# Tagged Questions

Questions about Gödel's incompleteness theorems and related topics.

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### Why a Riemannian manifold minus one point is not complete? [closed]

Could you give me a proof that a Riemannian manifold minus one point is ever complete? Thanks!!
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### Gödel numbering for sequences Using the Chinese remainder theorem [closed]

I looked into some youtube videos and got a simple idea about Gödel numbering and Chinese remainder theorem separately.... But can't see how to use them as one. Wikipedia giving a way or may be its an ...
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### How can logic talk about itself? [closed]

How can there exist theorems like Goedel's Completeness theorem or Incompleteness theorem? They all make some statements about logical theories, but don't we need a certain logical scheme first to be ...
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### Isn't it problematic to cite the Gödel sentence as a proposition asserting 'This sentence is unprovable' since it isn't really on point?

In the proof of Gödel's incompleteness theorem the Diagonalization Lemma is applied to the negated provability predicate $¬Prov_F(x)$: this gives a sentence $G_F$ such that $F ⊢ G_F ↔ ¬Prov_F(⌈G_F⌉)$...
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### What is the significance of Gödel's incompleteness theorems? [duplicate]

I have been trying to wrap my head around this for a couple of months now, it seems to me that a short informal stating would be this: You have the first incompleteness theorem, which proves that ...
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### Can one have a theory that includes its own consistency as an axiom?

Consider the theory with the following axioms: The axioms of ZFC The "axiom of consistency": "This theory, including this axiom and all of the theory's other axioms, is consistent." Phrased ...
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### I have read that complex axiomatics are Gödel complete, while naturals aren't. Why?

I have read in a book:(G. Martínez, G. Piñieiro: "Gödel para todos") that complex axiomatics are Gödel complete, while naturals aren't. How can this be if Naturals are a subset of Complexes, (or at ...
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### What would be the consequences if ZFC proved its own inconsistency, nonconstructively? [closed]

Let's say a nonconstructive proof was given in ZFC that ZFC was inconsistent. Note that this doesn't automatically make ZFC inconsistent. Given a consistent theory $X$, $X + \neg \text{Con}(X)$ is ...
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### Why is the proof of Gödel's first incompleteness theorem no contradiction?

I consider the following version of Gödel's first incompleteness theorem: Assume $F$ is a formalized system which contains Robinson arithmetic $Q$. Then a sentence $G_F$ of the language of $F$ can ...
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### Are Godel's incompleteness theorems proven non-trivial?

Godel's incompleteness theorem states there will be unprovable statements in some language. Can it be proven that the unprovable statements in some language $F$ are necessarily not just "trivially ...
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### Showing a metric space is not complete.

Consider the metric space $$B = \{ f \in C[0,1] : \int_a^b \left| f(x) \right| dx \leq 1\},$$ where $d(f,g) = \int_0^1 \left| f(x) - g(x) \right|dx$. I'm trying to show that this metric space is not ...
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### Is ZFC ω-consistent over ZF?

Gödel proved that if ZF is consistent, then ZFC is also consistent. He did this by showing that his constructible universe is a model of ZFC. Gödel also introduced the notion of an $\omega$-consistent ...
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### Is there a consistent arithmetically definable extension of PA that proves its own consistency?

Gödel's second incompleteness applies, for instance, to r.e. extensions of PA. I am wondering if it applies more generally to arithmetically definable extensions of PA. I see that there is a complete ...
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### Gödel's Incompleteness Theorem and Numerical Analysis

I am no expert in Mathematical Logic so I can't really express my question formally but I do hope that it will make sense. As far as I know the implication of Gödel's incompleteness theorem for ...
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### Does the existence of uncomputable functions imply that a theory is incomplete?

For example Kolmogorov complexity is uncomputable and Chaitin used that fact to prove incompleteness. If this is not the case, can you give me a counter example? Set of axioms is countable.