Re: An infinite debate
- From: "Randy Poe" <poespam-trap@xxxxxxxxx>
- Date: 7 Nov 2006 10:43:31 -0800
Ajeet wrote:
Randy Poe wrote:
Ajeet wrote:
Randy Poe wrote:
Ajeet wrote:
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.
You do not get an element of N by incrementing infinitely
many times.
Then what do you get?
You don't get anything. "Incrementing infinitely many times"
is not something you ever do.
I agree. This is more of thought experiment than an actual
implementation. For that matter the initial set that I constructed is a
union of infinitely many sets. No one would ever do that either.
Suppose a machine starts incrementing a printing
out natural numbers. At what point would they stop being natural
numbers?
Never.
The difference between the number at this point and the number
at the start would necessarily have to be infinite.
"This point" doesn't exist. They never stop being natural
numbers. The process of creating finite natural numbers
doesn't end.
Tell me if this does not make sense.
1. The machine prints out natural numbers each of which is greater that
the all the natural numbers it has printed earlier.
Yes.
2. This machine can theoretically do this infinitely.
It can run for arbitrarily long finite time. Every time you
check it, you will be at a finite time T.
3. After an infinite number of prints
Again, each time you get to a point of confusion we can
identify that you want to go to the end of something that
doesn't end.
"After an infinite number of prints" makes no sense. The
process of making prints doesn't end. There is no time
which is after that. All times are somewhere in the middle
of the print-making process.
ALL TIMES.
(the number of prints will be
infinite because there are infinitely many natural numbers), the number
printed by the machine will be "infinitely large".
The point here that a finite amount, when incremented by another finite
amount and "infinite" number of times, will result in an infinitely
large amount.
But there will never be a time at which an infinite number of
increments has occurred.
The reasoning you just mentioned is part of the conter-proof that I
gave in the original post. I agree with your post, except for one
thing. If a natural number can be infinitely large (which you dont
agree with, please ref to earlier in this post for my reasons), then
this point is again moot. Since S_i where i is infinitely large, will
contain infinitely large elements.
There is no S_i where i is infinitely large.
Note all of this is based on the contention that natural numbers can be
infinitely large. At this point I dont see any convincing argument as
to why that shouldnt be.
The natural numbers we work with, the ones defined by
Peano's axioms, are finite. Whether you believe that or not,
can you accept that there is something called "the set of
finite natural numbers"? If so, then we can reduce our discussion
to that: How big is the set of finite natural numbers?
- Randy
.
- References:
- An infinite debate
- From: Ajeet
- Re: An infinite debate
- From: Tim Peters
- Re: An infinite debate
- From: Ajeet
- Re: An infinite debate
- From: Randy Poe
- Re: An infinite debate
- From: Ajeet
- Re: An infinite debate
- From: Randy Poe
- An infinite debate
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