There's a subsection of my main argument (in my offline notes) that goes:
- ∃f(f(𝔼) = ♪)
- If we knew what f was in particular, then we could go to f -1(♪) = 𝔼
- But this would make 𝔼 knowable in a well-founded way, but the definition of 𝔼 is "the well-founded set of all sets knowable, currently, in a well-founded way," i.e. 𝔼 can't be known in a well-founded way or it would be an element of itself and thence not well-founded.
- Therefore ¬(we know f's character in particular)
Or is even the inverse of f, here, too general and/or abstract to violate the intended control parameter? For now, I've been assuming the existence of a function with an input below 𝔼 but which still goes to ♪ (don't worry about what ♪ is concretely/particularly, here#), so it doesn't seem like too much of a hassle to cordon 𝔼 off like so, but I suppose the subargument would go through more neatly/nicely if I didn't have to set up that guard.
#Just let it be known that f is not an identity function, and that its initial input is not a musical note per se, but is a large cardinal, i.e. a cardinal describable/definable in terms of exotic logic questions and variables in the outer darkness of our current set-theoretic knowledge. Assuming that f is not the identity, then, the issue is: if we could run the circuit f -1(♪) = 𝔼, this would itself constitute a slightly nontrivial, but still ever-just-so-substantive (then), picture of what 𝔼 is. That is to say, in knowing ♪'s nature (which we would beforehand), we could then define the trans-epistemic set as a large cardinal extrapolated by the inverse of f, from ♪, which would transform 𝔼 into a somewhat nontrivially known set, contrary to its intended definition.𝔼
𝔼It should also be mentioned that 𝔼, here, is somewhere in between (X) an intrinsically uncountable model of some set-theoretic universe and (Y) a von Neumann/Bernaysian proper class. More pointedly, it is meant to recapitulate much of Zermelo's vision of the open-endedness of the set worlds.
MAJOR EDIT (OF THEME): the other way to frame my question is to ask about how the knowledge-that/knowing-what distinction is handled in mainstream epistemic logic. We would say that we know that there is some f, here, but not what it is internally. Are that/what-knowledge inverses of each other, then, though? So that the question becomes: "If we know that a what-unknowable set exists, then if we knew what the inverse of that set is, would we end up knowing what the what-unknowable set is, contrary to the introductory definition?"