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Questions tagged [incompleteness]

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Is there any concrete relation between Gödel's incompleteness theorem, the halting problem and universal Turing machines?

I've always thought vaguely that the answer to the above question was affirmative along the following lines. Gödel's incompleteness theorem and the undecidability of the halting problem both being negative results about decidability and established…
Marc van Leeuwen
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Can proof by contradiction work without the law of excluded middle?

I was recently thinking about the validity of proof by contradiction. I’ve read for the past few days things on intuitionistic logic and Godel’s theorems to see if they would provide me answers to my questions. Right now I still have questions…
Charlie Parker
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Why does soundness imply consistency?

I was reading the question Consistency and completeness imply soundness? and the first statement in it says: I understand that soundness implies consistency. Which I was quite puzzled about because I thought soundness was a weaker statement than…
Charlie Parker
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Relation between "undecidability of arithmetic" and "godel's incompleteness theorem"?

There is a theorem that states that arithmetic is undecidable: i.e. $Th(\mathcal N)$, the set of all sentences in the standard arithmetic structure $\mathcal N=(\mathbb N,+,\cdot , 0,1)$ where the symbols are interpreted in the standard way, is…
user56834
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Understanding of Turing's Answer to the Entscheidungsproblem

I apologize if this question has been asked before, but I was not able to find a duplicate. I have just finished reading The Annotated Turing and I am a bit confused. From what I understand, the Entscheidungsproblem is whether or not an algorithm…
andars
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Analogy between Gödel's incompleteness proof and Richard's argument

If we take a look at Gödel's paper “On formally undecidable propositions”, the first self referential proof given in the paper, with the following formula: $$n \in K \equiv \overline{\textit{Bew}}[R(n); n]$$ Which look like this in more modern…
Greg Peckory
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Is there a decidable problem which has a proof that it cannot be proved to have a particular deciding Turing machine?

I came across the question Is there an algorithm that provably exists although we don't know what it is? I was able to follow the example "Given an integer $n\ge0$ is there a run of $n$ or more consecutive 7s in the decimal expansion of $\pi?$"…
advocateofnone
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What are the conditions necessary for a programming language to have no undefined behavior?

For context, yesterday I posted Does the first incompleteness theorem imply that any Turing complete programming language must have undefined behavior?. Part of what prompted me to ask that question in the first place is that, awhile ago, someone…
Mikayla Eckel Cifrese
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Does godel's incompleteness theorem still hold if we have a TM that can do an infinity amount of computations?

I have heard whisperings that if we have a turing machine that is allowed to compute infinitely many steps in finite time, then we can solve the halting problem. This made me wonder, if we have such a TM, do Godel's incompleteness theorems then not…
user56834
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Does Gödel's first incompleteness theorem apply to quantifier-free arithmetics?

Gödel's first incompleteness theorem states roughly that "for any axiomatization of arithmetic, there are statements that can neither be proven to be false nor true." Does this still hold when it comes to quantifier-free statements? I.e. if we have…
user56834
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Gödel's theorem and machines' power

I was studying AI and when a question came to my mind. I know that one of the objections to the possibility of a thinking machine examined by Turing is the so called mathematical objection, highlighting the fact that machines are subject to the same…
Amanda Wealth
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Does the first incompleteness theorem imply that any Turing complete programming language must have undefined behavior?

If I understand correctly, the first incompleteness theorem says that any "effectively axiomatized" formal system which is consistent must contain theorems which are independent of the axioms. In other words, there are models of the system where…
Mikayla Eckel Cifrese
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Did Wheeler really believe that physics was undecidable?

John Archibald Wheeler was a famous physicist. It has been stated that he thought that there was a strong connection between undecidability and quantum physics: This idea was given an early formulation by Wheeler himself: in unpublished notes to a…
jerard
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Self-referential systems cannot predict their future behaviour? (reference request)

Seth Lloyd claims in this video that free will stems from our impossibility of determining how a system capable of self-reference will behave in the future (e.g. whether a human being will have chosen to drink tea or coffee in 5 minutes). He relates…
smalldog
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Why we can't use deduction theorem on soundness to contravene second incompleteness with lob's theorem

I'm starting to learn modal logic and there is something that's bothering my mind for a while. we know from deduction theorem that $((\vdash q) \rightarrow (\vdash p)) \Leftrightarrow(\vdash (q \rightarrow p))$ and also from soundness we know that…
asha soroushpoor
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