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Topic: [HM] Infinity of primes in Euclid
Replies: 1   Last Post: Dec 26, 2005 5:30 PM

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Alexander Zenkin

Posts: 50
Registered: 12/3/04
[HM] Infinity of primes in Euclid
Posted: Dec 16, 2005 1:33 PM
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If my memory is accurate, recently somebody (either at [FOM] or [HM]
list) cited a similar attitude of Euclid to a line.

Something like that: Euclid never used the term 'infinite line', but
used the term 'an unlimited line'.

This is also an example of Aristotle's "Potential Infinity" as opposed
to an "Actual (Completed) Infinity".

Could anybody remind an exact citation of this point in Euclid?

Alexander Zenkin



-----Original Message-----
From: Mark Bridger
Sent: Thursday, December 15, 2005 5:33 PM
To: fom@cs.nyu.edu
Subject: [FOM] Infinity of primes in Euclid

Euclid does NOT say that there are infinitely many primes. Rather, he
proves that for any number of primes there must be another. The
reference is: Book IX, Proposition 20: "Prime numbers are more than any
assigned multitude of prime numbers." ("The History of Mathematics - A
Reader" ed. J. Fauvel, J. Gray.)

This is an example of Aristotle's "Potential Infinity" as opposed to a
"Completed Infinity."

M. Bridger




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