Today, I happened to see a link regarding M-Theory, describing how it was a solution of sorts: several incompatible Super-Symmetric “theories of everything” seem to nicely make up “facets”
of super-geometric “gemstone” bdmi4.png (800×800) (sstatic.net)
That got me thinking: What if the “Is true of itself” predicate and the “Is NOT true of itself” counter-predicate are, in fact, heads and tails of the same coin?
Can one logic-construct be formulated by two predicates describing opposite ends of a “spectrum”?
For normal algebraic geometry, one uses Zermelo–Fraenkel set theory, plus for example Tarski’s axiom (so you have choice and so on). Russell’s paradox is, at any rate, not a problem.
To continue the library analogy for Russell’s Paradox, whether a book mentions itself cannot be the exclusive criterion for determining whether it is included in a particular library, because it’s not always possible to determine whether a book mentions itself. One might, of course, make self-mention one of the criteria for inclusion in a library, but it can’t be the only one.
You have a Set and Predicate “Is true of itself”.
All fine and dandy.
The paradox arises when the Set and Predicate “Is NOT true of itself” is stated.
What, instead, if the set is “True of itself” and the predication is structured “Is”:/:“NOT”
Two predicates, describing TWO states of ONE set.
The output of this set’s spectrum is binary…since the stated predication is designated true or not true, so is the output.
The two predicates are co-joined instead of independent, thus the possibilities of the output are a co-joining of the two predicates.
One predicate “is true” grabs up all the possibilities, while the predicate “is NOT true” ends up with none.
Since the “Is NOT true” half of the output without possibilities is cojoined with the “Is True” half with all the possibilities…does that eliminate the paradox?
I can imagine sets with MULTIPLE predicates…and outcomes not necessarily binary, but continuous.
No problem because a club is not a possible member. Similarly, there is not problem having a library consisting only of books that mention themselves, nor having a library consisting only of books that do not mention themselves because a library is not a book.
No, but that is not paradoxical. You cannot put such a book in that library. So?
Here is a simple paradox. Take a sheet of paper and one side write: The sentence on the other side of this sheet is false. And on the other side write: The sentence on the other side of this sheet is false. Now you cannot assign a truth value to either of these apparently well formed sentences.
Yes, if you have different predicates, then you have different sets. A predicate can, however, be compound, made up of multiple other predicates joined together by logic operators. “This is the set of all numbers that are prime and of the form 2^{2^n}+1”, for instance.
OK, before we go any further, don’t worry about the “is” and the “not”. Worry about the “true of itself”. How do you express that mathematically? And to answer that, you’ll need an entire framework of mathematics.
Mind you, I’m not asking you to come up with an entire mathematical framework yourself. That’s way too much work, even for most professional mathematicians. But if you’re going to be exploring questions like this at all seriously, you’ll need to know what framework you’re using, and understand how to use that framework. The usual choice is Zermelo-Fraenkel set theory, better known as ZF. And, notably, Russell’s paradox does not exist in ZF, because ZF is constructed in such a way that it’s impossible to describe the set in Russell’s paradox.
Agree completely. The problem is having a predicate to itself. There is no such operator in predicate calculus. Gödel did pull it off by employing his number system and making it a statement of arithmetic. You can find most of the details in readable form in Gödel, Escher, Bach by Hofstadter.
