IPP - proof and question

This is a paraphrase of Chris's infinitude of primes, indirect
method. Any errors are entirely mine.

I hope you can follow it thru to the end, because there
I have one of those nagging questions that I think has
an obvious answer, but it seems so obvious I
can't think of it.

First, the proof uses the following rule of inference:

P => Q, not-Q
__________
not-P

where

P: the primes are finite
Q: the primes can be listed p1, ... , pk for some finite k.

SO:

suppose P. Then since any finite set is countable, there's
a 1-1 mapping from the finite set of primes to the integers.
So we can list the primes p1, ... , pk for some finite
k. (this needs proof but the proof is not necessary to
show for my purposes).

Then we can form N = p1*p2* ... *pk + 1, by the closure of the
integers with respect to multiplication and addition.

Then because of the unique factorization theorem, N is the
unique product of at least 1 prime > 1.

There are two possibilities:

(1) the unique factorization of N includes a prime p < N.
(2) the unique factorization of N includes no prime p < N.
So N itself is prime.

Assume (1). Then either p is one of the p1, ..., pk, or not.
If it is, then it divides p1*p2*....*pn AND N. But this
means it divides 1, which is absurd. So p can't be one of
the p1,....,pk. Therefore Q is false - the primes cannot be
listed p1,...,pk for some finite k.

Assume (2). Then N is not one of the original primes,
since it is greater than all of them, and
again Q is false - the primes cannot be listed
p1,...,pk for some finite k.

So since Q is false in all possible cases, P also is false
(by the rule of inference) and the primes are infinite.

Here's the question:

Why can't we follow AP's indirect proof and claim that
the listing p1, ...., pk forbids the existence of a prime
p < N that isn't one of p1,....,pk, but ALLOWS the
existence of the prime N not one of the p1,...,pk. It
seems obvious that we can't but I can't put my
finger on WHY we can't.

Obviously the only reason for this reasoning by AP
is to preserve his twin primes infinitude proof, but
STILL it needs to be clearly argued against by
a valid mathematical/logical argument, which
I personally can't come up with.

Thanks.

Ken

.

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