Again, you bring up someone else's quote, and it turns out that it contrasts with your post on account of the quote being fairly clear, though it is incorrect:

For a denumerable language, there are the same number of unprovable statements as provable ones, viz. denumerably many.

And he switched from 'statements' to 'facts', thus throwing off the count.

Why would that be an "overbroad mischaracterization"? — Tarskian

Too bad I'm not getting paid for correcting your stuff:

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.

"know its complete truth" is vague. What we could say is, "there are true sentences that are not provable". And no person can know individually infinitely many true sentences anyway.

"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".

You write slop. Though, that post is not your worst.

Incompleteness doesn't pertain to systems in general. Only to systems of a very certain kind. — TonesInDeepFreeze

It was an answer to the relevance of Godel's theorem. Of course, it only applies to systems in which it is provable.

"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

The system is a theory with a language. So, that is the "construction logic".

It emphasizes that you are in control of its true nature.

It's a bit like building a machine according to plan but you cannot predict everything it will be doing when it is running.

It was an answer to the relevance of Godel's theorem. Of course, it only applies to systems in which it is provable. — 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.

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.

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

It was an answer to "So what exactly did Godel add to our body of knowledge?".

My answer was a combination of what Hawking had said on the matter along with Yanofsky's take on the matter, without going into the nitty gritty details of when Godel's theorem is applicable because that was not the question to begin with.

Leibniz allegedly got his concept of a logic machine from the mystic Ramon Llull, who in turn may have been influence by Kaballah. He abandoned the this latter in life when perhaps he realized that the mind in not computational. Philosophers had known this however for ages.. Are it being said that Godel finally proved this fact about the human mind from pure mathematics?

Are it being said that Godel finally proved this fact about the human mind from pure mathematics? — Gregory

No, though some argue something like that.

Ramon Llull, — Gregory

Thank you for making mention to Ramon Llull.
I hardly see references or allusions from my country's philosophers, thinkers, and mystics. I respect your cultural knowledge. Another significant Peninsular author is San Isidoro de Sevilla.

Francisco Suarez, one of the greatest.

:up: :100:

a statement declares a fact; it does not in addition instantiate that fact to a given truth value. — Devans99

Right. A statement is about something, and, as a statement is separate from that something.

But “this statement is false” is about itself.

So it has no content to refer to other than the fact that it is a statement.

a statement is associated with but distinct from a truth value. — Devans99

Right. And there is nothing distinct from this statement to adjudge its truth value, to adjudge its content.

I agree it is not a statement, meaning it is not about anything. Not in the normal use of statements. It’s a fun logical puzzle, where the exercise of playing with it can yield some content about logic and language and truth. But if you don’t bring that content with you, it says nothing about anything, like saying “this statement is Fred”.

without going into the nitty gritty details of when Godel's theorem — Tarskian

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.

Are it being said that Godel finally proved this fact about the human mind from pure mathematics? — Gregory

A superb book that explains the theorem and discusses various reactions to it: 'Godel's Theorem' by Torkel Franzen.

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

The question was not about how to state the theorem. The question was about the value of the theorem.

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

In the context of the question "So what exactly did Godel add to our body of knowledge?", everything you say may be perfectly correct, but what answer does that give to the question at hand?

The "value of a theorem" is a philosophical question and not a technical one. It is not about giving precise technical details about what the terms "theorem", "theory" or "system" mean. The real answer will simply be lost amidst technical details that are irrelevant to the question at hand.

According to Stephen Hawking, Gödel's theorem is valuable because it shows us why positivism is futile. Yanofsky's work shows that Gödel's theorem is omnipresent. It is not an "overbroad mischaracterization" to state that Gödelian facts -- true but not provable -- are most likely omnipresent in our physical universe. Gödelian facts most likely massively outnumber predictable facts.

We most likely live in a Gödelian universe and not in a universe of Laplace's demon. That is the real metaphysical implication of Gödel's theorem.

The reason why Gödel's theorem has not yet been absorbed into modern metaphysics, almost a century after its discovery, is because of the incessant use of impenetrable language, meant to insist on irrelevant technical details, instead of dealing with the metaphysics that it implies.

I agree it is not a statement, meaning it is not about anything. — Fire Ologist

Yes, it seems to me that this is just another case of "philosophers" confusing themselves.

However you slice it, the intent of the "liar" determines whether he is lying. He either is or he is not. There is no both-and. The same goes for someone who claims to be speaking falsely rather than lying. Either they intend to speak falsely or they do not. 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.

Literally everything I say is a lie, therefore, not literally all things others provide are true.

This is assuming lie as equivalent to some opposite forms.

How we interpret the sentence matters. Some will do so more literally than others and use their own methodology. If it is paradoxical then try to make it not so and see if any meaning can be established.

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

Yes. I think Kripke's solution, as described in the Wikipedia article Liar Paradox, seems to be the most reasonable.

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.

A statement can only be true or false as it refers to the world. The statement "snow is white" is true IFF in the world snow is white.

As the statement "this statement is false" doesn't refer to the world, but only refers to itself, the words "this", "statement", "is" and "false" have no sense, as sense only comes from reference to the world, meaning that the statement "this statement is false" is senseless.

As the statement "colourless green ideas sleep furiously" expresses a nonsense proposition, then so does the statement "this statement is false".

As you say, a statement such as "this statement is false" may declare something without giving a truth-value to that something it has declared.

If the mind is pure matter (brain, nervous system) and it can compute Godel's truths, then maybe Hawking is right. And Penrose about microtubules?

Clarification:


I'm not saying that Godel proved the mind is non-computational or that the brain is not solely a type of machine. The middle ground is to ask whether understanding Godel or having the capacity in the brain to think it proves the brain is not a computational machine. Yep

The question was not about how to state the theorem. The question was about the value of the theorem. — Tarskian

Your answer to the question includes a terribly misleading characterization of the theorem.

everything you say may be perfectly correct, but what answer does that give to the question at hand? — Tarskian

The question was:

So what exactly did Godel add to our body of knowledge? — Gregory

As far as "exact", I responded:

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

Those are exact and include the most salient implications of the theorem. Of course, a lot more came in the wake of the theorem, but a single post could not be exhaustive even as to mathematics. Also, in other threads I've commented on the matters of the incompleteness theorem in philosophy of mathematics. I don't have comments at this time on the incompleteness theorem in connection with science, epistemology, ontology and metaphysics. But I did recommend an excellent book as a starter kit. If that is not sufficient for you, then so be it; I'm not on retainer to answer all your questions.

The "value of a theorem" is a philosophical question and not a technical one. — Tarskian

The question didn't ask about the "value" of the theorem. It asked "what exactly did it add to our body of knowledge?" It added exact mathematical knowledge. Much of that is technical. But also, I answered in terms that are both technically accurate and also easily understandable if one took just a bit of time to understand the basic terms such as 'consistent', 'axiomatization', etc. Anyone is free to provide an even more informal explanation, but if the liberties taken in that endeavor lead to egregious mischaracterization, then that deserves be noted. And when the theorem is discussed in general, it is worthwhile to correct and clear up misconceptions about the theorem - both general and technical.

It is not about giving precise technical details about what the terms "theorem", "theory" or "system" mean. — Tarskian

You are remarkable!

The system is a theory with a language. — Tarskian

You gave your definition. So I responded with what I consider to be better a explanation. You felt a need to add your notion of 'system' in terms of 'language', but you fault me for providing better information. And you complain about "precise technical details" when my explanation was not even so technical:

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

language
axioms
inference rules
sentence
provability
closed under provability
theorem
theory

An understanding of those very basic rubrics is needed to even start discussing what the implications of the incompleteness theorem are. And if one doesn't know, then one can look in any introductory text or article on mathematical logic, or perhaps find one of many posts I've written that explains various terms, or even just ask me now about them, as given time, energy and interest, I'll respond.

You're continual insistence that discussion should be held only at the level of technicality and detail that you personally prefer is arrogant and irrational. Especially as often the subject does require being clear on certain technical matters, indeed as the subject concerns a technical subject, even if a particular discussion is centered on general, less technical ramifications of the technical mathematics. And especially as you often enough post your own technical formulations (often enough, they're botched).

You are really too much! How did you get so mixed up and unreasonable?

The real answer will simply be lost amidst technical details that are irrelevant to the question at hand. — Tarskian

You arrogate to yourself what is "relevant" and what is the "question at hand". And you arrogate to yourself what level of technical detail should be mentioned.

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

Yep. :up:

Or the so-called "Liar's paradox":

• "I am lying."
• "Lying about what? You haven't yet managed to construct a coherent sentence."

incessant use of impenetrable language — Tarskian

Such things as 'consistent', 'system', 'proof' are not "impenetrable". But sometimes more technical terminology needs to be mentioned - so that the incompleteness theorem is not incorrectly overgeneralized, misconstrued or misrepresented. If one is claiming implications in science and philosophy from the incompleteness theorem, then those claims should not be based on incorrect characterizations of what the theorem actually is.

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.

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?

This sentence has five words.

Not true? — TonesInDeepFreeze

It is true that we can treat sentences as objects of predication, but the difference is that the number of words that a sentence contains is a material property, not a formal property. So could say, "Colourless green ideas sleep furiously," has five words, but he could not say, "Colourless green ideas sleep furiously," is true (or meaningful). Counting words and affirming a truth value are two different things.