Questions about Gödel's incompleteness theorems and related topics.

Questions about Gödel's theorems and related topics.

Questions about Gödel's incompleteness theorems and related topics.

Questions about Gödel's theorems and related topics.

775 questions

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Can you explain it in a fathomable way at high school level?

Hyperbola

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In the most recent numberphile video, Marcus du Sautoy claims that a proof for the Riemann hypothesis must exist (starts at the 12 minute mark). His reasoning goes as follows:
If the hypothesis is undecidable, there is no proof it is false.
If we…

Sriotchilism O'Zaic

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Most of the systems mathematicians are interested in are consistent, which means, by Gödel's incompleteness theorems, that there must be unprovable statements.
I've seen a simple natural language statement here and elsewhere that's supposed to…

Michael Harris

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In a recent question, it was discussed how LA is a foundation to other branches of mathematics, be they pure or applied. One answer argued that linear problems are fully understood, and hence a natural target to reduce pretty much anything to.
Now,…

leftaroundabout

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I'm making my way through Thomas W Hungerfords's seminal text "Abstract Algebra 2nd Edition w/ Sets, Logics and Categories" where he makes the statement that the Continuum Hypothesis (There does not exist a set with a cardinality less than the reals…

ŹV -

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I am trying very hard to understand Gödel's Incompleteness Theorem. I am really interested in what it says about axiomatic languages, but I have some questions:
Gödel's theorem is proved based on Arithmetic and its four operators: is all…

sova

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Here's the picture I have in my head of Model Theory:
a theory is an axiomatic system, so it allows proving some statements that apply to all models consistent with the theory
a model is a particular -- consistent! -- function that assigns every…

Abhimanyu Pallavi Sudhir

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Gödel's first incompleteness theorem states that "...For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system".
What does it mean that a statement is true if it's not…

TROLLHUNTER

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Completeness is defined as if given $\Sigma\models\Phi$ then $\Sigma\vdash\Phi$.
Meaning if for every truth placement $Z$ in $\Sigma$ we would get $T$, then $\Phi$ also would get $T$. If the previous does indeed exists, then we can prove $\Phi$…

JAN

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I have been thinking about the difference between provability and truth and think this example can illustrate what I have been wondering about:
We know that Goodstein's theorem (G) is unprovable in Peano arithmetic (PA), yet true in certain extended…

Sam Mayo

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Gödel states and proves his celebrated Incompleteness Theorem (which is a statement about all axiom systems). What is his own axiom system of choice? ZF, ZFC, Peano or what? He surely needs one, doesn't he?

Dave

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I am reading Peter Smith's An Introduction to Gödel's Theorems. In chapter 10, he defines "baby arithmetic" $\mathsf{BA}$ to be the zeroth-order version of Peano arithmetic ($\mathsf{PA}$) without induction. That is, $\mathsf{BA}$ is the…

WillG

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How would the Godel/Rosser incompleteness theorems look like from a computability viewpoint?
Often people present the incompleteness theorems as concerning arithmetic, but some people such as Scott Aaronson have expressed the opinion that the…

user21820

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Stephen Hawking believes that Gödel's Incompleteness Theorem makes the search for a 'Theory of Everything' impossible. He reasons that because there exist mathematical results that cannot be proven, there exist physical results that cannot be proven…

Haskell Curry

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I am very interested in learning the incompleteness theorem and its proof.
But first I must know what things I need to learn first.
My current knowledge consists of basic high school education and the foundations of linear algebra and calculus which…

Starless

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