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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Why is Entscheidungsproblem problem undecidable? [duplicate]

I have been trying to wrap my head around Gödel's Completeness Theorem, Incompleteness Theorem(s) and undecidability of Entscheidungsproblem. Turing wrote: It should perhaps be remarked what I shall ...
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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 ...
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Gödel's Incompleteness Theorems and Implications for Science

A few days ago, I heard a biologist mention that one implication from Gödel's Incompleteness Theorems is that an unlimited number of general statements can account for a given set of observations. ...
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What motivated Gödel to arithmetize syntax?

What were the benefits of arithmetizing syntax for Gödel? What did the arithmetization of syntax allow for Gödel that was otherwise not possible?
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Is there a way to avoid Gödel's incompleteness affecting mathematics as a whole?

I have been thinking about Gödel's incompleteness theorems and their ramifications for the whole of mathematics. In this question I assume some fixed formal system F expressive enough for the ...
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Poignancy because of Gödel's theorems - why?

Why do Gödel's incompleteness theorems make mathematicians so sad? There are complete, decidable and consistent fragments of mathematics like the arithmetic of real numbers, complex numbers, ...
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Can Gödel's incompleteness theorems be applied to ethics?

It seems, that simplistically, that Gödel's incompleteness theorems can be applied to ethics in a very straightforward way by: Replacing "True" and "False" with "Right" and "Wrong" Assuming real ...
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Was Kant anticipating Gödel's incompleteness in his antinomies?

Kant's attempts to prove that there's a limit to pure reason based on the existence of antinomies, i.e. pairs of propositions where each one is rational, but the propositions contradict each other. ...
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Did Gödel oppose or agree with the Logical Positivists?

Gödel was a member of the Vienna Circle, whose philosophical position as a group was Logical Positivism, or Logical Empiricism. The SEP article on him states that among his philosophical views were ...
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Quantum Mechanics has formally undecidable problems. What is the philosophical significance of this?

A certain problem is quantum mechanics has been shown to be uncomputable. This means that although it is in a certain sense making a prediction, there is no systematic way to determine what prediction ...
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What are the philosophical consequences of the undecidability of the spectral gap in quantum theory?

An article published in Nature yesterday proves that finding the spectral gap of a material based on a complete quantum level description of the material is undecidable (in the Turing sense). One of ...
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Do Gödel's incompleteness theorems have any consequences for epistemology?

Do Gödel's results have any impact on the theory of knowledge: realism vs anti-realism, the existence of the noumenal, the existence of synthetic a priori truths, etc... or are they just relevant to ...
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Does Gödel's argument that minds are more powerful than computers have the inconsistency loophole?

In "Raatikainen, P., 2005, “On the Philosophical Relevance of Gödel's Incompleteness Theorems,” , the author argues that Penrose's and others use of Gödel's theorem as an argument against mechanism (...
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What sources discuss Russell's response to Gödel's incompleteness theorems?

In his book My Philosophical Development Russell writes, In my introduction to the Tractatus, I suggested that, although in any given language there are things which that language cannot express, ...
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If the ZFC axioms cannot be proven consistent, how can we say for certain that any theorems in mathematics have been proven?

The ZFC axioms are the basis of modern mathematics. But Gödel's 2nd Theorem says that it is impossible to prove that these axioms are consistent. Hence, it is possible (if ZFC is inconsistent) that ...
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Where did Gödel write that first-order logic is the “true” logic?

In "On How Logic Became First-Order" Matti Eklund writes (p. 2/148): It appears to be widely held today that arguments from Skolem and Kurt Gödel, both alleged proponents of the thesis that ...
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Was Wittgenstein anticipating Gödel?

The Tractatus 6.123: 6.123 Clearly the laws of logic cannot in their turn be subject to laws of logic. (There is not, as Russell thought, a special law of contradiction for each 'type'; one law is ...
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Are there any work around after Godel's incompleteness theorems?

It seems to me that mathematicians or computer science theorists are still trying to come up a proof for conjecture they made in first or second order logic. By Godel's incompleteness theorems there ...
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How did the logical positivists respond to Gödel's incompleteness theorem?

In a lecture on philosophy of science I recently listened to, it was stated that Quine was the one who decisively refuted the logical positivist program. I've also read that Quine and Popper were ...
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Is this enough to conclude that G is false?

Let us assume a logical system with only 3 axioms (laws of thought): Axiom 1: A statement that is true will remain true till a change is made in the system. Axiom 2: A statement may not be true and ...
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How does Gödel's theorem apply to daily life?

I came across a simplified description of Gödel's theorem and the discussion touches on a concept of honesty (truth?) and completeness. How does Gödel's theorem apply to everyday interactions?