In Euclid's Elements Book XI proposition 20 (http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html), Euclid proves that:

Prime numbers are more than any assigned multitude of prime numbers.

I know that this is supposed to say something similar as there are infinitely many primes, but I don't really see this from this wording.

In my mind, this sentence means something like:

There are more prime numbers than any amount of prime numbers.

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    $\begingroup$ Euclid is not originally in English, so this phrasing is an artefact of the translation. $\endgroup$ – Mark Bennet Sep 5 '14 at 12:38
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    $\begingroup$ There was a conscious avoidance of the inf****y word. $\endgroup$ – André Nicolas Sep 5 '14 at 12:38
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    $\begingroup$ "There are more prime numbers than in any given set of prime numbers" $\endgroup$ – Taemyr Sep 5 '14 at 14:30
  • $\begingroup$ @AndréNicolas why? $\endgroup$ – YoTengoUnLCD Jun 26 '16 at 5:53
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    $\begingroup$ @YoTengoUnLCD: Because they were aware that reasoning about inf****y could lead to difficulties, for example the Zeno paradoxes. We also avoid saying that "at" infinity, $x^2/(x^2+1)=1$, replacing it by for every $\epsilon\gt 0 \dots$. $\endgroup$ – André Nicolas Jun 26 '16 at 6:00

You can see Aristotle and Mathematics and Actual infinity.

According to the Aristotelian philosophy, we cannot legitimately "handle" actual infinity; i.e. we have no experience of an infinite "collection" but only of an unlimited iterative process (the potential infinity).

Euclid's statement must be understood in this context : we never have a "complete" infinite set of prime numbers, but we have a procedure that, for a finite collection of prime numbers whatever, can "produce" a new prime which is not in the collection.

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    $\begingroup$ Right. So I conclude that (1) Euclid never said there are infinitely many prime numbers, (2) he saw no reason to qualify "multitude" by "finite", since that is implicit in the notion of multitude, and (3) he would probably have been horrified by the idea that posterity would attribute to him a proof of the existence of infinitely many primes. $\endgroup$ – Marc van Leeuwen Sep 6 '14 at 8:15
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    $\begingroup$ @MarcvanLeeuwen - I'm not sure about so "drastic" a conclusion ... I'm quite sure that you cannot find the word "infinite" in Euclid's Elements, but "mature" Greek math was not "horrified" by it (see Archimedes). My understanding is that if a "mainstream" Greek philosophers has to comemnt Euclid's proof, he should say that the proof show that the prime numbers are potentially infinite... $\endgroup$ – Mauro ALLEGRANZA Sep 8 '14 at 9:31

There are more prime numbers than any (finite) list of them can contain.

cantor's diagonal argument for the uncountability of the reals follows the same pattern: given any list - even infinitely long - of real numbers, he can prove that there are many elements missing; there are more real numbers than any (even countably infinite) list of them can contain.


The statement means the following:

No matter how many primes you have already found, you can always find a new prime number.


I disagree with your interpretation at the end, of what he meant to say. I think if you interpret his statement, you should come up with something similar to this idea. "Prime numbers" changes to "there exists prime numbers", "are more" should be read as "greater than", and finally "than any assigned multitude of prime numbers" meaning then any prime number that actually takes a non variable value. In full my interpretation is this.

"There exists prime numbers greater than any prime number that actually takes on a non variable form."

This is what I 'beleive' to myself he said, further I wonder if part of what he actually said has been removed? Reguardless I think it shows knowledge of the idea of prime numbers as a variable for representing all of them. I think it also is clear evidence of the understanding that there are infinite number of prime numbers.

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    $\begingroup$ What is a "non-variable form"? $\endgroup$ – Milo Brandt Mar 6 '15 at 1:30
  • $\begingroup$ I have no idea. If you are referring to my interpretation above, it refers to, or explicitly does NOT refer to something that varies or is not permanent. A prime number may can be 2 different things at different times. We may or may not know what it actually is. $\endgroup$ – marshal craft Mar 6 '15 at 1:50

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