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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 ...

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Is Propositional Logic Sound and Complete

I hope this is not a silly question, and I am not a logician by any means; however, after reading Godel's incompleteness theorem, I wonder if there are any systems who's axioms are sound and complete. ...
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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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191 views

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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97 views

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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1answer
98 views

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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1answer
193 views

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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429 views

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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1answer
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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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3answers
672 views

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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1answer
273 views

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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377 views

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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170 views

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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113 views

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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4answers
643 views

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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2answers
102 views

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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4answers
634 views

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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733 views

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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2answers
270 views

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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212 views

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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572 views

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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Godel's Incompleteness Theorem [duplicate]

To What Extent Can Gödel's Incompleteness Theorem Be Applied To Real Life Explanations And Proof Of The Existence Of Things?
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5answers
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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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677 views

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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814 views

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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3answers
1k views

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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1answer
87 views

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?
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About Godel and Anselm

Was Godel's work on trying to make Anselm's Ontological Argument more 'feasible' with modal logic successful or has this work just been lost in the many abstract debates that confuse the issues?
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Regarding Godel's theorems

Given a formal logic system W with it's set of axioms, if it is capable of 'handling' basic arithmetic then W can not be used to 'prove' its own consistency. In other words there will always be a ...
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238 views

Provable and contradictory?

For each consistent formal theory T having the required small amount of number theory, the corresponding Gödel sentence G asserts: "G cannot be proved within the theory T". This interpretation of G ...
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776 views

Does Gödel's second incompleteness theorem interact with logical positivism?

Gödel's second incompleteness theorem (GSIT), informally stated, says: For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal ...
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Gödel's ontological proof and the incompleteness theorem

In Gödel's ontological proof, he concludes: "necessarily, God exists" (Theorem 4) Does Gödel's second incompleteness theorem apply to this proof?
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Does the finitary proof of the consistency of relevant PA shows that first order PA is irrelevant?

Relevance logic takes a closer look at the implication operation in first-order logic. It suggests that implications such as: p and not p -> q cannot hold; in ordinary English, an example of this ...
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Can we expand the notion of truth in Godels incompleteness theorem?

Godels incompleteness theorem, which really should be called the undecidability theorem given that the paper of Godels which this theorem is taken from is named 'On formally undecidable propositions ...
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417 views

What is the connection between provability logic & Gödel's first incompleteness theorem?

Provability logic is a modal logic that interprets the modal operator of K as provability and an additional axiom derived from Löb's theorem. Now the SEP shows that it's possible to derive Gödel's ...
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1answer
198 views

Is there a finitistic set theory, and if there is, is it provably consistent?

ZFC is the mainstream set theory. It has an axiom of infinity which claims that there is at least one infinite set. Now suppose like Aristotle, we object and say that there are no actually infinites ...
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588 views

What is the impact of paraconsistency on Gödel's theorem?

Russell's paradox forced a restriction of the natural abstraction principle (that every predicate determines a set) so that set theory could be consistent; the standard one being ZF. However ...
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556 views

Is Gödel's incompleteness theorem still valid if one uses a higher-order logic?

Gödel's incompleteness theorem is wholly formal (in my understanding), and relies on a proof system that I assume is first-order. Does it make any difference to the theorem if higher-order logic is ...
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1answer
429 views

In Gödels Incompleteness theorem what is the notion of truth?

The entry on Gödels Incompletenss theorem in Wikipedia says: Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for ...
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1answer
245 views

What happens when we drop the condition that proofs are finite strings of inferences in Gödel's incompleteness theorem?

Gödel's incompleteness theorem shows that there are sentences that are undecideable, that is they nor their negation can be proved. This theorem operates purely syntactically or formally, that it ...
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How is Gödel's incompleteness theorem interpreted in intuitionistic logic?

Classically, one sets up an axiomatic system with a formal deduction system & an interpretation in a model. Generally it is sound, that is: a formally deduced theorem is also true when interpreted ...
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Can we add to PA a new predicate T such that for every sentence A of the old vocabulary the new theory proves T(Godel numeric number of A) iff A

I am new to logic but I believe this is not a difficult problem, yet I am still soo confused, and the reason for that is because there are so many gaps in my knowledge or maybe I have overlooked so ...
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Gödel's theorem and God

I have seen it argued that Gödel's Incompleteness Theorems have implications regarding the existence of God. Arguments for the existence of God run mostly along the lines: "Because of Gödel's Theorem, ...
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Was Gödel the first person to bring up that truth always exceeds the grasp of proof?

Was Gödel the first person to pose and solve this question in mathematics? In the larger philosophical debate, has this question been posed before? Say by Plato or Aristotle? One could interpret for ...