24
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How does Gödel's theorem apply to daily life?
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 ...
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14
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Was Wittgenstein anticipating Gödel?
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 ...
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12
votes
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Gödel's incompleteness theorems - what are the religious implications?
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 ...
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11
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What are the philosophical consequences of the undecidability of the spectral gap in quantum theory?
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 ...
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11
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Is there a way to avoid Gödel's incompleteness affecting mathematics as a whole?
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 ...
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10
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How does Gödel's incompleteness theorem apply to materialism and the mind
(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 ...
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10
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What did Gödel mean by "positive property" in his ontological argument?
"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" ...
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9
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Does Gödel's argument that minds are more powerful than computers have the inconsistency loophole?
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-...
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9
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Was Kant anticipating Gödel's incompleteness in his antinomies?
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 ...
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8
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Where did Gödel write that first-order logic is the "true" logic?
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-...
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8
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Does Gödel's argument that minds are more powerful than computers have the inconsistency loophole?
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 ...
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8
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Are axioms tautologies?
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 ...
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8
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Why did Gödel believe that there was a conspiracy to suppress Leibniz's works?
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, ...
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8
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How to show (in a hand waving manner) that the Godel sentence is true
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"...
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7
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Did Russell understand Gödel's incompleteness theorems?
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 ...
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7
votes
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Did Gödel oppose or agree with the Logical Positivists?
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 ...
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6
votes
Did Russell understand Gödel's incompleteness theorems?
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 ...
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6
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Do Gödel's incompleteness theorems have any consequences for epistemology?
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 ...
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6
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Can Gödel's incompleteness theorems be applied to ethics?
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 ...
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6
votes
Are axioms tautologies?
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 ...
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5
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Is this enough to conclude that G is false?
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 ...
5
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Does Gödel's argument that minds are more powerful than computers have the inconsistency loophole?
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 ...
5
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Can Gödel's incompleteness theorems be applied to ethics?
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 ...
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5
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Gödel: Why is a proposition undecidable?
This answer is a bit technical, but I think the OP will find it interesting.
Let me first recall a bit about the history of the incompleteness theorem (IT). How IT is usually stated is:
If PA (or any ...
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5
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Can you list examples of problems that can not be solved within a formal system but human beings have solved through construction or creativity?
There is no agreed upon example of this kind.
Let us explore the issues:
First, we cannot even come up with a decent example of a problem not solvable by any formal system. If you state the problem ...
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5
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Is it valid to prove the axioms of a system from themselves? How does it square with Gödel's incompleteness?
Yes, the axioms do trivially prove themselves. Your last derivation, however, is not valid: "A=A" can not be substituted for A because the latter is a symbol in a formal system, while the ...
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5
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Did Gödel do philosophy beyond logic?
The most oft-quoted of Gödel's philosophical works might be Is Mathematics a Syntax of Language? It is not available online, unfortunately, but excerpts are read in this YouTube video. Tait wrote a ...
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5
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The origin of a particular self-reference paradox
The origin is with the so-called Whiteley Sentence.
See C.Whiteley, “Minds, Machines and Gödel: A Reply to Mr. Lucas (1962)”, Philosophy 37:61-62 :
It is possible to devise a formula which will ...
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