User blog comment:P進大好きbot/Whether Rayo's number is well-defined or not/@comment-25601061-20180606221005/@comment-27513631-20180611224757

You mention "metatheory" exactly once.

If you're the sort to believe in a platonist universe, you usually have some sort of iterated transitive model consistency, likely with elementarily equivalent initial sections.

And you mention "A maximal consistent set" (alternative definition section, first bullet) without specifying which - this is my gripe.

My interpretation of how Rayo's Number is defined: We work in a two-sorted language. One sort "is" \(\mathbb N\), the other "is" \(V\), and the language has \(0, 1, +, \times, \in\). Rayo's number axiomatises the relation \(\text{Sat} \subset \mathbb N \times V\) (through an appropriate, implicit, irrelevant encoding of predicates into \(\mathbb N\)), and then uses \(\text{Sat}\) and quantification over \(\mathbb N\) to define the number in question. Why doesn't this work? And moreover, why are all of your theories set theories, as opposed to theories about \(\mathbb N\)?