[FOM] The uncountability of continuum. 1.
Alexander Zenkin
alexzen at com2com.ru
Mon Mar 7 17:22:37 EST 2005
Some FOMers were uttering an opinion that there are two different
proofs of the uncountability of continuum:
1) the direct proof given by Cantor in 1873/4 and
2) the indirect proof (by Reductio ad Absurdum (RAA)) given by
Cantor in 1890/1
(as a special case of the common proof of the inequality |P(Z)| > |Z| for
any set Z).
For example, S.Kleene ("Introduction to metamathematics") gives the
direct proof in the following quite strange (from the classical logic point
of view) form.
THEOREM 1. The set of real numbers, X=(0,1], is uncountable.
(It should be noted that Kleene even does not formulate explicitly this
statement as a THEOREM).
PROOF. Let {A:} "x1, x2, x3, . . . (1) be an infinite list or
enumeration of some but NOT NECESSARY ALL of the real numbers belonging to
the interval X=(0,1]". The application of Cantor's diagonal method (CDM) to
the list (1), generates a new real number, say x*, not belonging to (1).
Consequently, {A:} "the given list (1) is an infinite list or enumeration of
some but NOT ALL of the real numbers belonging to the interval X=(0,1]".
>From the arbitrariness of the list (1), it follows that there is not a
1-1-correspondence between X and N={1,2,3,. . .}, i.e., X is uncountable.
The proof really produces no RAA-contradiction, but has a form of a
quite senseless "deduction" A -- > A.
However it is obvious that Kleene's condition A is in reality a
composition of two opposite cases.
Case 1.
Let {B:} "a list x1, x2, x3, . . . (1) contains NOT ALL real numbers from
X=(0,1]".
The application of CDM to (1), generates a new real number x*, not belonging
to (1). Consequently, {B:} "the given list (1) contains NOT ALL real numbers
from X=(0,1]".
The "deduction" B -- > B is the vicious circle error and therefore
the Case 1 proves nothing from the classical logic point of view. In this
case Cantor's new real number x* is simply a concrete specimen (one of many)
of a real number, not belonging to the list which, according to the
condition B, "contains NOT ALL real numbers from X=(0,1]".
Case 2.
Assume that {C:} "a list x1, x2, x3, . . . (1) contains ALL reals from
X=(0,1]".
The application of CDM to (1), generates a new real number x*, not belonging
to (1). Consequently, {not-C:} "the given list (1) contains NOT ALL reals
from X=(0,1]". From the contradiction (of a very specific form, C -- >
not-C), it follows that C is false. From the arbitrariness of the list (1),
it follows that there is not a 1-1-correspondence between X and N, i.e., X
is uncountable.
In this case Cantor's new real number x* is a counter-example which
disproves a common statement C.
It is obvious that the Case 2 is an indirect proof based on Reductio
ad Absurdum method.
So, the only logical basis in order to state the uncountability of
continuum is Cantor's INDIRECT proof-1890 basing on the Reductio ad Absurdum
method, i.e., the Case 2 above.
We have thus the following important methodological
THESIS-01. There is not a direct proof of the uncountability of continuum.
There is only Cantor's indirect RAA-proof-1890 of the uncountability of
continuum, i.e., the proof by means of Reductio ad Absurdum method.
P.S. Without Thesis-01 I can't explain to my students the fact that some
known set-theorists are deeply wrong stating that there is a direct proof of
the uncountability of continuum. An important philosophical meaning of the
Thesis-01 is obvious, I think.
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