What if I do a proof like this?
If the set of twin prime numbers is finite then we can make a list, let S be the list of twin prime numbers,
S=P1,P2,P3,P4,PN
Now let us construct a number Q such that Q=P1×P2×P3×P4×PN+1
If Q is a twin prime then there exists a larger twin prime then on S
If Q is composite then non of the twin primes of S will divide Q.
Both of this conclusions yield to a contradiction, therefore there are infinitely many twin prime numbers
