Green-Tao theorem consequence is not contradictory? Why?

Question

According to the Green–Tao theorem:

there exist arbitrarily long sequences of primes in arithmetic progression.

However, one of the properties of Green-Tao theorem is:

Any given arithmetic progression of primes has a finite length.

I can only read two contradictory sentences. What am I missing here?

Answer 1

The theorem says that, for each $n$, there is a sequence of $n$ primes in arithmetic progression. This in no way contradicts the fact that there is no infinite arithmetic sequence such that all of its terms are prime numbers.

Answer 2

These statements mean that for any number $n$, there are infinitely many arithmetic progressions composed entirely of primes, with no primes between consecutive members of the progression, of length at least $n$.

It does not mean that there exist arithmetic progressions of infinite length comprised only of primes – it is easy to show that no such progression exists by simple modular arithmetic.