Re: An infinite debate


Randy Poe wrote:
Ajeet wrote:
Tim Peters wrote:
[Ajeet]
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).

Then this consequence follows:

If r is an element of S, r's decimal representation is finite.

Simply because there must be an i such that r is an element of S_i, in which
case r's decimal representation has i digits.

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

Good luck ;-)

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.

Why? This effectively assumes what you're trying to prove. What /does/
follow is that r's decimal representation is not finite.

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.

Whereas what actually obtains is that every finite prefix of r is contained
in S. The idea that there must be some k such that r's first k digits
appear in S while r's first k+1 digits don't is simply false.


Yes ... I have been given this argument. I understand it but dont get
the premise. For example, you will probably argue that the set of all
finite strings will not contain an infinite string.

Correct.

But then what is
the length of the largest finite string in the set?

Why does there have to be a largest finite?

If it is finite,
then shouldnt there be such a maximum finite length string?

No. Why must the process of adding 1 to finite numbers
to get new finite numbers end?

There is no
such maximum finite number .. i.e. it is infinitely large.

Eh? Because a finite number can be arbitrarily large it
can't be finite?


I am not assuming a largest finite. In fact im arguing for the
opposite. Let me ask you, since N is the set of natural numbers, what
number of N do you get when you infinitely increment any number by 1.
This number would clearly be infinitely large.

If the above statement is true, then the proof I mentioned must be
true. And frankly I cannot see why the statement should be false.

- Randy

.

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