User blog comment:P進大好きbot/Whether Rayo's number is well-defined or not/@comment-25601061-20180606221005/@comment-27513631-20180609011005

Much of the musing is about how Rayo's number is precisely defined - I'm not familiar with the exact definition for Rayo's number, but believe that the definition of Little Bigeddon is precise enough to not be open to interpretation.

There's also much musing about a platonist universe. It's uncontroversial to work in a manner which is valid internal to any (countable) model of ZFC. That's what's normally done (possibly with some LCAs added), so objections based on that idea are invalid.

There is no maximal consistent set of statements - as the theory is incomplete, there are independent statements which are as valid choices as their negations.

Look up the concept "metatheory" - you seem to be trying to reinvent it.