Question

There's a subsection of my main argument (in my offline notes) that goes: ∃f(f() = ♪)If we knew whatfwas in particular, then we could go tof-1(♪) = But this would make knowable in a well-founded way, but the definition of is the well-founded set of all sets knowable, currently, in a well-founded way, i.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 knowf's character in particular) Or is even the inverse off, 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 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 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 thatfis not an identity function, and that its initial input is not a musical noteper 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 thatfis not the identity, then, the issue is: if we could run the circuitf-1(♪) = , this would itself constitute a slightly nontrivial, but still ever-just-so-substantive (then), picture of what 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 off, from ♪, which would transform into a somewhat nontrivially known set, contrary to its intended definition. It should also be mentioned that , 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 knowthatthere is somef, here, but notwhatit 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? If f-1(k) = ?, what can we know about f? We know that f(?) = k. And weknow?can't be known. Can we thenknowthe 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] thenimpossibleto 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.

Answer 1

The student's question exists at the intersection of mathematical and philosophical domains, focusing on epistemology and set theory. It questions whether knowing the existence of a function can lead to understanding a what-unknowable set through its inverse, amid a discussion of Russell's distinction between knowledge by acquaintance and propositional knowledge.

Step-by-step explanation:

The student's question deals with set theory, large cardinals, functions, and epistemic logic within the realm of philosophical mathematics and touches upon issues of knowledge-theoretic character of certain mathematical concepts, particularly functions and their inverses. When discussing the nature and knowledge of sets and functions, which are mathematical constructs, the student is engaged in a form of philosophical inquiry into mathematics. The student raises a question about propositional knowledge, which is knowledge of facts that can be true or false, and aligns with the domain of traditional epistemology, which focuses primarily on propositional knowledge. The student also inquires about the possibility of knowing the character of a function if its inverse can lead to a set that is inherently what-unknowable. The dilemma presents a parallel to classical philosophical problems, such as whether the inverse function, which is an abstraction, can violate defined boundaries and control parameters. Understanding the student's question involves recognizing Bertrand Russell's distinction between knowledge by acquaintance—which is direct, non-inferential knowledge—and propositional knowledge, which includes truths arrived at by reasoning or experience. Philosophers have long contended that while one may know of the existence of something (knowledge that), the specific nature or description (knowledge what) may remain elusive especially when the subject is abstract, such as large cardinals in set theory, where Russell's analysis becomes relevant.