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Questions tagged [incompleteness]

Questions about Gödel's incompleteness theorems and related topics.

Questions about Gödel's theorems and related topics.

913 questions
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Can someone explain Gödel's incompleteness theorems in layman terms?

Can you explain it in a fathomable way at high school level?
Hyperbola
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What is a simple example of an unprovable statement?

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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Decidability of the Riemann Hypothesis vs. the Goldbach Conjecture

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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Is linear algebra more “fully understood” than other maths disciplines?

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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Why is the Continuum Hypothesis (not) true?

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…
zetavolt
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Understanding Gödel's Incompleteness Theorem

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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I don't understand Gödel's incompleteness theorem anymore

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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What is the difference between Completeness and Soundness in first order logic?

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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True vs. Provable

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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Computability viewpoint of Godel/Rosser's incompleteness theorem

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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Difference between provability and truth of Goodstein's theorem

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…
user522866
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6 answers

Are there more truths than proofs?

Noson Yanofsky is a theoretical computer scientist at Brooklyn College. He presents the following argument on pages 329-330 of his book The Outer Limits of Reason, published by the MIT Press. The set $\mathbb{N}$ of natural numbers has uncountably…
simple jack
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What axioms Gödel is using, if any?

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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Can Peano arithmetic prove the consistency of "baby arithmetic"?

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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Does Gödel's Incompleteness Theorem really say anything about the limitations of theoretical physics?

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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