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Questions tagged [incompleteness]

Questions about Gödel's incompleteness theorems and related topics.

Questions about Gödel's theorems and related topics.

868 questions
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Consistency hierarchies and Gödel’s theorem

We know that for any theory $T$ that can interpret arithmetic, the recursive theory $T_{bad} = T + \mathop{Con}(T_{bad})$ is inconsistent, by Godel’s second incompleteness theorem. However, $T_{okay} = T + \mathop{Con}(T)$ may be consistent, and is…
Demi
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Assumption of Gödel's Incompleteness Theorem

I am not a mathematician and have not read Godel's Proof. My knowledge about Godel's theorem is limited to this. So my question is this (please correct me if I have misinterpreted or misunderstood the meaning): According to Godel's theorem, every…
Yash Jain
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Godel theorem: confusion on "PM is consistent -> PM's consistency is not PM-provable"

I came to a good summary of Godel's incompleteness theorem proof (http://www.jamieyorkpress.com/wp-content/uploads/2012/04/G%C3%B6dels-proof-summary.pdf) but I'm confused on one step which leads to the conclusion that "PM is consistent -> PM's…
ios learner
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Conculsion of Godel's proof alternative view

I've recently become very interested in Godel, having just read "Godel, Escher, Bach" and subsequently "Godel's Proof". One thing that strikes me as a flaw/alternative explanation in the (albeit non-rigorous) argument: does not the proof just mean…
Matt
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Arithmoquine function in Gödel's proof

Could someone explain as detailed as possible how the Arithmoquine{a,a'} function works or is the defined in Gödel's proof of the incompleteness theorem? To describe my question better... In this video after 9:30 there is a definition of an…
João
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Questions about godel's first incompleteness theorem

I'd rather not get into the formal proof of godel's first incompleteness theorem. But I have 2 general questions. Looking at the statement from…
Ameet Sharma
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Proof of Incompleteness - Existence of Gödel number $g$

I was watching a video: https://youtu.be/HeQX2HjkcNo?t=1171 and had a quesion on the incompleteness proof. In the video, he introduces us to the statement: "There is no proof for the statement with Godel number $g$" Then he says that the trick is…
B2VSi
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Regarding Godel's arithmetic technique in Incompleteness Theorem.

This question is about the number system Godel used in his Incompleteness Theorem. It seems that the result of his theorem is that somethings (in mathematics) may never be provable, although they will still be true. I was listening to a lecture…
Juel Herbranson
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Can the functions in Gödel's Incompleteness proof be expressed as Gödel numbers?

I'm trying to understand the entirety of Gödel's incompleteness theorem, and Gödel's proof. Going by this English translation Gödel defines 45 functions (relations) which build on each other to make the function of provability, there called…
Jonny Hyman
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Existential arithmetical formula independent of ZFC

In Gödel's incompleteness theorem, the Gödel formula is in the language of arithmetic, so adding it as an axiom changes the properties of $\mathbb{N}$. To me that's already difficult to grasp, because the natural numbers seem a primitive and…
V. Semeria
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If a system is complete, must it be inconsistent?

Godel's incompleteness theorem basically says that a set of axioms cannot prove everything. But you can add those unprovable truths to your set of axioms to expand it. Suppose you keep expanding your set of axioms until it contained all truths…
Croolsby
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Are there alternative formalizations of consistency that bypass the Second Incompleteness Theorem?

Gödel's Second Incompleteness Theorem expresses the consistency of a formal system within the system itself using a rather carefully designed proof checking predicate. The conclusion of Gödel's argument is that this formalization of consistency…
Sebastian Oberhoff
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Same number of provable as unprovable statements?

Building on this question: Infinitely many nonequivalent unprovable statements in ZFC because of Gödel's incompleteness theorem? is the infinity of provable statements the same infinity of unprovable statements, or are there more unprovable…
dashnick
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Are there complete formal systems?

I am relatively new to logic, so forgive my ignorance. My question is: Are there any formal systems that are deemed complete? If so, how did we prove they were complete and which are they?
Moises Issi
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Need help understanding Godel's incompleteness Theorem Proof

So I've been trying to understand this theorem for the past week and there is a proof (Not rigorous) that was given to me that I just cannot understand. I'm not gonna lie I'm not so good at mathematics but, I've read it like 50 times+ and it just…
AsianPro
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