From: Hal Finney <hal@rain.org>

Date: Tue Dec 09 1997 - 14:58:37 PST

Date: Tue Dec 09 1997 - 14:58:37 PST

Perry E. Metzger, <perry@piermont.com>, writes:

*> Lets say there is an ever expanding set of problems that we want to
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*> calculate, and the set of problems we wish to calculate expands faster
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*> than the amount of computation we have available per unit time.
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*>
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*> As someone with a mathematical background and a background in C.S.,
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*> I find a lot of the arguments unconvincing because of just such
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*> difficulties. The problem is never having unlimited computation. In
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*> theory, if the universe lasts infinitely and we have enough
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*> electricity my laptop will perform an infinite amount of
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*> computation -- but it can't do enough per unit time to be of interest.
*

This is one of the paradoxes of infinity. It is like the hotel with

an infinite number of rooms, all occupied. Now a new guest comes along.

The manager makes space for him by moving the person from room 1 into

room 2, the one from room 2 into room 3, etc. This leaves room 1 empty

and the new guest can stay.

Next, an infinite number of guests come along. Again the manager finds

space, this time moving the person from room 1 to room 2, from room 2 to

room 4, from room 3 to room 6, ... from room n to room 2n. This frees

up all the odd numbered rooms, of which there are an infinite number,

making enough rooms for the infinity of new guests.

In the case of the computation, suppose every time you solve one problem

you generate 10 new ones. You still are able to get to every problem

eventually. You generated problems 0-9 at time 0, problems 10-19 at time

1, problems 20-29 at time 2, and in general problems 10n - 10n+9 at time n.

You are still able to retire problem x at time x, so every problem gets

dealt with in its turn and there are no problems which never get solved.

This reasoning obviously does not apply to any finite number, only to

infinity, which is why it is so important to have an infinite amount of

computation and not just a very large but finite amount.

Hal

Received on Tue Dec 9 23:06:48 1997

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