# 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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### Can Internal Set Theory provide a complete system of arithmetic?

Internal Set Theory (IST) is a conservative extension of ZFC that adds three axioms that serve to define a predicate standard such that all numbers are either standard or not. There are finitely many ...
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### Is this a paradox or a mistake? [closed]

Zhang Hong recently asked "is there a paradox lurking in Godel's 1931 incompleteness proof" (paraphrase)? This can be answered in two ways: First, by proving quite generally that there are ...
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### Is there a paradox in the proof of Godel's incompleteness theorem?

The Gödel Incompleteness Theorem was a major discovery in modern logic that has consistently attracted the attention of scientific and philosophical circles. However, since the Gödel Incompleteness ...
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### Was Gödel actually convinced that his ontological proof was correct?

The proof is obviously logically valid, but it is as obvious that it isn't logically sound. For instance, the second axiom states that ¬P(φ) ⟺ P(¬φ), take φ(x) ⟺ x is a male human being. Then either ...
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### Completeness in finished system

Gödel's incompleteness theorems addresse formal axiomatic théories. Incompleteness of arithmetic of natural numbers is an example. My question is if a theory regarding a finite class of numbers cannot ...
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### How does Gödel’s encoding of mathematical statements into natural numbers enable self-referential propositions?

As part of his proof for the first incompleteness theorem, Gödel encoded mathematical expressions into unique numbers. These were used to construct statements exhibiting self-referentiality, such as ...
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### Is it a problem for arithmetic or our representation (or both) that there is incompleteness?

Is this a settled (as much as it can be) philosophical area? I feel like I understand that there will always be incompleteness for a finite set of axioms trying to capture all of arithmetic. But I ...
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### Do Gödel sentences (even indirectly) assert only their own unprovability?

Sometimes the basic Gödel sentence is said to mean something like, "This sentence is unprovable in system F." Perhaps more correctly, it is sometimes said to mean something like, "There ...
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### Do the incompleteness theorems need the provability predicate to be expressed, or can they be expressed via just ⊢?

In his "Epistemic Set Theory," William Reinhardt says: It is the purpose of this paper to formulate axioms for Gödel's modal operator B for provability (see [3], [8]) in the context of set ...
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### Is Gödel's incompleteness theorem based on premisses bearing a contradiction? [closed]

In a 7-page article, Chaim Perelman provides an argument supporting the idea that Gödel's premises for the incompleteness theorem bear a contradiction. Has this ever been refuted? If yes, how? Does ...
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Here is my argument: One of the incompleteness theorems is “If a system is noncontradiction, it is incomplete” Incomplete means that there are propositions that are true but cannot be proven. The ...
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### Can you mathematically prove the existence of God?

So I came across this video (https://www.youtube.com/watch?v=z0hxb5UVaNE), which claims to prove the existence of God using math. I then searched and found stuff like this: mathematician Kurt Gödel's ...
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### A technical question about the limitation of z of "jointing together" or "zus(x,y)" in Gödel Arithmetization

I am recently reading Professor Carnap's Logical Syntax of Language. In p.61 D18.1., the limitation of z is not greater than: pot [prim (sum[lng(x), lng(y)]), sum(x,y)]. Remarks: z is the series-...
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### Prerequistes for mathematical logic

I have a working knowledge of calculus and linear algebra. But when I pick up books on mathematical logic (for example the ones listed in the logic study guide by Peter Smith), they often use ...
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### Kant's commentary on the faculty of judgment: did he anticipate things like incompleteness/halting/truth-undefinability?

First, to cite the (Meiklejohn) version of the argument: If understanding in general be defined as the faculty of laws or rules, the faculty of judgement may be termed the faculty of subsumption ...
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### Could God be a Gödelian sentence?

first of all I must specify that I am a mathematician so I hope I am using the appropriate words to formulate my question. Recently, while talking with some colleagues, a question arose; Based on the ...
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### How is the completeness of first order logic reconciled with the incompleteness of set theory?

First Order Logic (FOL) is complete in the sense that: there is a proof procedure for FOL such that just the statements(/wffs) of FOL that are true and remain true under any re-interpretation of their ...
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### In line with Roger's Penrose argumentation, why is human mind not computable, when large language models are?

Roger Penrose famously from Gödel's Incompleteness Theorem, that human mind is not computable, because mathematical intuition is not computable (a mathematician can prove more than any formal system, ...
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### Can, "This problem is unsolvable," be used to formulate the first incompleteness theorem in erotetic logic specifically?

Assumptions/definitions: the Gödel sentence is informally equivalent to, "This sentence can't be proved in system X," where X is appropriately specified. Since that sentence can itself be ...
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### Can humans symbolically manipulate X that describes themselves?

Consider the following premise: Any statement regarding the physical world can be proven within the system of X (assume X to be something like Quantum Field Theory) by humans. One may argue that this ...
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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 Gödel 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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