Prime Gap Definition: $Prime(p) \land Prime(p + a) \land (\forall n \in \mathbb{N}, p < n < p + a \Rightarrow \lnot Prime(n))$
A) “There are infinitely many prime gaps.”
B) "Eventually, all natural numbers are not prime gaps."
I'm having a hard time wrapping my head around translating infinite and eventually and applying the prime gap definition accordingly.
You can capture claim A) by claiming that for every number that is a prime gap, there is a greater number that is also a prime gap. Now, in order to do so, it helps to first state exactly the formula that states that '$a$ is a prime gap'. That would be:
$\exists p (Prime(p) \land Prime(p + a) \land (\forall n \in \mathbb{N}, p < n < p + a \Rightarrow \lnot Prime(n)))$
OK, so then the claim that for every number that is a prime gap, there is a greater number that is also a prime gap becomes:
$\forall a (PrimeGap(a) \to \exists b (b > a \land PrimeGap(b))$
i.e.:
$\forall a (\exists p (Prime(p) \land Prime(p + a) \land (\forall n \in \mathbb{N}, p < n < p + a \Rightarrow \lnot Prime(n))) \to \exists b (b > a \land \exists p (Prime(p) \land Prime(p + b) \land (\forall n \in \mathbb{N}, p < n < p + b \Rightarrow \lnot Prime(n))))$
Now, B) is a strange claim ... I am not sure what they mean by it ...