Incompleteness: Rebecca Goldstein on the Life and Work of Kurt Gödel
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- เผยแพร่เมื่อ 18 ต.ค. 2020
- Best known for his Incompleteness Theorem, Kurt Gödel (1906-1978) is considered one of the most important mathematicians and logicians of the 20th century. By showing that the establishment of a set of axioms encompassing all of mathematics would never succeed, he revolutionized the world of mathematics, logic, and philosophy.
Rebecca Goldstein is the author of Incompleteness: The Proof and Paradox of Kurt Gödel and, most recently, Betraying Spinoza: The Renegade Jew Who Gave Us Modernity. She is a fellow of the American Academy of Arts and Sciences, and has received many awards for her fiction and scholarship, including a John D. and Catherine T. MacArthur Foundation Fellowship.
She shares her insight into the life and work of Kurt Gödel.
Beautiful; very insightful. Eloquently portrayed.
So fascinating. Reading about Godel and Cantor lately and they were both so brilliant but both on the verge of insanity. I have Rebecca Goldstein's book on Godel, among other books on these topics. She is amazing - this was a great talk and well produced. Thank you.
What a great insights into Gödel's world. Thank you!
How utterly fascinating a presentation by this gloriously brilliant, and beautiful woman. Her presentation is indeed enhanced significantly by every movement of her flashing eyes, and her cocked head, ever framing her hair and face all the more attractively, albeit unconsciously. Her vocal enthusiasm and facial animation are so contagiously compelling, and her obvious love for Godel so captivating that one can only speculate how much more speedily, and far his work would have advanced had she been of his generation, and fallen in love with him, and become his life partner. It appeared to me that she had indeed fallen in love with him, even in his physical absence, and separated by so many years.
By the end of the video, I was entirely smitten with her brilliance, and her beauty., and her love for her life's work. Her apparent love of Godel himself became a most compelling element of her presentation. Her delivery is as eloquent as one might hope to find in the entire world of the media (talking heads) at large, never mind within academia (including the world of authors). I must order two of her books to start, that dealing with Godel, and then the biography of Spinoza. If she were a university professor, I would wish myself sufficiently intelligent to partake of her font of academic brilliance, and remain a student for life! But meanwhile, I will make do with reading her books (and enjoying her video appearances...she is so eloquent, the world cries out for her to grace an academic venue, somewhere!).
I would of liked to asked Godel what it felt like to show Einstein that time, in fact, does not exist. Awesome work, and thanks Mrs. Rebecca.
Loved the way, You express the words
Thank you for this video, great! Her book on Gödel must be a really good reading.
At university I was twice asked was I Jewish, in fact the second time I was told I was Jewish. I’m not, but it’s great to know I have something in common with Gödel.
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.
Excellent thanks 😊
What Godel showed vis a vis the continuum hypothesis is that, if assuming Zermelo Fraenkel axioms of set theory plus the continuum hypothesis would lead to a contradiction, that contradiction would exist without the continuum hypothesis and that, thus, one is free to assume the continuum hypothesis. Cohen proved this also to be the case for the negation of the continuum hypothesis, thus completing the process of showing that the continuum hypothesis is undecidable within the Zermelo Fraenkel formulation of the axioms of set theory. Godel also proved that assuming the axiom of choice would not create an inconsistency in ZF that wasn’t already there and that, hence mathematicians could safely assume it. The axiom of choice is, arguably, much more important than the continuum hypothesis. Assuming it to be true leads to some incredibly non-intuitive results.
When I was at university studying computer science, every time we got onto logic, I would get a headache. Looking at Gödel, I think I can see why.
I have always thought that in mathematics a proposition, unless it is an axiom, can only be considered true if it can be proved by logical steps from the axioms. So how can we say that the Goldbach conjecture is true when it has not been proved?
You are making a categorical error when you speak of “truth” and “provability”. Provability is a feature of the syntax of a formal language, and truth is a feature of its semantics. The study of a formal language’s syntax in a formal system is the object of proof theory, which by itself does not consider how any of the expressions in its language are to be interpreted(a semantic feature). In proof theory, expressions of a language are meaningless uninterpreted symbols with no sense of being true. A proof theorist is only interested in the analysis of a formal language’s syntax and the manipulation of symbols into well-formed expressions of the language from other well-formed expressions in accordance with rules of symbol manipulation(or the rules of inference).
Truth on the other hand is studied in model theory, which DOES consider the semantics of a formal language. An expression can only be true when it is interpreted by a model. The main objects of model theory are models(structures or sets with a domain of elements we wish to study together with relations on those elements). An expression is said to be true when it is interpreted in a model and if the elements of that model make the expression true. It is possible to have an expression in a formal system that is true in one model but false in another. For example, consider the formula φ = ∃x: x+x=1. In plain English, this just says that there exists some number x such that its sum with itself is equal to one. Let us consider two models to interpret this formula: the set of real numbers ℝ and the set of integers ℤ. Which model makes this formula true? Well it can’t be the integers ℤ, because there doesn’t exist an integer such that its sum with itself is 1. This formula is false when interpreted in the integers ℤ. However, it is true when interpreted in ℝ, because there does exist a real number such that its sum with itself equals 1, namely, 0.5.
It is true that provability implies truth. This property of a formal system is called soundness. This means that indeed proving an expression of a formal system implies that it is true when interpreted in a model. The proof of soundness is trivial, and it is a result that belongs not just to proof theory or model theory, but to both. It is the intersection of both theories and finds a way to relate the syntax of a formal language with its semantics. The converse, however, that truth implies provability isn’t so trivial. This property of a formal system is called completeness. Gödel was able to prove that any model will interpret any expression in systems of first-order logic to be true. This is Gödel’s completeness theorem of first-order logic.
For formal axiomatic systems strong enough to contain a model of the natural numbers, there will always be expressions which are interpreted to be true by its model(Goldbach’s conjecture possibly, for instance), but which nonetheless can not be demonstrated to be a finite sequence of syntactic manipulations of symbols. This is Gödel’s (first)incompleteness theorem.
I hope this answers your question.
The provability of Godel's Incompleteness Theorem is subject to the limitations of its own propositions, right?
Gödel’s life could be used to prove his own theory!
No maybe it could prove itself. You just didnt give it a date in time. Maybe the system is modifying itself in cosmic time.
I switched off at 3:03, because it seemed to be all about her.
Hmm... “most important mathematician ....” That’s a big claim and very dependent on your definition of important.