An infinite debate

Hi All,

Consider the following set :

S = U S_i (0<=i<infinity and U denotes the set operation of union)

where S_i is the set of all numbers, whose decimal representations are
of length i. For example the number 3.4 belongs to S_2. (assume the
decimal does not add to the length, so 34 would also belong to S_2).

Argument : S = R (set of all real numbers)
========

Proof:
=====
Suppose a real number "r" does not exist in S. let the decimal
representation of "r" be d_0.d_1d_2d_3......(to infinity) where d_i is
the ith digit.
+Since r does not belong to S, there will be some digit k which will be
off. More precisely

the prefix d_0.d_1d_2d_3....d_k-1 belongs to S but,
the prefix d_0.d_1d_2d_3....d_k-1d_k does not belong to S

But this is not possible because that prefix would have been added in
all sets S_j where j >= k+1.

Therefore no such k exists.

Therefore r belongs to S.

Counter - Proof:
=============
Consider the real sqrt(2). The decimal representation of this real
would be infinite.

Now all the sets in the union are disjoint. Therefore there must exist
a set S_m which contains sqrt(2) and no other set can contain this
number.

However, no such m can exist, because then m would need to be infinite.
(there is no natural number "infinity")

Hence S cannot be R.

Paradox anyone?
Ajeet

.

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