarguments in metamathematical proofs
belong to S.Kleene (see his proof of the CantorBernstein 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 mirrorstatements (MirrorTheorems) based on the cognitive visual
image of the 11correspondence between binary trees T_{R}
and T_{L} shown in Fig. 2.
MTHEOREM 1. IF the geometrical point x
of the segment [0,1] is an individual object THEN the corresponding
infinite path x of the tree T_{R} attaines
its w level.
COROLLARY 1. All infinite paths x of the tree T_{R}
attaine its w level.
COROLLARY 2. The path x^{} = 0.000...1_{w}
(the geometric point in usual sense) is the maximal transfinite
small number (maximal infinitesimal): since
x = Card { x } = 2 ^{ w}.
MTHEOREM 2. IF an infinite path x of the tree
T_{R} attaines the w level
THEN the corresponding infinite path x^{}
of the tree T_{L} attaines the w
level of the tree T_{L}.
COROLLARY 1. All infinite paths x^{}
of the tree T_{L} attaine its w
level.
MTHEOREM 3. IF a path x^{}
of the tree T_{L} attaines the w
level THEN Ord{x^{}}= w.
COROLLARY 1. The path x^{} =
1_{w} ... 000. is the minimal transfinite
large number: since x^{}
= Card {x^{}} = 2 ^{w}
= À_{0} .
COROLLARY 2 By virtue of Y , Card{all x^{}Î
T_{L} } = Card{all xÎ
T_{R} }.
MTHEOREM 4. IF the geometrical point x of the
segment [0,1] exists as an individual object THEN there exists
the Cantor least transfinite integer w .
MTHEOREM 5. IF a path x^{}
of the tree T_{L} attaines the w
level THEN there exists (by Peano!) (w+1)th
level in the tree T_{L} and an infinte path x^{}
which attaines the (w+1)level.
COROLLARY 1. All infinite paths x^{}
of the tree T_{L} attaine its (w+1)level.
MTHEOREM 6. IF the transfinite (w+1)level
in the tree T_{L} exists THEN there exists the
corresponding transfinite (w+1)level
of the tree T_{R}.
COROLLARY 1. All transfinite paths x of the tree
T_{R} attaine its (w+1)level.
COROLLARY 2. There is the infinitesimal x = 0.000...0_{w}
.1_{w+1}. of the order (w+1)
and Card{x}=2^{(w +1)}
COROLLARY 3. There exist infinitesimals x of any
transfinitesmall order a , so that
Ord{ x } = a and Card { x } = 2^{a}.
Remark here that all the MirrorTheorems are condition statements.
Therefore all objections and doubts concerning the tree T_{L}
(particularly, the existence and rather unusual properties of the transfinite
integers x^{}ÎX ^{})
are mirrorlikely reflected into the same objections and doubts concerning
the tree T_{R} (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.
