I came across this problem in a test:
Twin primes are pairs of numbers $p$ and $p+2$ such that both are primes—for instance, 5 and 7, 11 and 13, 41 and 43. The Twin Prime Conjecture says that there are infinitely many twin primes.
Let TwinPrime(n) be a predicate that is true if $n$ and $n+2$ are twin primes.
Which of the following formulas, interpreted over positive integers, expresses that there are only finitely many twin primes?
a) $∀m.∃n.m ≤ n \land \,\lnot(TwinPrime(n))$
b) $∃m.∀n.n≤m \rightarrow TwinPrime(n)$
c) $∀m.∃n.n≤m \land TwinPrime(n)$
d) $∃m.∀n. TwinPrime(n) \rightarrow n≤m$
I've tried to solve this by converting the sentences into their English equivalents, but the fact that we have to express "finiteness" using quantifiers eludes me. I have the answer key with me, but I am unable to work back from that either.
Some help/hint would be appreciated, thank you.