Use of infinitude of primes in the Green-Tao theorem [closed]

In a video I watched last night on nuking mathematical mosquitos, Matt Parker gave the following proof of the infinitude of primes: suppose there are finitely many primes. The Green-Tao theorem says there are arbitrarily long arithmetic progressions in the primes, hence there cannot be finitely many primes. Contradiction.

Leaving aside the slightly dubious and unnecessary use of proof by contradiction, it made me wonder whether or not this proof was circular (and Parker himself remarks: "Green and Tao took it as a given that there are infinitely many prime numbers and my pithy proof may very well be circular!"). Namely, is there some fact about the infinitude of primes that is used deep in the proof of the Green-Tao theorem? For instance, in some density arguments or similar?

closed as off-topic by Włodzimierz Holsztyński, András Bátkai, Alex Degtyarev, Ryan Budney, JoëlFeb 12 '16 at 11:41

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• I think this proof is not circular. On the other hand, as I recall, Green and Tao do use basic facts like $\zeta(s)$ has a pole at $s=1$ which implies readily that there are infinitely many primes. – GH from MO Feb 11 '16 at 23:01
• mathoverflow.net/questions/42512/… – Peter Humphries Feb 11 '16 at 23:12
• What they use about primes (Goldston-Yildirim result) is of course much much more than infinitude. – Fedor Petrov Feb 11 '16 at 23:14
• In a way the question appears to be answered in the question you link to as pointed out by @PeterHumphries Why is that answer not sufficient? – user9072 Feb 12 '16 at 2:10
• I'm voting to close this question as off-topic because it has been answered in a different question (see comments). – András Bátkai Feb 12 '16 at 7:36

[Green and Tao] need to embed $[1,N]$ in $Z_p$ for some prime bigger than $N$ to get a nice field structure for some arguments to work.