First Order Logic: Prove that the infinitely many twin primes conjecture is equivalent to existence of infinite primes

Solution 1:

The missing implications are clever uses for the fact that $T$ is a complete theory.

$(2)\implies(1)$: In $M$ there is an infinite prime which has a twin, $M$ satisfies the following sentences, for every $n$, "There exists $p>c_n$ such that $p,p+2$ are both prime". But since $T$ is complete it must prove those sentences, and therefore they must hold in $N$, which exactly means $(1)$ is true.

$(1)\implies(3)$: Suppose that $(3)$ fails, and $M$ is a model without infinite twin primes, but with infinite members. Then in $M$ it is true that "There is $c$ such that there are no twin primes larger than $c$", by the virtue of completeness of $T$, this must be true in $N$ which means that there are only finitely many twin primes (since a bounded set in $\Bbb N$ is finite). And therefore $(1)$ must also fail.