Why would that be an "overbroad mischaracterization"? — Tarskian
You can perfectly know the construction logic of a system but that does still not allow you to know its complete truth. — Tarskian
Incompleteness doesn't pertain to systems in general. Only to systems of a very certain kind. — TonesInDeepFreeze
"know the construction logic of". What is a "construction logic"? Maybe you mean the construction of the syntax? Better yet, just to say "the syntax rules". — TonesInDeepFreeze
It was an answer to the relevance of Godel's theorem. Of course, it only applies to systems in which it is provable. — Tarskian
The system is a theory with a language. — Tarskian
But you start out by mentioning logic in general, thus giving the impression that systems in general are incomplete, thus adding to the general confusion so prevalent on this point. — TonesInDeepFreeze
Ramon Llull, — Gregory
a statement declares a fact; it does not in addition instantiate that fact to a given truth value. — Devans99
a statement is associated with but distinct from a truth value. — Devans99
without going into the nitty gritty details of when Godel's theorem — Tarskian
Are it being said that Godel finally proved this fact about the human mind from pure mathematics? — Gregory
You don't have to go into details merely to avoid egregiously mischaracterizing the subject. I stated the theorem in just one sentence, and using only ordinary words. Plus the other dangling sloppiness in what you wrote. — TonesInDeepFreeze
I usually take a (logistic) system to consist of a language, axioms and inference rules. And I take a theory to be a set of sentences closed under provability. A theory may be the set of theorems in a language and derivable from axioms with rules. So, with each system, there is the theory induced by that system. — TonesInDeepFreeze
I agree it is not a statement, meaning it is not about anything. — Fire Ologist
I think the word ‘declarative’ is important; a statement declares a fact; it does not in addition instantiate that fact to a given truth value. — Devans99
Kripke proposes a solution in the following manner. If a statement's truth value is ultimately tied up in some evaluable fact about the world, that statement is "grounded". If not, that statement is "ungrounded". Ungrounded statements do not have a truth value. Liar statements and liar-like statements are ungrounded, and therefore have no truth value.
The question was not about how to state the theorem. The question was about the value of the theorem. — Tarskian
everything you say may be perfectly correct, but what answer does that give to the question at hand? — Tarskian
So what exactly did Godel add to our body of knowledge? — Gregory
The incompleteness theorem is: If a theory is formal, sufficient for a certain amount of arithmetic and consistent, then the theory is incomplete. That is highly informative: It tells us that there is no axiomatization of arithmetic such that every sentence of arithmetic is a theorem or its negation is a theorem. It tells us that there is no axiomatization that proves all the true sentences of arithmetic. It tells us that there is no algorithm to determine whether any given sentence of arithmetic is true. And the methods of the proof lead to profoundly informative results such as the unsolvability of the halting theorem and that there is no algorithm to determine whether a given Diophantine equation is solvable. — TonesInDeepFreeze
The "value of a theorem" is a philosophical question and not a technical one. — Tarskian
It is not about giving precise technical details about what the terms "theorem", "theory" or "system" mean. — Tarskian
The system is a theory with a language. — Tarskian
I usually take a (logistic) system to consist of a language, axioms and inference rules. And I take a theory to be a set of sentences closed under provability. A theory may be the set of theorems in a language and derivable from axioms with rules. So, with each system, there is the theory induced by that system. — TonesInDeepFreeze
The real answer will simply be lost amidst technical details that are irrelevant to the question at hand. — Tarskian
Same goes with "This statement is false", not all statements that can be uttered in a language are meaningful, and I agree it's not much use to spend much time pondering about them — leo
it says nothing about anything, like saying “this statement is Fred”. — Fire Ologist
As the statement "colourless green ideas sleep furiously" expresses a nonsense proposition, then so does the statement "this statement is false". — RussellA
incessant use of impenetrable language — Tarskian
Many "philosophers" mistakenly hold that sentences have meaning apart from speakers — Leontiskos
There are formulations in which there is no speaker nor reference to "I' or things like that. — TonesInDeepFreeze
Many "philosophers" mistakenly hold that sentences have meaning apart from speakers, and when one reifies sentences in this way they have taken the first step towards this sort of self-confusion. They strangely believe that a sentence can self-negate itself because they have taken their eye off the ball: the speaker. — Leontiskos
This sentence has five words.
Not true? — TonesInDeepFreeze
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.