I think the Boolean algebra formula for the paradox might be right – perhaps it is redundant, but I think it is perhaps sufficient to code the paradox as a logic circuit.
The Formalism, Restated – Part 9 of 10
Earlier: Regarding the Translation
Later: Plain Explanation
I’ve come to the decision to categorize boolean algebra as an assumption, and to defend assuming boolean algebra on its own merits. I think assuming binary logic is probably defensible. So, schematically, I would assume boolean algebra, defend that assumption, and use the assumption of boolean algebra to derive a paradox. Obviously there’s work that needs to be done defending my assumption, but on the other hand, I think that that work can be done. I think that the idea that “logic is true” is sound.
That being said, my hypothesis raises the possibility of three-value proofs.
The general liar’s revenge sentence shows that logic is inconsistent even if it is incomplete, based on a hypothetical scenario predicated on the assumption of logic gates. Importantly, the assumption of logic gates as hypothesis suffices to show logic to be inconsistent.
That being said, when Gödel talks about incompleteness, he is referring to something tantamount to the incompleteness of logic gates.
Why completeness is preferable:
To talk about logic as a distinct object, we need sets. However, I do not know how to go about this, because I do not find how to go about accessing sets to be immediately obvious. I’m tempted to simply use proof by contradiction, which is an attractive option because to do so wouldn’t reduce my theory to triviality, because set theory would structure logic. Can there be a set of “true”?
Of course, perhaps changing a computational framework into a set-theoretic framework so that the system can comment on itself more readily isn’t what I want to do, because to do so would move away from logic as such.
Toward set theory – Part 3 of 3
Earlier: The Predicate
Because the predicate to the formalism is similar to “assume the proof”, it could be asserted that the proof proves itself, obviously.
I also think Tarski may have commented on the predictate.
Toward set theory – Part 2 of 3
The predicate of the formalism can be postulated as hypothetical, so the conclusions of the formalism according to that hypothetical could come to bear on that hypothetical.
Toward set theory – Part 1 of 3
Later: The Predicate