I am not a Platonist, as I keep pointing out. Existence, even not in mathematics, is relative and confusing to begin with, so I don’t pretend to try and understand it in a meaningful way.

However, we have a proof, a constructive proof that large cardinals are consistent. And they exist in an inner model of our universe.

Recall that $0^\#$ exists if and only there exists a non-trivial mouse. Now recall that such mice exist. Vacanati mice.

I’m sorry to all those who claim that inaccessible cardinals are inconsistent. Your claim is that reality is inconsistent. Which might just be the case…

Now you can ask whether or not large cardinals are a human construct. Here we run into a problem, as these non-trivial mice are a human construct themselves…