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There's a subsection of my main argument (in my offline notes) that goes:

  1. ∃f(f(𝔼) = ♪)
  2. If we knew what f was in particular, then we could go to f -1(♪) = 𝔼
  3. 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.
  4. 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?"

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  • Why would f^-1 even exist? f may not be injective and map multiple inputs to ♪, or it may not be surjective. And why do we need to "know" what f is in particular to go to f^-1(♪) = 𝔼? As long as we know that f is bijective, so that f^-1 exists, we can go whether we "know" it or not.
    – Conifold
    Sep 26 at 23:18
  • @Conifold I should have realized to say that I'm using a more abstracted concept of functions than the usual set-theoretic/ordered-pair account. I've never settled my mind on this score, to be sure, though my intuition tends towards trying to abstract over the ordered-pair definition. And I should've added what I had in my notes, that the given f is set apart from an identity function, since it is moot whether that function would map back-and-forth here.
    – Kristian Berry
    Sep 26 at 23:59
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    Your Russell paradox like well-founded set of all (countable) well-founded sets was indeed used in the Cantor diagonal argument proof of the uncountability of the smallest ordinal ω₁. But most consistency proof including ZFC only transfinitely inducted up to ε₀ which is a fixed point of the exponential function with base as the smallest initial ordinal ω in an ordinal arithmetic fashion. Veblen function is the next normal function beyond exponential to arrive at the very large Feferman–Schütte impredicative ordinal. But since all these are...
    – Double Knot
    Sep 27 at 6:26
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    well within ω₁ ergo if your ♪ is just ε₀, your E could be said to be just ε₀ assuming it's an order type itself. And your inverse function is just logarithmic function in an ordinal arithmetic sense, not an identity function per your spec. The reflection principle and cofinality function have not entered here yet, and of course I'm just using ordinal here instead of cardinal while you can regard this as a simplified special case without changing your main thesis...
    – Double Knot
    Sep 27 at 6:52
  • @DoubleKnot you bringing up ordinal analysis is illuminating my memory, here, since in my rambling notes, it was such a function system that led, in major part, to the details of E and the "musical note" term. If you could adapt your considerations to an inherently uncountable input, do you think your comments could serve as an answer to the question? You always have insights into what I'm saying but I rarely get a chance to accept your answers, if ever :P I know you have a good score here so are not minded to look for the extra points but in this case I think you've earned them.
    – Kristian Berry
    Sep 27 at 10:00

1 Answer 1

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If f-1(k) = ?, what can we know about f? We know that f(?) = k. And we know ? can't be known. Can we then know the "specific nature" of f? My little grey cells inform me that IF knowing f requires us to know ?, as is usually the case [extricating an input-output coordinate (x, y) from which to extract a pattern] then impossible to know f.

BEGIN EDIT

Perhaps we can sample the k's. For example, if we get k = {1, 4, 9, 16, ...}, we can say f(x) = x². No? A classic math problem.

END EDIT

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  • So would you say that knowing the particulars of k, here, means adverting to something lower than ?, even if we can know of ? "from the outside"? I wish I could stabilize the distinction between knowledge-that and knowing-what, in the epistemic logic notation I'm using...
    – Kristian Berry
    Sep 27 at 10:01
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    Hair-splitting isn't my strong suit, mon ami. God gave me only so much rope ... to die trying. I can however say, with a certain degree of uncharacteristic confidence I might add, that at these depths (😅), how shall I put it now?, our problems have doubled, as in not just ?? but also f?
    – Agent Smith
    Sep 27 at 11:49

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