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24

Here's what Jordan Ellenberg, a professor of mathematics at the University of Wisconsin, has to say about this topic in his Does Gödel Matter? article: What is it about Gödel's theorem that so captures the imagination? Probably that its oversimplified plain-English form—"There are true things which cannot be proved"—is naturally appealing to anyone ...


13

Not on this at least. Wittgenstein is alluding to Frege on logical syntax. From Tractatus:"Frege says that any legitimately constructed proposition must have a sense. I say that any proposition is legitimately constructed". Laws of syntax are similar in form to ethical laws: thou shalt not (form such and such sentences). Wittgenstein's response in both cases ...


12

Gödel's theism is discussed by Franzen in Gödel’s Theorem: An Incomplete Guideto Its Use and Abuse. He penned a version of the ontological argument, and in 1961 ranked the worldviews “according to the degree and the manner of their affinity to or, respectively, turning away from metaphysics (or religion)... Skepticism, materialism, and positivism stand on ...


11

What the result means, essentially, is that in certain toy models there can be no algorithm deriving some macroscopic characteristics (spectral gap) from microscopic parameters of the models. The main import is that we get a Gödel sentence that unlike the original has some explicit mathematical meaning. Let's be generous and assume that the situation extends ...


11

It is a natural idea, but unfortunately the answer is no, it is not feasible. The root of incompleteness is not numbers, but the possibility of (implicit) self-reference, arithmetic is just the simplest structure that already realizes that possibility. In fact, one does not even need the Peano arithmetic, but a much weaker Robinson arithmetic without even ...


10

(Disclaimer: All of what follows is explicitly done under the assumption that the mind is a Turing machine and that we can formally axiomize our mathematical thinking. This has, of course, never been shown to be possible. What I'm describing is the state of affairs that would obtain if that were possible, given the metamathematical and metalogial results of ...


10

"Positive" is what Leibniz and other proponents of the ontological argument called qualities that make something "better" than it is without them (Anselm spoke of "good" as in summum bonum). In particular, Leibniz defined "perfections" ascribed to God as "simple, positive qualities in the highest degree", see ...


9

The following is from a late paper of Russell's titled "Logical Positivism". It can be found in "Logic and Knowledge" It appeared that, given any language, it must have a certain incompleteness, in the sense that there are things to be said about the language which cannot be said in the language. This is connected with the paradoxes - the liar, the class ...


9

Godel's theorem says nothing about human understanding. It only places limits on certain formal axiomatic systems. Humans have ways of understanding that transcend formal axiomatic systems; for example, we can extend a given axiomatic system to prove the truths that were unprovable in the unextended system. As an example, Zermelo-Fraenkel set theory (ZF) ...


9

For detailed discussions of the so-called Lucas-Penrose arguments, see : Torkel Franzén, Gödel's theorem : An incomplete guide to its use and abuse (2005), Ch.6 Gödel, Minds, and Computers, page 115-on and Francesco Berto, There's Something about Gödel : The Complete Guide to the Incompleteness Theorem (2009), Ch.11 Mind versus Computer: Gödel and ...


9

This reminds me of the older question Was Wittgenstein anticipating Gödel? There is more to it in the case of Kant than there was in the case of Wittgenstein though, at least in spirit. One could say that Kant pioneered in epistemology the stratification into levels of discourse, which Gödel later applied to formal semantics. When the Gödel theorem ...


8

He did not write it anywhere. The quote itself only calls Gödel and Skolem "alleged proponents", and later in the article Eklund remarks that "the (supposed) evidence that Skolem adhered to first-order logic is that Skolem held that set theory and arithmetic should be given first-order axiomatizations, whereas... evidence that Gödel adhered to first-order ...


8

We can construct a computer that implements an inconsistent formal theory to which Gödel theorem does not apply just like we can construct a computer that implements Peano arithmetic. A simple example is Meyer's relevant arithmetic R# that allows some contradictions, but uses paraconsistent logic (without the law of explosion) to limit their effect. In R# ...


8

It is worth separating the logic from the epistemology. Let's start with the logic. A (first order) theory is a set of sentences. Usually we are interested in deductive systems, so we require a theory to be closed under the relation of provability. A theory T is axiomatizable if there exists a subset of T, the axiom set, such that all of the sentences in T ...


8

The OP information probably comes from Wikipedia's article on Characteristica Universalis "The logician Kurt Gödel, on the other hand, believed that the characteristica universalis was feasible, and that its development would revolutionize mathematical practice (Dawson 1997). He noticed, however, that a detailed treatment of the characteristica was ...


8

Gödel’s Incompleteness Theorem is a result about formal systems. Its proof requires certain assumptions about the properties of specific formal system F: basically, about its "expressive capabilities". In a sense that can be specified rigorously, system F must have the capabilities to manufacture the provability predicate for F, i.e. a suitable formula PrF(...


7

The two notions (completeness and incompleteness) are not opposites but very much connected (not only by Godel's name in the name of the two theorems). Do take into account that Godel's Completeness Th of First-Order Logic is : if a sentence is true in all the models of the axioms (i.e. it is a logical consequence of the axioms) then it is also formally ...


6

As mentioned in a comment, Alasdair Urquhart has written a paper, Russell and Gödel (Bull. Symb. Logic 22 (2016), 504–520), that discusses a number of different topics, including Russell’s view of Gödel’s results. He provides many of the Russell quotes that other respondents here have given, as well as the following quote from an “Addendum” that was written ...


6

Russell's comments on Gödel were scanty, but it was very unlikely that Russell did not understand what Gödel was talking about. The paradox presented by Gödel sentence was nothing new; it was the same old vicious circle paradox, which had been abundantly dispelled by Russell's Theory of Types[source 1]. Russell discovered the Theory of Types in 1906. The ...


6

"Is anything essential lost" or are the shortcomings of Meyer's system in "the exotic outer reaches of what is possible in traditional PA"? PA proves that formula if p > 2 is prime, then there is a positive integer y which is not a quadratic residue mod p; that is, ∃y ∀z: ¬(y ≡ z^2 (mod p)). That looks like a pretty unexotic bit ...


6

V-omega, the set of hereditarily finite sets is a model of the theory you get by taking ZFC and replacing the axiom of infinity with its negation, and it is bi-interpretable with Peano Arithmetic (so indeed, Gödel's incompleteness theorem still applies). For more, see e.g. https://math.stackexchange.com/questions/107639/what-are-the-consequences-if-axiom-of-...


6

Gödel's theorems only apply to specific theories. In particular, they must be capable of proving all the provably true statements of arithmetic. Gödel's theorems have very strong implications if you presume the universe you live in is capable of proving all true statements of arithmetic. This creates neat tail-chasing loops if you believe that the ...


6

Gödel was a young man in search of a place to belong, many young intellectuals were attracted to the Vienna Circle for its pluralism and tolerance. But it wasn't purely social. Gödel was clearly interested in mathematics, in 1925-26 Schlick, the circle's founder, gave lectures on philosophy of mathematics which Gödel attended. But it was Carnap, who really ...


6

There are a few limitations that are worth mentioning: Arithmetic is not a trivial thing. In particular, one has to deal with the axiom of induction, which is metaphorically quite similar to a tower of Babel argument. It took a lot of work to develop meanings of logical concepts which could reach to infinity with the finesse mathematics does. For many, "...


6

An axiom is something you assume to be true without proof. A tautology is a statement which can be proven to be true without relying on any axioms. An axiom is not a tautology because, to prove that axiom, you must assume at least one axiom: itself. If you wanted to be more pedantic (which is always fun), the idea that you can prove a tautology without ...


5

Unless I'm missing the point, none of your axioms offer any definition of proof or state that all true statements in your system must be provable. Given that, there's no reason a statement might not be both true and not provably true --or, for that matter, false, but not provably false. In my opinion, your axiomatic system would need a formal definition of ...


5

As such, Turing machines are not consistent or inconsistent - formalized systems are. However, once we fix a coding of (e.g.) the syntax of the language of arithmetic, we can view some Turing machines as enumerating the theorems of some formal systems (and every formalized system (its theorems) is enumerated by some machine). Then, when the coding kept fixed,...


5

I'd say this is your largest concern: Assuming real world situations display a minimum amount of complexity - analogous to the "capable of proving statements of basic arithmetic" clause Real world situations usually display an amazing degree of complexity, unlike basic statements of arithmetic. Unfortunately, most attempts to extend Gödel's theorems ...


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