The Theorem's of Gödel (Noam Chomsky)

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[rodolforesende]

Some people believe that the inability to measure the diagonal of a square with a side length of one using rational numbers proves the existence of irrational numbers.This is not correct.
The proper understanding is that you need to construct irrational numbers to measure the diagonal of a square with a side length of one. The fact that you do not have a rational number to measure this diagonal simply shows that no rational number corresponds to the length of that diagonal. It doesn't automatically prove the existence of irrational numbers; >>>it highlights the need for them.<<<
This idea relates to subtle points in understanding Gödel's results. Even the renowned philosopher Wittgenstein misunderstood Gödel's work. Essentially, what Gödel's incompleteness theorem tells us is that if a formal system uses "usual" logic and can express "usual" arithmetic, THEN it is either incomplete (meaning there are true statements that it cannot prove) or inconsistent (meaning it could prove both a statement and its negation). Goedel shows that we need systems beyond the "usual"... and mr chomsky, goedel's results certainly have strong impacts in linguistics!!!!

[sashafalcon]

Would've been great to hear his answer 30 years ago, when his mind was still active enough.

[jamestagge]

As a follow up to my recent posts on (Goedel’s incompleteness theorem) the architecture of materiality and that of the realm of abstraction, the two structurally linked, which prohibits for formulation of conceptual contradictions, I present the following for critique.

After watching several video presentations of Geodel’s incompleteness theorems 1 and 2, as presented in each I have been able to find, it was made clear that he admired Quine’s liar’s paradox to a measure which inspired him to formulate a means of translating mathematical statements into a system reflective of the structure of formal semantics, essentially a language by which he could intentionally introduce self-referencing (for some unfathomable reason). Given that it is claimed that this introduces paradoxical conditions into the foundations of mathematics, his theorems can only be considered as suspect, a corruption of mathematic’s logical structure. The self-reference is born of a conceptual contradiction, that which I have previously shown to be impossible within the bounds of material reality and the system of logic reflective of it. To demonstrate again, below is a previous critique of Quine’s liars' paradox.

Quine’s liar’s paradox is in the form of the statement, “this statement is false”. Apparently, he was so impacted by this that he claimed it to be a crisis of thought. It is a crisis of nothing, but perhaps only of the diminishment of his reputation. “This statement is false” is a fraud for several reasons. The first is that the term “statement” as employed, which is the subject, a noun, is merely a place holder, an empty vessel, a term without meaning, perhaps a definition of a set of which there are no members. It refers to no previous utterance for were that the case, there would be no paradox. No information was conveyed which could be judged as true or false. It can be neither. The statement commands that its consideration be as such, if true, it is false, but if false, it is true, but again, if true, it is false, etc. The object of the statement, its falsity, cannot at once be both true and false which the consideration of the paradox demands, nor can it at once be the cause and the effect of the paradoxical function. This then breaks the law of logic, that of non-contradiction.

Neither the structure of materiality, the means of the “process of existence”, nor that of the realm of abstraction which is its direct reflection, permits such corruption of language or thought. One cannot claim that he can formulate a position by the appeal to truths, that denies truth, i.e., the employment of terms and concepts in a statement which in its very expression, they are denied. It is like saying “I think I am not thinking” and expecting that it could ever be true. How is it that such piffle could be offered as a proof of that possible by such a man as Quine, purportedly of such genius? How could it then be embraced by another such as Goedel to be employed in the foundational structure of his discipline, corrupting the assumptions and discoveries of the previous centuries? Something is very wrong. If I am I would appreciate being shown how and where.

All such paradoxes are easily shown to be sophistry, their resolutions obvious in most cases. What then are we left to conclude? To deliberately introduce the self-reference into mathematics to demonstrate by its inclusion that somehow reality will permit such conceptual contradictions is a grave indictment of Goedel. Consider;

As mentioned above, that he might introduce the self-reference into mathematics, he generated a kind of formal semantics, as shown in most lectures and videos, which ultimately translated numbers and mathematical symbols into language, producing the statement, “this statement cannot be proved”, it being paradoxical in that in mathematics, all statements which are true have a proof and a false statement has none. Thus, if true, that it cannot be proved, then it has a proof, but if false, there can be no proof, but if true it cannot be proved, etc., thus the paradox. If then this language could be created by the method of Goedel numbers (no need to go into this here), it logically and by definition could be “reverse engineered” back to the mathematical formulae from which it was derived. Thus, if logic can be shown to have been defied in this means of the introduction of the self-reference into mathematics via this “language” then should not these original mathematical formulae retain the effect of the contradiction of this self-reference? It is claimed that this is not the case, for the structure of mathematics does not permit such which was the impetus for its development and employment in the first place. I would venture then that the entire exercise has absolutely no purpose, no meaning and no effect. It is stated in all the lectures I have seen that these (original) mathematical formulae had to be translated into a semantic structure that the self-reference could be introduced at all. If then it could not be expressed in mathematical terms alone and if it is found when translated into semantic structures to be false, does that not make clear the deception? If Quine’s liar’s paradox can so easily be shown to be sophistry, how is Goedel’s scheme not equally so? If the conceptual contradiction created by Goedel’s statement “this statement has no proof” is so exposed, no less a defiance of logic than Quine’s liar’s paradox then how can all that rests upon it not be considered suspect, i.e., completeness, consistency, decidability, etc.?

I realize that I am no equal to Goedel, who himself was admired by Einstein, an intellect greater than that of anyone in the last couple of centuries. However, unless someone can refute my critique and show how Quine’s liar’s paradox and by extension, Goedel’s are actually valid, it’s only logical that the work which rests upon their acceptance be considered as invalid.

[BehroozCompani-fksx]

Russel's paradox is a man made nonsense sentence that has no real world importance or value. You can make many more such paradoxes. Godel used a similar paradox to show mathematics incompleteness.
Godel's incompleteness theorems are misinterpreted. What it really shows is that if you use a nonsense paradox and classical logic what happens is that the logic tells you it can not handle nonsense. That is not surprising nor profound.

[DornigeChance]

At least a few people got, that math is cool but still made up stuff.