By adding a large cardinal axiom to ZFC, one can prove statements of arithmetic that wasn't provable from ZFC alone. I need a little help understanding exactly how that works.
Here is what I (think I) understand: If LCA is some large cardinal axiom, then any model of ZFC+LCA contains a set which is a model of ZFC, and the existence of such a model implies the consistency of ZFC. Also, there is a Gödel number $a$ for the "conjunction" of all the axioms of ZFC+LCA and a Gödel number $b$ for the statement that ZFC is consistent, and there is an arithmetical relation $R$ that represents "... implies ...", and $aRb$ is provable.
So far so good, but here is what I don't understand: How does $aRb$ and the assumption of LCA interact to produce a new arithmetical theorem? $a$ is a number and thus cannot follow from LCA; nor can $aRb$ be used for modus ponens. In other words: while I understand how there can be provable arithmetical statements that some axioms imply something else, I don't understand how it makes any difference to arithmetic whether those axioms are actually adopted.
How does this work? What have I misunderstood?