And if nothing else, this is the clue, "next higher order...". It appears you want to get to the point where there is no higher order. And that would seem to lead to a set-of-all-sets type of contradiction.Thus a single formal system have every order of logic giving every expression of language in this formal system its own Truth() predicate at the next higher order of logic. — PL Olcott
And if nothing else, this is the clue, "next higher order...". It appears you want to get to the point where there is no higher order. And that would seem to lead to a set-of-all-sets type of contradiction. — tim wood
Your question (again, if I understand it), is can there be a super-strong formal system that is not incomplete. I am guessing not. And I'm sure a rigorous discussion would be well, rigorous. — tim wood
↪PL Olcott Which is to say - just between us in case we're both wrong - that each system being itself deficient requires a successor system to fix it, but that simply creating a new deficiency. Ordinal arithmetic being formidable, I don't see an escape. — tim wood
You're off to the races into transfinite-order logics. If I understand the question of the title, it is equivalent to asking if Godel's incompleteness (theorem) is entirely resolved at some higher level of logic. My guess is not. — tim wood
I cannot give a rigorous answer, but I agree with this. If Tarski's undefinability theorem is basically that "arithmetical truth cannot be defined in arithmetic", or that true Gödel numbers are not definable arithmetically, meaning there’s no first-order formula for this, I think it does go for higher order logics. For those higher order logics there is their own true but unprovable Gödel number. — ssu
I imagine Tarski's indefinability theorem would. AFAIK second order logic already has diagonalization results - so it's either inconsistent or incomplete. You don't need to go above first for it. So long as you put enough arithmetic in, you're going to get the self referential bullshit that sets up these paradoxes. — fdrake
Um, no.When all orders of logic are included in the same formal system then such a system cannot be incomplete. — PL Olcott
A formal system having only one order of logic is like the "C" volume of an encyclopedia only having articles that begin with the letter "C".
— PL Olcott
And a complete set would have everything from A to Z. But in our case, you can't have a complete set. — tim wood
Anything that can't be proved in one order of logical can be proved or refuted in the next. A formal system having every order of logic cannot be incomplete. A formal system having only one order of logic is like the "C" volume of an encyclopedia only having articles that begin with the letter "C". — PL Olcott
seems to follow the principle that every simple idea can be made convoluted enough that it can no longer be understood.
— PL Olcott
That's a convenient principle. Btw, how do you know when an idea is just that simple? — tim wood
Now consider that you're taking the set of all provable statements of all logics up to the nth order. That will then be the set of provable statements of the nth order logic, due to the hierarchy. — fdrake
there'd need to be a single unifying metalanguage in which the formulae of all the levels could be expressed. — fdrake
The truth and provability symbols in the metalanguage would thus apply for theorems applying to the big union logic, rather than having a plethora of distinct symbols in different metalanguages — fdrake
"[They, (the ideas here presented)] implicitly justify the generalization that every symbolic logic is incomplete...." (316) — tim wood
There are many ways to further extend second-order logic. The most
obvious is third, fourth, and so on order logic. The general principle,
already recognized by Tarski (1933 [1956]), is that in higher order
logic one can formalize the semantics—define truth—of lower order logic.
https://plato.stanford.edu/entries/logic-higher-order/ — Stanford
↪PL Olcott What is your point? We suppose - that's the best I can do - that a proposition undecidable in L is decidable in L', and one in L' in L'', and so forth. But apparently there is no Lωω...ω that is itself complete. — tim wood
Higher-order logic is the union of first-, second-, third-, ..., nth-order logic; i.e., higher-order logic admits quantification over sets that are nested arbitrarily deeply.
— PL Olcott
Do we need more than first and second order logic in practical uses? — Corvus
By the theory of simple types I mean the doctrine which says that the objects of thought ... are divided into types, namely: individuals, properties of individuals, relations between individuals, properties of such relations, etc.
https://en.wikipedia.org/wiki/History_of_type_theory#G%C3%B6del_1944 — History of Type Theory
For formalizing the entire body of human knowledge that can be expressed using language we need this: — PL Olcott
Let's try this. Suppose you have your "this," call it T. Now suppose you have some expression. Is it or its negation in T? If so, great! You're done. If not, then you have to figure out if it should be or not. And using existing knowledge, you cannot (if you could, it or its negation would already be in T). — tim wood
This at least seems true. Mainly because such a listing lacks the power of systems that are incomplete in the Godel sense, and in fact have nothing to do with it. Your groceries list can stand for such a body of knowledge, and nothing incomplete about it.A knowledge ontology inheritance hierarchy capable of formalizing the entire body of human knowledge that can be expressed using language need not be incomplete in the Gödel sense. — PL Olcott
Properly and correctly qualified as unprovable within the system out of which it arose, but proved true by other means.True and unprovable — PL Olcott
Btw, true is an adjective indicative of a quality that true statements have. Good luck with any attempt to comprehensively further define just what that quality exactly is. — tim wood
There seems to be a finite limit to the number of orders of logic needed. — PL Olcott
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