An anomaly of Set Theory: (4) russell’s paradox - YouTube
Due to a convolution of the logic, trying to include the predicate “is true of itself” into the set of all "is true of itself"s is sensible, logical, and true.
BUT…
Including the predicate “is NOT true of itself” into the set of all "is NOT true of itself"s gives rise to the paradox: including it in it’s set would imply it is true…when it is false.
But omitting it from the set because it is false implies it is true.
It is a personal hypothesis of mine that a paradox is not a self-contradictory component of reality…just the example of the limitations of the language used to define environment the paradox arises from.
Suppose we rewrite the paradox as a math problem:
F(x)= {-F(x)}
and, conversely,
-F(x)={F(x)}
Something that can make a function the negative of it’s own set.
Most likely something more complex than a negative times a negative equals a positive.
Something that involves “i” …the sqrt of-1…in a multi-dimensional, hyper-geometric framework found in
quaternions Quaternion - Wikipedia
and octonions Octonion - Wikipedia
where, I believe, properties such as Distributive, Associative, and Commutative identities no longer hold the way they do using Real Numbers?
What I note about Russell’s Paradox is that it’s structured in such a way that the first case can go either way: if for example you postulate a book of all books in a library that mention themselves, then it could either mention itself with no paradox, or equally might not mention itself and therefore be listed in the book of all books that don’t mention themselves with no paradox. And there’s no a priori reason it should be one way or another.
While the other case- the book that lists all books that do not mention themselves- then it can be listed in neither book without paradox.
I don’t know how to express this more formally, but hopefully someone can follow what I’m grasping at: that somehow the situation is set up so the first case “grabs” all the possibilities, leaving none for the other?
It is the set theory you have when you’re not having sets. (Older Australian’s will get the joke.)
Otherwise, you can try to generate some other sort of set theory. But overall, in order to be called that, it is going to have to provide the kinds of characteristics we generally associate with set theories. It is going to have to have interesting and useful properties.
In some sense this is correct. But mathematics is, by intent, more restrictive. The entire point of Russell and Whitehead’s work was to avoid limitations of language and to put things on a solid formal foundation. They sought to provide a constrained expressive system upon which mathematics could be built. So they hoped to do it by restricting the language. Famously, it didn’t work.
Most paradoxes one hears about aren’t actually paradoxes. But Russell’s is.
Something as simple as the Liar’s Paradox is an example of the limitations of language - simply because the expression of the paradox “This sentence is false” does not provide enough context in and of itself to be meaningful. The paradox assumes the sentence’s interpretation and context. A sentence of the form “Within the context of my expressing this sentence, this sentence is false” doesn’t have quite the same ring.
I don’t have time to elaborate, but I know that the mathematician Louis H. Kauffman has investigated imaginary truth values in the connection with set-theoretical paradoxes, see here (link goes to .pdf). The basic point is that you can exchange (the imaginary unit) i for -i, as long as you do it everywhere, without changing the structure of the complex numbers.
Hmm. I was hoping to find some useful insight in that paper. I’m really not sure what to make of it. Halfway through it seems to veer off into woo-woo land, and sounds more like a modern version of Kabalistic Majik mixed with quantum woo than mathematics. He fixes the paradoxes by redefining them as a recursive series in time - basically replacing equality with assignment to form an infinite series. \sqrt -1 is only useful as a way of creating an alternating series.
Since your link goes to a bunch of videos, I highly recommend this particular video for anyone interested in learning about what the OP is talking about: https://www.youtube.com/watch?v=ymGt7I4Yn3k
It is, admittedly, a bit of an odd duck, rooted in Spencer-Brown’s Laws of Form, and notions from early cybernetics and control theory. But I think there’s bits and pieces that have merit, and using the fact that conjugation is an automorphism of the complex numbers to add a fixed point to the negation operation might be one.
There is an interesting theorem by Lawvere in the setting of category theory that exposes the basic structure of many of these diagonalization schemes—Russell’s paradox, but also Gödel’s incompleteness, the undecidability of the halting problem, Cantor’s theorem, Tarski’s undefinability of truth, and so on. Here is a somewhat accessible survey, working with ordinary set theory.
One takeaway is that these results require that a certain function has no fixed point, i.e. an x such that f(x) = x. But negation has no fixed point—f(0) = 1 and vice versa. This is what yields the inconsistent assignment of truth value to propositions like ‘the set of all sets that are not members of themselves’. But with negation no longer being fixed-point free, this self-negation is no longer paradoxical.
Another approach is to appeal to quantum theory. There, we may define truth values as the two basis states of a qubit in the computational basis, |0\rangle and |1\rangle. The negation operation (given by the Pauli x operator) takes one to the other, but the equal superposition |0\rangle + |1\rangle is left invariant, again producing a ‘truth value’ consistent for these self-negating propositions.
Because one wishes to assign a truth value to every proposition for which a truth value is meaningful, and one cannot finitely determine whether every proposition has a meaningful truth value.