## "problem" with Euclid's proof of the Infinitude of Primes

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• I am having a problem with Euclid s proof of the Infinitude of Primes. Sorry for the not so short post ahead :) Please see this link for the modern
Message 1 of 1 , Jan 31, 2004
I am having a "problem" with Euclid's proof of the Infinitude of Primes.

Sorry for the 'not so short post' ahead :)

Please see this link for the modern translation of the proof (note there are 2 versions there, don't forget the one in the bottom of the page :).
These versions are considered true (of course) and can be found in most textbooks.
http://primes.utm.edu/notes/proofs/infinite/euclids.html

Now the "problem" is that I came across this "ShortCut Version" of proof which looks like a real shortcut to Euclid's proof.
I am wondering if there is any mistake/hidden assumption/etc. I am missing here:
The proof gos like that:
"
(1). The prime numbers are the numbers p1,p2,...,pn,... of set S.
Suppose S is finite [supposition].

(2). If S is finite then: p1,p2,...,pn is the complete series set, so that pn is the largest prime.

(3). Form W+1 = (p1xp2x,...,xpn) + 1.
W+1 is not in the set S so it is not prime.
W+1 is not divisible by any of primes in S_P so it
is prime.
Reverse supposition and primes are infinite.
"

P.S.
I do not think the 'counter example': 2*3*5*7*11*13+1= 30031 = 59*509 => hence a composite W+1, work here.
Why?
because 2,3,5,7,11,13 isn't the Actual list of All primes. its a partial list and for that reason doesn't necessarily has the same attribute as the complete list.
To me this looks the same as this:
say 1,2,3,4,5 are all the "favorite" numbers.
["favorite" - a set I just made up].
My Great Theorem: the sum of all "favorite" numbers is odd.
proof: 1+2+3+4+5=15. QED :)
Now one would say: "but look - 1+2+3=6" so there is a partial sum that is Even.
And I would answer: but my theorem is about the sum of ALL the favorite numbers, so we are talking about different cases!.
* would I be correct? If not, why?