Questions tagged [incompleteness]

Questions about Gödel's incompleteness theorems and related topics.

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Relation between Halting problem and Godel's incompleteness Theorem

I've heard that these are closely related to each other. However, while the former looks totally trivial to me, the latter still puzzles me even after I read the proof of it. So I made up a scenario ...
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Proving $PAE ⊢ (Pr_S(\#(X → Y)) → (Pr_S(\#(X)) → Pr_S(\#(Y))))$, where $Pr_S(n)$ holds iff $n$ is the Gödel number of a formula provable from $S$

I'm trying to solve the following question set by my professor: Show that if $S$ is a definable set of sentences, and $Pr_S$ is an associated proof predicate, and $X$ and $Y$ are any formulae, then $...
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Proving a result similar to the diagonal lemma/fixed-point theorem

As the title explains, I'm trying to solve the following exercise that was left for the reader in my lecture notes: Show that for any two formulae $F(v_1)$ and $G(v_1)$ in $L_E$ (language of ...
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Prove that ever $\Sigma_0$ sentence is provable from PA (Peano Arithmetic, with exponentiation)

I am trying to study the intuition and steps leading up to proving Godel's Incompleteness theorems. In a text I am studying it asserts that every true $\Sigma_0$ sentence is provable from PAE (where ...
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Is there any way to avoid Gödel incompleteness theorem?

I was wondering whether there have been successful attempts to avoid somehow the incompleteness theorem. Many point out that every language that is powerful enough to model natural numbers can produce ...
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26 views

Showing that formulae which are provably $Σ_n$ and $Π_n$ ($n > 0$) in the arithmetical hierarchy are closed under $∃$ and $∀$ respectively

As the title explains, I'm trying to solve the following problem: Show that for $n > 0$, formulae provably $Σ_n$ with respect to PA are closed under existential quantification, and formulae ...
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36 views

Can a finite Turing machine model itself?

Given the following constraints, can a finite Turing machine (FTM) model itself? An FTM receives only data initially stored on its tape; The amount of initial data on the tape is limited to $D_{m}$ ...
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On the Possibility of Godel Numbering

I've gotten to the point of the book Godel, Escher, Bach which really gets into demonstrating the mechanisms of Godel's incompleteness proof. I've had to go back ...
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Axioms which are used to prove Gödel's second incompleteness theorem [duplicate]

Gödel's second incompleteness theorem concerns the question of proving consistency of certain axiomatic systems. But what set of axioms is used to prove Gödel's second incompleteness theorem itself? ...
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Defining a consistent complete system in terms of decidable statements.

Say we take a consistent but incomplete formal system such as ZFC, which has theorems (provably true statements) and negated-theorems (provably false statements). I want to define a system, T, the ...
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Can the existence of some positive integers satisfying some equations ever be independent of ZFC?

In a subject like number theory there are often open questions along the lines of 'Are there positive integers $a_1 , \ldots , a_n $ satisfying the following (finite number of) equations'. For example ...
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Why doesn't the Continuum Hypothesis show that $\text{ZFC}$ is simply consistent? [duplicate]

Where is my error? A system is simply inconsistent is every formula is provable (if $\text{A} \wedge\neg \text{A}$ were provable for some formula $\text{A}$, then by $\neg$-introduction, any formula ...
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34 views

How does Ramsey Theory relate to Godel Incompleteness theorem?

My knowledge in math is not too advanced but I am super interested in this topic and although I think I know part of the answer, I do not know for sure how these two theories are related. Also, if ...
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In what formal system is Godel's Incompleteness Theorem (or similar statements of undecidability) proven?

For example, consider the proof using Rosser's trick as shown on wikipedia. https://en.wikipedia.org/wiki/Rosser%27s_trick#The_Rosser_sentence That proof isn't inside the arithmetical theory T, but in ...
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Existence of proof of consistency of stronger axiomatic systems

Assume $\mathsf{ZFC}$ is consistent. Are there axioms $\mathsf{P}$ and $\mathsf{Q}$ independent from $\mathsf{ZFC}$ such that?: $\mathsf{Q}$ is a stronger axiom than $\mathsf{P}$ There is a proof ...
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Peano Arithmetic - Decidability

I need to show that if Peano Arithmetic does not decide a sentence $\varphi$ then the standard model of Peano Arithmetic satisfies the negation of $\varphi$. I know this partly has to do with Godel's ...
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Godel Escher Bach: Why is a Godel proof-pair representable in TNT?

On page 441, fundamental fact 2 asserts that: The property of forming a proof-pair is testable in BlooP, and consequently, it is represented in TNT by some formula having two free variables. Why is ...
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1answer
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Question about a proof of Gödel's 1st incompleteness theorem

I am confused about one of the standard proofs of this theorem that I've seen. I believe it's a small thing that I missing as authors seem to skip straight through it. The proof I am referring to is ...
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Could mathematics be based on CWA?

https://en.wikipedia.org/wiki/Closed-world_assumption Would proposition with free variables still be writable in such a system ? If not, would it be a problem since theorems are usually closed ...
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Does Godel 1st theorem make sense?

It seems to me that there is 2 family of statements that I call "logically undecidable" (to distinguish with computationally undecidable which is define by turing machines), i.e. of ...
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Proving a theorem in GL modal logic

I'm trying to prove a number of theorems in the system GL using the basic axioms of system K and the following axiom schema from GL: $\vdash_G \Box A \rightarrow \Box\Box A$ for every A. One of ...
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Can't understand definition of system $P$ given by Kurt Godel in his “On formally undecidable propositions…”

I am reading On formally undecidable propositions of Principia Mathematica and related systems by Kurt Godel. And I am not quite sure I am getting his definitions right. See the screenshot beneath: .....
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Incompleteness Theorems for Limit-Computable Formal Systems

Godel’s First and Second Inconpleteness Theorems are about Peano Arithmetic, but their punchlines are respectively that “Any sufficiently expressive computable formal system cannot be complete and ...
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Boolos’s proof of the first incompleteness theorem. Predicate $C(x,y)$ and assumption of completeness.

I am trying really hard to understand how Boolos’s proof works, but I keep having doubts about it. Can you spot any logical flaws in my reasoning? This question arose from a previous question of mine, ...
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Why does math work if there are too many paradoxes?

I'm a newbie that studies applied maths and I have been learning about measure theory for the past few weeks and I came across things like Banach–Tarski paradox, Gödel's incompleteness theorems, axiom ...
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Avoiding self reference and proving incompleteness via elementary means.

I am in process of reading Logic and Godel incomplete proof (not completed) when the following two thought pop up in my head. I think I can show incompleteness through elementary reasoning A Proof ...
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286 views

Boolos’s proof of Gödel’s first incompleteness theorem. What am I getting wrong?

Sorry in advance, the following is quite messy, I didn’t find a way to express myself more clearly and rigorously. I would appreciate suggestions on how to make the following better and try to act ...
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Gödel's first incompleteness theorem. What did I get wrong? [closed]

I'd like to point out that obviously I don't claim the following to be right, and I now recognise that the way my question was phrased before editing could have been interpreted as very arrogant, and ...
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How can I prove that primitive recursion “preserves” representability in Peano Arithmetic?

I'm working on my thesis about Gödel's Incompleteness Theorems, and at some point I need to prove that the $\textsf{PA}$ system is able to represent all the recursive functions. By recursive function ...
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Constructive nature of Gödel's Incompleteness Theorem

Let $E_n$ be an expression with Godel number $n\in\mathbb{N}$. Then we denote the Gödel number of $E_n(n)$ as $d(n)$. Now for a set of natural numbers $A$, we define a set $A^{\ast}$ as follows: $$\...
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Proving a correct system satisfying certain conditions is incomplete

I was reading R. Smullyan's book, Gödel's Incompleteness Theorems and got stuck at an exercise: In the following problem $P$ denotes the set of Gödel numbers of provable statements in system $\mathcal{...
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How do “we” know the incompleteness of second-order logic?

The incompleteness of first-order arithmetic is relatively easy to wrap your head around -- there are non-standard models of PA in which Godel's sentence has a non-standard Godel number and so is &...
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How are the Hilbert-Bernays-Löb derivability conditions checked for the modus ponens condition?

To recall, the Hilbert-Bernays-Löb derivability conditions state that $\operatorname{Bew}$ is a derivability predicate if for any pair of sentences $S, T$: If $S$ is provable, so is $\operatorname{...
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How to universalize $\text{Prov}(\ulcorner y < K(x)\urcorner) \to y < K(x)$ in a paper of Kikuchi

In Kikuchi's paper Kolmogorov complexity and the second incompleteness theorem he defines for $\Sigma_1$ binary predicates $R(x, y)$ the condition $$ \Gamma_{1}(R) \Leftrightarrow \forall x\forall y(R(...
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The philosophical significance of Chaitin's Theorem

In a book review of Torkel Franzén's "Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse" in the Notices the reviewer (Raatikainen) writes: Franzén also devotes a brief chapter to ...
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Gödel: If a statement of existence is neither provable nor refutable, doesn‘t that imply that the statement is false? [duplicate]

I‘m not very versed in Gödel‘s incompleteness theorem but in a naive way: If a statement of existence is not provable, there you cannot find an example which fulfills the statement (otherwise the ...
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Does Gödel’s theorem preclude the existence of all means of proving all statements in a formal system, or preclude only computable means?

It is necessarily the case that you can eventually generate all possible finite proofs on a UTM. It also necessarily the case that you can eventually generate all possible theorems on a UTM. This ...
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Construct Gödel sentence with diagonalisation function?

In How is a Godel sentence constructed? and here, we construct a Gödel sentence $G$ by using the diagonalisation lemma to find a sentence $G$ such that $$G \leftrightarrow \lnot\mathop{Prov}(\...
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Proof of Separation Lemma by Raymond Smullyan in Gödel's Incompleteness Theorems

In Gödel's Incompleteness Theorems book, Smullyan presents a proof of the Separation Lemma for sets. Let be S a system, where all formulas of $\Omega_4$ and $\Omega_5$ (axiom schemes of (R)) are ...
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What subsystem of second-order arithmetic proves the weak Godel’s Theorem?

Godel’s Incompleteness Theorem states that any omega-consistent recursive theory in the language of first-order arithmetic is incomplete. But there is a weaker version of Godel’s Theorem that is ...
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The arithmetized completeness theorem

In Kikuchi's paper Kolmogorov complexity and the second incompleteness theorem he states the "arithmetized completeness theorem" as follows: Let $T$ be a recursively axiomatizable theory in ...
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Can inconsistent systems be mathematically interesting/useful?

According to the top answer to this question: Doing mathematics we often have an idea of an object that we wish to represent formally, this is a notion. We then write axioms to describe this notion ...
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Is there any metric on R with which it is incomplete.

I read the definition of complete metric space. A metric space (X,d) is complete if every cauchy sequence in X converges in X. So, from this definition it looks like if a metric space is not complete ...
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Does the independence of the axiom of choice imply Gödel's incompleteness theorem?

I recently wrote this answer describing Gödel's completeness and incompleteness theorems, in which I came to the conclusion that a theory is (syntactically) complete if and only if all its models are ...
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The set of Godel numbers of axioms of Robinson arithmetic is recursive

Prove that the set of Godel numbers of axioms of Robinson arithmetic is recursive. I'm studying Godel incompleteness theorems and I want to prove The set of Godel numbers of axioms of Robinson ...
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Recursive essentially for Godel incompleteness theorem

I'm reading Godel incompleteness theorem from A mathematical introduction to logic by Enderton. GÖDEL INCOMPLETENESS THEOREM (1931) If A ⊆ Th N and #A is recursive( the set of Godel number of ...
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What is it about natural numbers that makes them a prerequisite for Godel's Incompleteness Theorem to hold?

I came across an old question on here asking about Gödel's Incompleteness Theorem and the theory of Real Closed Fields, specifically about why the former doesn't apply to the latter. The top answer ...
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Proof of the theorem “Any recursive relation is representable in Cn A$_E$.”

I'm reading Godel incompleteness theorem from Mathematical introduction to logic by Enderton. There is something about recursive functions I don't get. In the text it first says: "DEFINITION. A ...
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Does it make sense to define the length of a line segment in terms of addition of infinite points?

Since a point has zero length, how can a line segment of, say, 1-unit length—which is a collection (addition) of infinite points, that is $0 + 0 + \cdots$—have 1-unit length? Does it make sense to say ...
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How important is first-order logic to Gödel's incompleteness theorem?

I'm trying to understand Gödel's incompleteness theorem, and I'm not sure if the theorem only applies to cases where the syntactic logic is first-order logic, or more generally. On one hand, I've seen ...

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