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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 ...
Conifold's user avatar
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11 votes
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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 ...
Conifold's user avatar
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11 votes

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 ...
Conifold's user avatar
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10 votes
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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 ...
Not_Here's user avatar
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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" ...
Conifold's user avatar
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9 votes
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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 ...
Conifold's user avatar
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8 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 ...
Timothy Chow's user avatar
8 votes

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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8 votes
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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 ...
Bumble's user avatar
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8 votes

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, ...
Conifold's user avatar
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8 votes
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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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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 ...
Conifold's user avatar
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6 votes
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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 ...
Panu Raatikainen's user avatar
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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 ...
Cort Ammon's user avatar
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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 ...
Cort Ammon's user avatar
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Why is ZFC not as susceptible to Gödel's incompleteness as was the Principia Mathematica?

ZFC is susceptible to Gödel's incompleteness as was the Principia Mathematica. See Gödel’s Incompleteness Theorems for an introduction to the theorem : Any consistent formal system F within which ...
Mauro ALLEGRANZA's user avatar
6 votes

Gödel’s Incompleteness Theorem: How can truth go deeper than proof?

At least in the case of arithmetic, the concept of arithmetical truth can be defined as a type of non-computable extension of the Peano axiomatic system. Basically the idea is to consider the set of ...
Hypnosifl's user avatar
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6 votes
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Why can Goedel's Incompleteness Theorem be proven?

"Godel thus proved an unprovable statement" - this is not quite the case, as you yourself recognize in the lines following. Rather, Godel proved that there exists an undecidable statement ...
emesupap's user avatar
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5 votes

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 ...
Derek Janni's user avatar
5 votes

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 ...
Arno's user avatar
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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 ...
Noah Schweber's user avatar
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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 ...
Conifold's user avatar
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5 votes

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 ...
Conifold's user avatar
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5 votes
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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 ...
Mauro ALLEGRANZA's user avatar
5 votes

Before Gödel, was undecidability of axiomatic systems an issue at all?

While it’s not quite a perfect parallel, a related concern about decision problems had been phrased some years before Gödel’s work: see https://en.m.wikipedia.org/wiki/Entscheidungsproblem . David ...
Sofie Selnes's user avatar
4 votes
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What motivated Gödel to arithmetize syntax?

Arithmetization of syntax allows Gödel to show that statements about number theory are also statements in number theory. This allows him to construct self-referential statements about number theory in ...
Alexander S King's user avatar
4 votes

How is ω-consistency different from ordinary consistency?

Yes, see ω-consistent theory. It plays a role in the study of formal theories in mathematical logic research, and is related to the so-called ω-logic needed to express the "true standard ...
Conifold's user avatar
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4 votes

Gödel's incompleteness theorem and non-standard logics/foundational systems

There are a large number of ways to sidestep Gödel's proof. The question is whether those systems have sufficient practical value to mathematics to be used for any purpose other than sidestepping his ...
Cort Ammon's user avatar
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