Newsgroups: comp.ai.philosophy
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From: alphonce@cs.ubc.ca (Carl Alphonce)
Subject: Re: how many distinct thoughts can a person have?
Message-ID: <1991Jun26.162710.18052@cs.ubc.ca>
Sender: usenet@cs.ubc.ca (Usenet News)
Reply-To: alphonce@cs.ubc.ca (Carl Alphonce)
Organization: University of British Columbia, Vancouver, B.C., Canada
References: <1991Jun19.033316.18773@athena.mit.edu> <1991Jun20.083708.13355@tygra.Michigan.COM> <2772@infinet.UUCP> <2773@infinet.UUCP>
Distribution: usa
Date: Wed, 26 Jun 91 16:27:10 GMT

In article <2773@infinet.UUCP>, sena@infinet.UUCP (Fred Sena) writes:
|> 
|> I think that there is some confusion here between "uncountable" and "infinite".
|> I don't believe that they are the same thing at all because I think that
|> there are many things that are uncountable yet finite.
|> 
|> For example, if I were to ask you the number of cars that are running on the
|> roads at a particular instant, the number would be uncountable because you
|> can't be everywhere at once to count them and the number of cars would change
|> as you were in the process of counting.  I think that everyone would agree
|> that the number would be finite as well, since there are a finite number of
|> cars.
|> 

I agree that there is some confusion about this.  The terms finite, infinite,
countable, and uncountable, all have precise mathematical definitions.  However,
they are also used in contexts where it is not clear whether the mathematical
meaning or the "common" meaning is the one which was meant to be conveyed.

Maybe some attempt can be made to indicate when a "technical" meaning of a
word is what is to be conveyed, so that these (fruitless) discussions can be
avoided in future.

For those who are familiar with the mathematicaal definitions of these terms,
feel free to stop reading here.  For those unfamiliar, here is a (hopefully)
non-technical (hopefully) brief summary:

Without getting technical, we may loosely define the terms as follows:

A set is finite if there are n elements in the set (where n is a natural
	number).

(Alternately, where one defines 0 to be {} (ie: the "empty" set - the set with
						no members)
				1 to be succ(0) = {0} = {{}}
				2 to be succ(1) = {0,1} = {{},{{}}}
				. . .
			      n+1 to be succ(n) = {0,1,...,n},

one can say that a set A is finite if there is a one-to-one mapping from
A to some natural number n.)

A set A is infinite if there is no natural number n such that there is a 
one-to-one mapping from A to n.

If we let N, the set of natural numbers, be the set {0,1,...,n,...},
then a set A is said to be countable if there is a one-to-one mapping
from A to N.

Note that (obviously) the set natural numbers is countable.  Furthermore,
the set of even numbers is countable (take the one-to-one mapping to be
f:x -> 2x), and the set of prime numbers is countable (let f be the 
function mapping n to the n-th prime number).  Even the rational numbers
are countable.

However, there are some uncountable sets also.  The set of real numbers,
for example, is uncoutable.  Thus, there is no one-to-one mapping from
the set of natural numbers to the set of real numbers.  For a proof of
this, see any good logic or set theory book.

Carl.
alphonce@cs.ubc.ca
