I am trying to understand Godel's Second Incompleteness Theorem which says that any formal system cannot prove itself consistent.
In math, we have axiomatic systems like ZFC, which could ultimately lead to a proof for, say, the infinitude of primes. Call this "InfPrimes=True". In that case, does Godel's Second Incompleteness Theorem mean that we cannot be sure whether there exists a proof for NOT("InfPrimes=True"), or "InfPrimes=False"?
Since Godel's Second Incompleteness Theorem says we cannot be sure the system is consistent, is there a way to know for sure whether any given statement is true AND there does not exist any proof in that system showing the statement is false?