A statement in second-order-arithmetic which proves second-order-arithmetic consistency

Question

Is there a statement in second order arithmetic which it's truth proves the consistency of second order arithmetic? Note that if such statement exists it must be unprovable in second order arithmetic.

Answer 1

I'm not 100%, but let's give it a try:

"There exists a smallest inaccessible cardinal larger than the size of set of sets of all the numbers in question."

Answer 2

The question doesn't mention a base theory - we want a sentence that proves the consistency of $Z_2$ relative to which theory?

Depending on the base theory there are several options, including:

• The Gödel sentence of $Z_2$ directly implies the consistency of $Z_2$ over primitive recursive arithmetic.

• The sentence "there is a countable $\omega$-model of ZFC" implies the consistency of $Z_2$ over $\text{ACA}_0$. We could replace ZFC with any theory that proves the consistency of second-order arithmetic. Replacing $\text{ACA}_0$ with a weaker base theory is complicated because weaker theories have trouble handling the satisfaction relation for countable models.