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.

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.

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 "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.

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 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.

The crank says, "When an idea is said to be an "object" this is Platonism, by definition. Platonism is the ontology which holds that abstractions are objects."

That's not a credible definition of 'platonism'.

https://plato.stanford.edu/entries/platonism-mathematics/ :

"Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. [emphasis added]"

"Mathematical platonism can be defined as the conjunction of the following three theses:

Existence.
There are mathematical objects.
Abstractness.
Mathematical objects are abstract.
Independence.
Mathematical objects are independent of intelligent agents and their language, thought, and practices [emphasis added]"

https://plato.stanford.edu/entries/platonism/ :

"Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and non-mental [emphasis added]."

The crank doesn't know jack about logic, mathematics or philosophy of mathematics. But still he's good for serially shooting his ignorant mouth off about them.

coming off as senile. — Lionino

"senile" is to guffaw. — TonesInDeepFreeze

I should not have honored that garbage even by laughing at it.

"senile" is juvenile. Worse, it's pernicious. One would think that such crude ageism wouldn't get into public past the lips of a putatively aware poster. People have mental difficulties for many different reasons. It's not a matter of age, but of the difficulties no matter their cause. Meanwhile, bigoted ridicule of people for their age is obnoxious and disgusting. Also pretty bad is to compound that bigotry by making it a term of general insult against targets whose age is not even known and not relevant no matter what it is.

if by irrational you mean things of the sort of believing the colour green is sweet and that the moon is made of cheese. — Lionino

Synesthesia does occur. And people have all kinds of false beliefs not derived by good inferences. But beyond those, people also have even more profoundly alternative states.

If a mystic experiences contradictions as being true, then he's not breaking the laws of thought?
— TonesInDeepFreeze

I don't think any such experiences are possible. — Lionino

Of course they're possible. Whether in absurdist day dreaming, insanity, dreaming or in mystic state, one can have all kinds of irrational thoughts and dispositions. Even in everyday experience, people often drift to sleep with disconnected nonsensical ideas and irrationality.

But, if it is the case that it is possible, definitionally there are no laws of thought that preclude from that happening, because it happened, therefore oen is not breaking laws of thought. — Lionino

Yes, and therefore "laws of thought" pretty much reduces to simply "conditions necessary for mentation". If whatever one thinks, no matter how irrational, is not breaking the laws of thought, then the notion of 'laws of thought' is so general that it is hardly worth mentioning. That suggests putting some more meat on the bones of your definition.

You can post or not post as you please. And I'll do the same.

I don't pretend to be a bully and I'm not one. And "senile" is to guffaw.

Meanwhile, no matter how you regard me as "coming off", I don't manufacture perceptions about you in that way. No matter how you "come off" to me, I regard the substance of your posts, good or bad, on their own terms, not personally.

I don't require your courtesy. And I don't require you not to post so that you don't wear out my patience as you do. Anyway, in general, many people in this forum will be discourteous quite soon after they are disagreed with.

[EDIT: "courtesy" from a guy who makes a ridiculous argument against the common courtesy of noting that emphases were added to a quote.]

I didn't mention you skills. I mentioned your knowledge.

And you don't have to feel they that my view is needed nor do you have to request it for me to state it.

Meanwhile, you lashed out at another with your characterization of his knowledge of language. Same applies to you in your knowledge of logic. You've made hundreds and hundreds of posts about logic that are a dead end as your gravamen can be neatly summarized in a couple of sentences (as I did for you) without the pointless variations all on the same pointless theme.

I'll try to combine your clauses into a defintion:

Laws of thought are facts about your mind such that those facts are necessary for the operation of the mind.

I don't know if that's what you mean, but it's my best guess.

Or maybe just say:

Laws of thought are the necessary mental conditions for the operation of the mind.

From that definition, it follows that they can't be broken.

/

So, when a person is utterly irrational, they are still obeying the laws of thought on account of the fact that there are mental conditions necessary for the operation of their mind?

A law of thought is necessary for the mind no matter what it is doing, ironising, dreaming, thinking, or whatever. All of these have subjacent operations that are necessary to them. — Lionino

Whatever is "subjacent", in those mentioned mental states, the laws of thought are broken in the sense of irrational thinking, believing or imagining. If a mystic experiences contradictions as being true, then he's not breaking the laws of thought? If one dreams that one's great-grandfather is both alive and dead at the same time, one is not breaking the laws of thought?

.

I thought they were two different definitions. But the second includes additional assertions beyond what I would have thought is a definition. Also, I don't know what 'instead' refers to.

Actually, easier just to list a three book course, which I've done several times in this forum.

Laws of thought are facts of the matter about your mind — Lionino

And a fact about minds is that they are often irrational.

I should list "prerequisites" for talking about logic.

The refuted person may not be disposed to accept that he's been refuted. But it doesn't follow that if a person points out that he's not been refuted (and gives clear argument about that), then that person is doing that because he doesn't want to admit to having been refuted.

I'm happy to read any definition you'd restate.

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.

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.

People operate mentally in all kinds of ways: Fictionally, absurdly, poetically, ironically, day dreaming, dreaming, mystically and insanely.
— TonesInDeepFreeze

And all of those operations are operations of the mind, therefore bounded by the rules of the mind, which we may call laws of thought. — Lionino

You just completely ignore the point, that I've made twice, now a third time:

In such mental states, people often break the laws of thought.

But your point reduces to the tautological: the mind can't operate rationally without operating rationally. No one disagrees with that.
— TonesInDeepFreeze

I am aware of that. The tautology therefore is about law of thought, not about laws of logic, a different concept, thus it does not follow that laws of logic are unbreakable. — Lionino

Yes, it doesn't follow. No one said otherwise. And yes, I was referring to your notion of the laws of thought. I'll say it again:

One can break the laws of thought on pain of being irrational. But you say that the laws of thought are unbreakable. But one can break the laws of thought. So you regroup by saying that one can't break them and be rational. But that is not at issue. My point is that one can break the laws of thought, contrary to your earlier claim.

Moreover, even that point is not required, since we know that people do break laws of thought
— TonesInDeepFreeze

Do I have to repeat my definition, which, if anything, is quite the appropriate definition? — Lionino

Definition of what? Of 'the laws of thought'? Repeat or not repeat whatever you like.

if there is a single law of logic that can be broken, and that law of logic corresponds with a law of thought, then there is a law of thought that can be broken
— TonesInDeepFreeze

If the law of logic is understood as expressing a law of thought — which in modern days that is not how it is understood — Lionino

Where is there a report that modern writers in general believe that laws of logic may not be understood as expressing laws of thought? And what period do you regard as modern?

hence my original comment to Leontiskos —, by definition it can't. If law of logic is understood as how we understand it today, laws of thought do not correspond to laws of logic because, as we have agreed, the latter may not be respected by some system, they may only allude to or be based on laws of thought. — Lionino

I'm uncertain whether I understand you. Certain systems don't respect certain laws of thought. That doesn't entail that laws of thought cannot be broken. Indeed, it evidences that they can.

Also, you say "the latter", which is 'laws of logic'. So 'they' also refers to 'laws of logic'. And you say 'they may only allude to or be based on laws of thought'. So that is saying that laws of logic may only allude to or be based on laws of thought. But that seems the opposite of anything we've agreed on. If the laws of thought require rejecting contradiction, then systems that allow contradiction do not adhere to that law of thought.

I'm not talking about guessing what post was quoted.
— TonesInDeepFreeze

I am. You constantly [emphasis added] mistake what post is being quoted. — Lionino

(1) In one case, I was unclear as to whether you were quoting in agreement with the quoted poster. And I overlooked that your recent lashing out was not directed at me. That is not even remotely constant (2) In this instance, I've been in exactly the right place about what was posts was referenced.

"Jack is happy" is grammatical even when the speaker misused the word 'happy' while thinking it meant 'doleful'.
— TonesInDeepFreeze

I have refuted that already. Talking of circles. — Lionino

Your replies don't even come close to a refutation.

It's plain as day: One can easily see that "The cat is black" is grammatical, without having to know anything about the person who said it, or even if it was not said by a person but formed randomly by a machine. You've not refuted that. One of your replies is that we assume the speaker knows the meanings of the words. But that is not necessary to see that the sentence is grammatical. We could say, "I have no idea whether the person who wrote "The car engine is noisome" knows that 'noisome' means 'offensive' not 'noisy' but that doesn't matter if all you want to know is whether the sentence is grammatical. I'll happily and without any reservation tell you that is."

/

Oh, and about nitpicking: Your objection to "If ___, then ___" is a doozy!

Then I overlooked that it did.

So, unlikely as it seems, you apparently don't know what "rules" means, or "language" for that matter.
— tim wood

What an actual dolt, my lord. Learn your own language first so foreigners don't have to teach it to you. — Lionino

Are you saying the poster's sentence is not adequate English?

One instance that I can see might be regarded as nitpicking was when I said saying "B is true" was extraneous. But I mentioned it in a stylistic sense that it's better not to include extraneous items so that the arguments can be seen more clearly, without the distraction of those items.

I don't know your point. Anyway, people may use the word 'logical' differently: (1) pertaining to logic or (2) logically correct.

And so, your incorrect nitpick about my use of 'grammatical' when obviously I mean 'grammatically correct' or 'according to the rules of grammar'.

.

You said that I nitpick. I don't. But then you turn around and incorrectly nitpick!

I responded exactly regarding the post of mine that you referred to.

every sentence is grammatical. Not every sentence is grammatically correct. — Lionino

Oh, please! Talk about inane nitpicking that isn't even correct! Obviously I'm using 'grammatical' in the sense of 'conforming to the rules of grammar'.

I don't understand that. — Lionino

I explained it when I first flagged you on it.

I am not reading them. — Lionino

But you are. Right now. Anyway, my posting is not based on whether you read or don't read.

Your argument stooped to the tactic of citing ambiguity as if we would not be discussing modulo certain ambiguities.

You skipped my point.

/

Morphology concerns form. And so also syntax. Especially in logic, syntax is a mater of form, hence 'well formed'.

It is a misspelt word. It has nothing to do with syntax. — Lionino

In logic, terms are formed by rules. If symbols are not in correct order or incorrectly omitted, then they are not syntactical. In that way too, if a string of letters doesn't even form a word, then the expression in which the string occurs cannot be syntactical.

Are the words in correct case, inflection, etc.
— TonesInDeepFreeze

English has no morphological cases. — Lionino

I didn't say 'morphological cases'.

Merriam Webster is not reliable neither is it competent. — Lionino

I have found Merriam to be good, especially unabridged, but some deterioration over the years. I read all of yours. I mentioned the others for emphasis.

The usual sense of 'grammar' is 'syntax'
— TonesInDeepFreeze

It is not. — Lionino

From definitions you posted yourself.

/

If I were to nitpick, it would be a whole other thing. Being careful to state things about logic accurately so that false conclusions about are not drawn is not nitpicking.

I still think the LNC overall articulates a law of thought — Lionino

That is not at issue. What is at issue is whether that law of thought can be broken. Yes it can. Of course, if we hold that it is required for rationality, then we may say it can't be broken rationally. But that doesn't refute that people break laws of thought often. Full circle again.

I have edited the post you are quoting. So now it reads "as the necessary conditions/operations for my/human/any mind". In this sense, I don't think it can be broken, as the mind, definitionally, cannot operate outside of these conditions. — Lionino

People operate mentally in all kinds of ways: Fictionally, absurdly, poetically, ironically, day dreaming, dreaming, mystically and insanely. But your point reduces to the tautological: the mind can't operate rationally without operating rationally. No one disagrees with that.

And so we've come around again full circle.