Questions tagged [goedel]

Questions related to the work of Kurt Gödel. Please mind the spelling of his last name: "Gödel". If you cannot or don't know how to create the "ö", you might also write his name as "Goedel". In all cases, please avoid "Godel". If you want to create a (hyper)link to, say, a Wikipedia entry, you might have to manually change the "ö" to "%F6".

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What is the definition of 'The System of all positive properties is compatible'?

A sentence at the end of goedel's 1970 proof of god's existence is: The possibility of god's existence means 'The System of all positive properties is compatible' But without knowledge what ...
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Is Norman Megill's view of Gödel's incompleteness theorem compatible with what philosophers have said about it?

Here is one recent and seemingly expert appreciation on the consequences of Gödel’s incompleteness theorem for mathematics: Gödel’s incompleteness theorem showed that it is impossible to achieve ...
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Why is Hofstadter referring to Gödel as "a young Turk from Austria"?

In I am a Strange Loop, Hofstadter describes the history of Gödel's incompleteness theorem, talking about how venerable Russel, after developing his supposedly all-encompassing and paradox-free ...
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Why can Goedel's Incompleteness Theorem be proven?

Preliminaries: The way I understand it, Gödel took Russel's and Whitehead's Principia Mathematica (PM) and mapped strings of symbols from PM onto the integers, their Gödel numbers. He then constructed,...
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Gödel's loophole [closed]

Gödel's loophole is an unidentified, alleged contradiction in the US Constitution, one that purportedly allows for America to transform into a tyranny on an internal "legal" basis. Since ...
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Necessity of arithmetic truths into Godel sentences

My layman but hopeful to understand self is slowly trying to understand some of Godel and the philosophical implications of his work (uh oh). Currently my understanding is that on some level: Godel ...
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On the consistency of logical arguments on ethics and morality in the light of Incompleteness theorem

My friend claimed that 'Any type of formal system is inconsistent' to support his another claim 'reason for each moral judgement will be different, situationally we can change ( for eg; consent will ...
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Did Godel think certain math could only be understood if platonism is correct? (and correspondence and nominalism)

I’m reading Shapiro’s Thinking About Mathematics, and there’s a quote by Godel which I would like to fully understand, both his intended meaning and how it’s viewed in the wider context of mathematics,...
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Realism as necessary for impredicative mathematics to avoid viscous circle, but not really?

Here is an quote from Godel from Shapiro’s Thinking About Mathematics: “…the vicious circle…applies only if the entities are constructed by ourselves. In this case, there must clearly exist a ...
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Are Gödel's Incompleteness Theorems a refutation of Rationalism?

According to Putnam, Gödel's theorems show that the set of truths in Number Theory (i.e., true propositions involving natural numbers and their properties) is not recursively enumerable, whereas all ...
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Meaning of quote by Goedel

"The meaning of the world is the separation of wish and fact. Wish is a force as applied to thinking beings, to realize something. A fulfilled wish is a union of wish and fact. The meaning of the ...
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Does Gödel’s findings boil down to part of classical mathematics (as opposed to computation) is flawed?

According to artificial intelligence researcher Joscha Bach, only classical mathematics is affected by Gödel’s incompleteness theorem however not computation where calculations are performed in a step-...
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Gödel’s Incompleteness Theorem: How can truth go deeper than proof?

My current understanding: Math starts with a set of basic (purportedly self-evident) statements that are taken as a given without the need to prove them true, like e.g., a + b = b + a etc. Such ...
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Seeking a quote of Russell on what he could conceive or discuss

At some point someone asked Bertrand Russell about formal logical language without distinctions of type. (I think it might have been Quine, who was developing for example kinds of algebraic logic ...
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Godel's incompleteness theorem when the cardinality of axioms is > ℵ_0?

So I was thinking of Godel's theorem (I am by no means an expert in this topic). Does Godel's only work when the cardinality of the number of axioms is the same as the cardinality of the number of ...
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Question on Godel's Remark on Algorithmic Nature of Mind

Gödel claimed that what the Theorems do entail (specifically, the Second Theorem) is that mathematics is inexhaustible: It is this theorem [i.e., the Second Theorem] which makes the incompletability ...
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Some doubts on Incompleteness Theorems

An important point to note about first incompleteness theorem is that while a certain formula is "true" but unprovable, it is "true" on the basis of my understanding (intended ...
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Is logic about a priori mind?

What is logic? One can imagine Turing, Godel or Post writing a paper on logic. What provides the "validity" to the content they write? One proper answer to this question is the a priori &...
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Is there a weaker/general version of Incompleteness Theorem which holds for every formal axiomatic system?

Is there a general version of Godel's Incompleteness Theorem which holds for any formal axiomatic system (and not just those capable of modelling basic arithmetic)? If no, is it absurd to ask why such ...
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Does Gödel believe in the existence of his rotating universe?

I am wondering whether Gödel believe ain the existence of his rotating universe since he is a mathematical Platonist. I am also wondering in what entities believe mathematical platonists. For example: ...
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Does Godel's second incompleteness theorem mean it's impossible to know whether a proven statement cannot also be disproven?

I am trying to understand Godel's Second Incompleteness Theorem which says that any formal system cannot prove itself consistent. In math, we have axiomatic systems like ZFC, which could ultimately ...
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Why is Turing claiming that a complete and computable axiomatization of arithmetic would imply the decidability of first-order logic?

So I'm reading the famous paper of Turing "On Computable Numbers, with an Application to the Entscheidungsproblem". At the beginning of his proof of the undecidability of first-order logic (FOL), he ...
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Non-consistent mathematical axioms

It is known that axioms are the building blocks of mathematics. Differents sets of axioms different "games". What I don't understand is how do we know that we pick axioms that are consistent? . Does ...
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Why does Gödel's incompleteness theorem apply to multiple formal systems?

Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor ...
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In non-platonism, can undecidable statements have truth value?

Most sources I can find about Gödel's incompleteness theorems summarize the result as "there exist true arithmetical statements that have no proof." It seems coherent to say that there exist ...
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Before Gödel, was undecidability of axiomatic systems an issue at all?

Before Gödel, was the issue raised that there may be undecidable statements within axiomatic systems of thought? Gödel managed to answer affirmatively by proving that the assumption of the ...
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What did Gödel mean by "positive property" in his ontological argument?

In his ontological proof, Gödel states (Axiom 1) If a property is positive, then its negation is not positive. What does he meant by this term? I have come across authors who replace this notion ...
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How to show (in a hand waving manner) that the Godel sentence is true

I have been reading Graham Priest's The Logic of Paradox, and there is a section where he tried to show that our informal proof argument (in Priest's terminology, naive proof procedure) is more ...
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What evidence is there that Gödel believed the mind to be non-physical?

On the Stanford Encyclopedia of Philosophy's article on Platonism in Metaphysics, the author writes that "Gödel's version of this view — and he seems to be alone in this — involves the idea that the ...
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SELF-REFERENCE AND THE LANGUAGES OF ARITHMETIC

Hello can someone explain me exacty how in this fragment of the paper (SELF-REFERENCE AND THE LANGUAGES OF ARITHMETIC, RICHARDG.HECK,JR.): (9) Tr(x) ≡∃y(rhs(x,y)∧¬Tr(y)), where rhs(x,y) is a formula ...
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Source of Godel's quote on materialism

In an interview, David Berlinski quoted Godel's view on materialism: It [materialism] succeeds to the extent that the materialists assign material objects all the properties that used to be ...
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The origin of a particular self-reference paradox

This is a simple reference request, for the origin of a particular type of paradoxical statement. The example I remember is Roger Penrose can't consistently claim this statement to be true. It's a ...
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Why does one need to specify that the language is "well-orderable" for first order logic to be complete?

While reading Wikipedia I noticed the phrase "well-ordered language" in the following related to Gödel's completeness theorem: The completeness theorem then says that for any first-order theory T ...
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What is the relationship between computation and Gödel's incompleteness theorems? [closed]

In what way do Godel's incompleteness theorems impact computers/hypercomputers? Do they somehow prevent them from being capable of computing everything (of computing literally all uncomputable/...
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Why is ZFC not as susceptible to Gödel's incompleteness as was the Principia Mathematica?

So, from what little I have read (such as this answer), it appears to be that one reason why the program of Logicism, as laid out in the Principia Mathematica, failed was that its goals (of finding a ...
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Gödel's incompleteness theorems - what are the religious implications?

Apparently Kurt Gödel believed that his incompleteness theorems have some kind of religious implications. Despite Gödel's belief in a personal God, this was still somewhat surprising to me. ...
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Gödel's Results and Philosophy of Mathematics [closed]

Gödel's results essentially conclude that there are True but Unprovable statements in arithmetic. My thoughts are as follows: Axioms form the foundation of mathematics -because we need to assume ...
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Is Propositional Logic Sound and Complete

After reading Gödel's incompleteness theorem, I wonder if there are any systems whose axioms are sound and complete. I realize that Gödel's arguments (at least from my sources) only apply to ...
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How does Gödel's incompleteness theorem apply to materialism and the mind

Assertion 1: Humans use some logical system to understand the universe Assertion 2: Gödel proved through a formal logic what is provable about a logical system is a subset of what is true about it, ...
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Gödel's incompleteness theorem and non-standard logics/foundational systems

I am amateur in the field of mathematical logic, so sorry for any confusing parts of this question. It is well known that Gödel's incompleteness theorem shows there are great limits to what first-...
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Godel's theorems

The result of Godel's theorems was that we knew for sure that a formal axiomatic system wasn't capable to derive all of mathematics. The math derived under the system cannot be consistent and complete....
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How is ω-consistency different from ordinary consistency?

I've read Gödel's explanation and others but my understanding is unclear. Answers to the followup questions below would help: does ω-consistency have any relevance to methods or ideas not connected ...
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Did Gödel do philosophy beyond logic?

On Wikipedia Gödel is described as a philosopher and since I do know of his logical works (as far as I am able to understand what I've read and heard of it). I wanted to know if he wrote more stuff, ...
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Why did Gödel believe that there was a conspiracy to suppress Leibniz's works?

It is known that Gödel was obsessed with Leibniz, and apparently he even believed that their was a worldwide conspiracy among academics to suppress Leibniz's works. Does anyone know where this came ...
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Are semantic inferences between undecidable sentences in a system possible?

For example from the Gödel sentence "G iff ¬P([g])", where g is G's Gödel-number, is it possible to make semantic inference (not syntactic, only at the level of truth between the undecidable sentences ...
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Is it valid to prove the axioms of a system from themselves? How does it square with Gödel's incompleteness?

I recently asked whether the axioms are tautologies, and got comments that seemed to me highly suspicious. Namely, that you can always prove an axiom from itself, that you can trivially say A ...
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Are axioms tautologies?

My understanding is that axioms are the unprovable statements upon which systems are built. Tautologies are in essence things that can't be false. Godel's Incompleteness Theorem, though, shows that ...
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Existence as a predicate and Godel's ontological argument

I am referring to this paper https://github.com/FormalTheology/GoedelGod/blob/master/GodProof-ND.pdf which has formalized the ontological argument. If I am not mistaken, watered down, the argument ...
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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?

This kind of problem is mentioned in a book I have read, but the book did not give a concrete example. If any such problem existed, this might help me understand human creativity. I think it would ...
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Gödel: Why is a proposition undecidable?

Gödel has proved the existence of undecidable propositions for any system of recursive axioms capable of formalizing arithmetic. But do we know the logical causes of this state of undecidability? In ...