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Questions tagged [incompleteness]

Questions about Gödel's incompleteness theorems and related topics.

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Undecidable statement that doesn't involve infinity

All statements independent of ZFC I'm aware of involve infinity. Do we know any statement independent of ZFC of the form $\forall n \in \mathbb{N}: P(n)$ where $P(n)$ is disprovable if false? (In ...
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1answer
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Ordering between formal theories by provability of consistency

I am studying proof theory with Girard's monograph from '87 ('proof theory and logical complexity'). 1.5.6. is an exercise called 'ordering between theories'. It reads as follows: " (i) Let $\...
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Are the HBL derivability conditions necessary for Gödel's incompleteness theorems? (For Löb's theorem?)

I am currently working with 'proof theory and logical complexity', a monograph on proof theory. In the exercises of the first chapter one was asked to prove Löb's theorem (https://en.wikipedia.org/...
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1answer
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An alternative formulation (or corollary) of Tarski's theorem? [Or just a typo?]

In my proof theory monograph (proof theory and logical complexity, Girard from '87) there is an exercise 1.5.4. on page 78 called 'Tarski's theorem'. It says: "Show that there is no truth predicate ...
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Why the 'natural' consistency proof of PA cannot be carried out $\textbf{in}$ PA

In my proof theory monograph there is this exercise: "The natural proof of PA cannot be carried out in PA. Why? (This proof consists in showing that all theorems of PA are ture.)" Apparently, by '...
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Is every theory meeting Gödels incompleteness, also incomplete below its consistency level?

Is it always the case that for any theory $T$ that meets Godel's criteria for incompleteness, there is a sentence $P$ such that neither $T \vdash P$, nor $T\vdash \neg P$; and such that $T+P$ is equi-...
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Example of incomplete, but decidable theory, and of complete and undecidable theory, question

On wikipedia it is written that Decidability should not be confused with completeness. For example, the theory of algebraically closed fields is decidable but incomplete, whereas the set of all ...
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How can a formal system ever be non-obviously unsound?

When reading about formal systems one is often warned that, even assuming the system is consistent, one can't be sure that any theorem proved in the system is actually true. For instance, if one can ...
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Gödel's completeness theorem and the undecidability of first-order logic

I'm working through this module, "Undecidability of First-Order Logic" and would love to talk about the two exercises given immediately after the statement of Godel's completeness theorem. First, ...
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Recursion Theory/Incompleteness Theorems: Computability of sets of formulas in first order logic

I am struggling with the following two problems: Suppose that $M$ is a structure with finite universe and finite alphabet. Show that the set of formulas $\{\varphi$ $\mid$ for every $M$-assignment $\...
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1answer
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Why is $\forall x (x = x)$ an example of a first order sentence that is neither provable nor disprovable?

In the first answer to this question: What is the difference between syntactic and semantic completeness? the sentence $\forall x (x = x)$ is given as an example of something that is neither ...
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Why can't we prove consistency of ZFC like we can for PA?

this might be a silly question, but I was wondering: PA cannot prove its consistency by the incompleteness theorems, but we can "step outside" and exhibit a model of it, namely $\mathbb{N}$, so we ...
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2answers
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Gödel's incompleteness 1: construction of the formula relating Gödel number of proof to Gödel number of proven statement

I'm trying to understand the proof sketch of Gödel's Incompleteness Theorem 1 on Wikipedia (https://en.wikipedia.org/wiki/Proof_sketch_for_G%C3%B6del%27s_first_incompleteness_theorem) In particular, ...
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Can the Incompleteness Theorems be proven in the context of other logics, for example with Paraconsistent Logic?

And, if they cannot be proven, does anyone take this as a good argument to adopt the alternate logic?
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Reference textbook for Godel's incompleteness theorems proved in PRA or PA metatheory

I think one of my recent misunderstandings and confusions is about how can metatheory be PRA or PA if those are "formal theories". This confusion arose when I started reading Godel's theorem which ...
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1answer
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Why Godel's theorem influences Hilbert's programme?

I have trouble understanding why it is being said that Godel's incompleteness theorem shows that Hilbert's programme is essentially impossible. So, if I understand Godel's second incompleteness ...
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Proving fundamental theorem of arithmetic using primitive recursive functions

I am reading Kleene's "Introduction to Metamathematics" and in chapter 45 he mentions that "a given positive integer can be factored into a product of prime factors which is unique to within the order ...
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1answer
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How can any proof of Gödel's incompleteness theorem be accepted considering systems of mathematics themselves are incomplete?

I believe there are at least several proofs of Gödel's incompleteness theorem. Nagel and Newman wrote a book (1958) that presents one in particular. But considering the theorem itself exposes ...
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1answer
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Is this “Simple Proof of Godel's Theorems” assuming some form of the Axiom of Choice?

I found this paper as a first result in Google after searching for "godel theorem proof". On page 3: Let $B_1(n), \,B_2(n), \,\dots$ be an enumeration of all formulas in $\mathcal{N}$ having ...
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1answer
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Usefulness of Gödel's incompleteness theorem

Given a first-order language $L$ and a theory $T$ in that language (a set of formulas of $L$), if $T$ is strong enough to prove arithmetic, then Gödel's second incompleteness theorem tells us that $T$ ...
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How to prove a generalized version of Gödel's Second Incompleteness Theorem?

Let's start from Gödel's Second Incompleteness Theorem (GST) in the following form: If $\mathcal{T}$ is a consistent, recursively axiomatizable first order theory that contains $\mathsf{PA}$ as a ...
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Are there any simple explicit examples of formal systems which prove their own consistency?

Godel's first incompleteness theorem states roughly that you can't write down a finite list of axioms that can decide all statements about arithmetic: any such formal system is incomplete. I feel like ...
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Does Gödel's incompleteness theorem invoke a Law of Excluded Middle contradiction? [closed]

Does Gödel's incompleteness theorem cause the Law of Non Contradiction to contradict its self? If so, would this be a considered a conjecture?
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George Boolos and Gödel's Second Incompleteness Theorem

In Mind, Vol. 103, January 1994, pp. 1-3, George Boolos wrote: And so on. In fact, if a claim can be proved, then it can be proved that the claim can be proved. And that too can be proved. ...
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Is consistency an axiom of mathematics?

I watched the numberphile video on Gödel's Incompleteness Theorem today, and I was wondering about something. It seems the key to accepting the truth of Gödel's Theorem is to demand that mathematics ...
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1answer
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Is Fermat's last theorem provable in Peano arithmetic?

The sentence $S$ which Gödel in his proof of the incompleteness theorem proves to be be unprovable in the system of Peano arithmetic can be proved (as a true theorem of PA) outside PA (and necessarily ...
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1answer
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I want a simple example of Gödel's incompleteness theorems as it applies to Mathematics [closed]

I want a simple example of Gödel's incompleteness theorems as it applies to Mathematics. I am looking for an example not a proof. If possible a simple example that involves little Mathematical ...
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1answer
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Is there a statement which can not be proved in any axiom systems

As we know a statement may not be proved in some axiom system according to the godel incompleteness theory, can we always solve it by some way that change the axiom system?
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1answer
113 views

Goedel's first incompleteness theorem, the omega rule, and Tennant's reflection rule

Typical discussions of Goedel's first incompleteness theorem note that PA can prove of each integer that it doesn't number the proof of the Goedel sentence $G$. They then note that using an omega rule ...
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Can an error be found in this proof of Gödel's incompleteness theorem?

Can you find a division by zero error in the following short proof of Gödel's incompleteness theorem? First a little background. $\text{G($a$)}$ returns the Gödel number of the formula $a$. As in ...
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Is Godel's incompleteness theorem unavoidable?

So after Godel's Incompleteness theorem and the fact that some theorems mathematicians are interested in are independent of ZFC (e.g. Continuum Hypothesis) is there some hope for some other ...
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What if someone comes up with a proof of consistency of arithmetic under the following conditions?

Suppose that one day, someone comes up with a proof of consistency of Peano arithmetic (PA) within itself. Then, does it mean that PA is actually inconsistent? Because, by the Gödel's incompleteness ...
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Minimal axioms necessary to prove the incompleteness theorems?

What's the minimal sufficient (plausibly) consistent system of axioms to prove the First incompleteness theorem? More interestingly can the First incompleteness theorem be proved in a consistent self-...
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Does it follow from Gödel's theorem that this world cannot be fully described by math?

What are the flaws in the following reasoning? By Gödel's theorem, for any non-contradictory formal system $F$, at least as complex as ordinary arithmetic, there exists a true statement $G(F)$ (or ...
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2answers
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Incomplete theory proving its incompleteness by a formula neither provable true nor false in the theory

Would it be possible that an incomplete theory had a formula that proved its incompleteness, but that same formula belonged to the set of formulae of that theory that can't be proven true or false, so ...
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Do Tarski's Axioms prove all of Euclid's Elements? [duplicate]

A fairly self explanatory title; do Tarski's first order axioms given in his famous 'What is Elementary Geometry?' Suffice to prove all of the theorems in Euclid's Elements? (Excluded non-plane ...
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Understanding variable replacement in Gödel's Incompleteness Theorem

I am a High School student and I am doing a school work on the Fundamentals of Math and in the moment I am reading Gödel’s 1931 article On Formally Undecidable Propositions. I am having a great ...
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1answer
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How do mathematicians know the problems they're trying to solve are not undecidable?

I'm not a mathematician so apologies if this question makes no sense. My understanding of modern mathematics is that it is built on a set of axioms known as Zermelo–Fraenkel set theory + the axiom of ...
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What do the Incompleteness theorems really say about the inexhaustibility of mathematics.

It seems that Godel himself believed that the incompleteness theorems seem to imply the inexhaustibility of mathematics; since he states you can simply add the consistency statement of the system as a ...
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On the clarification of Manin's remark about Gödel’s incompleteness theorems

In his paper Georg Cantor and his heritage Yuri I. Manin writes (see page 7, 3rd paragraph), Baffling discoveries such as Gödel’s incompleteness of arithmetics lose some of their mystery once one ...
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$\Sigma\subset Th(\mathbb{N})$ is finite, so $\Sigma$ has $2^{\aleph_0}$ non-isomorphic models

I want to prove the statement in the title using Godel's theorems. Without Godel's theorem, I may use the theorem about infinite model from cardinality $\kappa$ $\Rightarrow$ infinite models from any ...
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Negative Numbers and Gödel’s Incompleteness Theorem

I was examining Q (Robinson Arithmetic) when it occurred to me that Q contains no statements about the negative numbers or subtraction. No resource that I’ve been able to find has discussed Gödel’s ...
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Do we need to go in meta-theory for proving completeness?

Do I need to go in some meta-theory A for proving theory B is complete ? Or can I do it inside the theory B ? If I really need it, what is the reason for that ?
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True yet unprovable statement?

I have recently been studying Godel's Incompleteness Theorem. I am completely new to the study of logic, so I have been working to break down each component. Assuming I've understood the theorem ...
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1answer
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Can a theory be consistent but not $\omega$ consistent?

Say we have an axiomatizable theory $T$ extending $Th(A_E)$ where $A_E$ are the axioms of arithmetic. Is it possible to extend $T$ such that our extended theory is consistent but not $\omega$-...
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Every Turing machine corresponds to a formal system

Solomon Feferman, at page 138 of his 2006 paper "Are there absolutely unsolvable problems" says that each formal system of axioms can be made to correspond to a suitably designed Turing machine so ...
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Proving the impossibility of a proof

Given that according to Gödel's theorems there are propositions in any language equivalent to first order logic that cannot be proven right or wrong, is it possible to prove that some of such ...
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expression that cannot be “written down” re: incompleteness, logic

I apologize for the poor title. We are given a computer that writes down only expressions that are true. Let $\omega$ be an expression. Define the composition C of $\omega$ as $\omega(\omega)$. We ...
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How to understand Smullyan's notion of weakly adequate Gödel numbering

I was working on Smullyan's Diagonalization and Self-Reference (1994). In Chapter 9 (p.168), he defines the notion of a weakly adequate Gödel numbering as follows: Let $\mathscr{L}$ be a first-...
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Is First Order Logic + Arithmetic semi-decidable?

I understand that it is not complete, but is it decidable, semidecidable or not decidable? Also, does something have to be complete for it to be considered decidable or semidecidable? Meaning, can ...