Infinitude of primes

We want to show that there are infinitely many primes. We will argue this by assuming the opposite and deriving a contradiction.

[edit] Argument

We introduce a finite set $P$ of all primes, and its product $m$ .

Introduce $P,m$ .

Assume for all $n \in \mathbb{N}$ , if $n$ is prime then $n \in P$ .

Assume $P$ is a finite set, $P$ is non-empty, and $P \subset \mathbb{N}$ .

Assume $m = P_0 \cdot \ldots \cdot P_{|P|-1}$ .

The above notation for the product of a subset of $\mathbb{R}$ has equivalent alternatives.

Assert $m = \prod P$ .

Assert $m = \prod_{x \in P} x$ .

We now demonstrate that the above assumptions allow us to derive a contradiction.

Assert $m \in \mathbb{N}$ .

Assert for any $p \in \mathbb{N}$ ,

if $p$ is a prime factor of $m+1$ then

$p$ is not a factor of $m$ ,

$p$ is prime,

$p \in P$ ,

$p$ is a factor of $m$ ,

there is a contradiction.