[FOM] Infinity of primes in Euclid

Alexander Zenkin alexzen at com2com.ru
Fri Dec 16 13:33:40 EST 2005

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: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf
Of Mark Bridger
Sent: Thursday, December 15, 2005 5:33 PM
To: fom at 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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