# Tagged Questions

1answer
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### Expressibility of Gödel's Incompleteness Theorem

Can Gödel's Incompleteness Theorem be expressed as a formal sentence in ZFC and be proven formally or is it inherently meta-mathematical? (Note: I am referring to the theorem itself, not the ...
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### Can two distinct formulae (or series of formulae) have the same Gödel number?

As I am studying Gödel's incompleteness theorem I am wondering if two distinct formulae or series of formulae can have the same Gödel number? Or the function mapping each formula or series of formulae ...
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### Gödel's proof method and fundamental theorem of arithmetic

I am a novice to Gödel's proof (the theorem that consistency contradicts completeness), and, as it seems to me, he uses the fundamental theorem of arithmetic to uniquely number any formula. My ...
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### Could it be that Goldbach conjecture is undecidable?

The result closest to Goldbach conjecture is Chen's theorem [Sci. Sinica 16 157–176], the proposition 1+2''. It is natural to ask if it is likely that under our arithmetic axioms the Goldbach ...
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### If a statement holds for all standard models of PA, then does it hold for all models?

Suppose that $\varphi$ is a consequence of every standard model of PA. Then is it provable from PA?
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### Gödel's way of teaching non-standard models to Takeuti.

In Memoirs of a Proof Theorist, Gaisi Takeuti relates how Gödel taught him about nonstandard models in an "interesting" way: It went as follows. Let T be a theory with a nonstandard model. By ...
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### particular property and completeness?

I was puzzeling with the almost standard definition of completeness: In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula ...
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### Criticism on truth of Gödel sentence in standard interpretation

Mendelson in his book mentioned the Gödel sentence and argued that in standard interpretation it is true. But Peter Milne in his article (On Godel Sentences and What They Say) criticized: "But we know ...
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### Truth of Godel's sentence in standard interpretation

It is siad that the Godel's sentence: g is true in the standard interpretation. But I have problem in truth of g in the standard interpretation. We proved that if theory K is consistent g is not ...
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### Gödel incompleteness theorem [closed]

Gödel incompleteness theorem states that any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.So what are some Gödel sentences about ...
0answers
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### Undecidable sentence in Godel's incompleteness theorem? [duplicate]

At first I want to apologize to you for repeating my question, since I didnt get satisfying answer. And because answering to this question is very important to me, I have to repost this question. ...
0answers
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### What follows from Incompleteness about provability of partial correctness?

A colleague and I can't figure out what our professor is getting at with this question: What follows from the incompleteness theorems about the provability of partial correctness assertions? What ...
2answers
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### Undecidable sentence in Godel's incompleteness theorem

In Godel's incompleteness theorem, the undecidable sentence is g: I am not provable. Ok. I accepted it and realized that in satandard interpretation it is true. So we found a true sentence which ...
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### Scapegoat theory and PA

A theory T is scapegoat if for every formula A with only one free variable there exist a closed term s such that T proves: (∃x(¬A(x)))⇒¬A(s) I think it is an expectable property for each theory since ...
5answers
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### Understanding ω-consistent and ω-incomplete theory

A theory $K$ is said to be $ω$-consistent if, for every formula $B(x)$ of $K$, if $﹁ B(n)$ is a theorem in $K$ for every natural number $n$, then it is not the case that $(∃x)B(x)$ is a theorem in ...
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### Concrete example for diagonal lemma

Diagonal lemma says that in a theory with enough assumption for any formula $A(x)$ there exist a sentence $B$ such that $B$ $\iff$ $A(\#(B))$ is a theorem in that theory, in which $\#(B)$ represents ...
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### Is Douglas Hofstadter's version of Godel's proof utter nonsense?

Is Douglas Hofstadter's version of Godel's proof, which he offers in his book Godel, Escher, Bach, utter nonsense? Hofstadter goes to great length to disguise the fact that there are two distinct ...
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### If $T$ proves any incorrect $\forall$-rudimentary sentence, then $T$ is inconsistent

A theory $T$ in the language of arithmetic is called $\omega$-inconsistent if for some formula $F(x)$, $\exists x F(x)$ is a theorem of $T$, but so is $\neg F(n)$ for each natural number $n$. ...
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### Gödel's incompleteness theorems: where to learn? Is there a straightforward relation between the two?

What would be a good textbook or paper to learn the proofs of the two Gödel's incompleteness theorems from? I would prefer it to be as close to the original proofs as possible. I have not tried to ...
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### Modern book on Gödel's incompleteness theorems in all technical details

Is there a modern book on Gödel's incompleteness theorems that goes into each and every technical aspect of the proof of them (a classical one, if such exists)? I'm not interested in popular ...
1answer
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### Must every decidable theory be axiomatizable?

Note: By "theory" I mean a set of sentences, not assumed to be closed under logical consequence (otherwise the question would be trivial). Comments/ideas: There's a well-known result that every ...
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### Peano's theory of arithmetic and Gödel's 1st Incompleteness Theorem

Let $\mathcal{N}$ be Peano's 1st order theory of arithmetic and $\mathscr{A}$ it's standard model (which we assume exists). Infer from Gödel's 1st Incompleteness Theorem that there exists a closed ...
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### Problem with completeness theorem and $\mathsf{Con(ZFC)}$

Using the reflection theorem we can prove in $\mathsf{ZFC}$ that for any finite $\Lambda\subseteq\mathsf{ZFC}$ there is some limit ordinal $\gamma$ such that $R(\gamma)\models\Lambda$. Then by the ...
1answer
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### Are forcing techniques possible to automate for mechanized reasoning?

Looking at Cohen's success at proving independence of the Axiom of Choice and the Continuum Hypothesis, I was wondering if it was possible to mechanize forcing techniques for the purpose of proving ...
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### What is the notion of truth used in Godels incompleteness theorem?

First-order logic is complete & sound. The notion of truth used here is model-theoretic. Informally Godels incompleteness theorem says that for a sufficiently strong formal language there are ...