Questions about Gödel's incompleteness theorems and related topics.

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How to show that F is recursive?

In the road to Godel's incompleteness theorem, Problem saying: Show that Frame For $G:\omega\times\omega\times\omega^{n}\rightarrow\omega$ a recursive function, the function $F:\omega\times\omega^{...
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1answer
49 views

Can any statement about natural numbers be written in TNT?

I was trying to understand Godel's theorem from here Godel's First Incompleteness theorem. I still do not completely understand the theorem, but have a broad idea about it. As the link mentions Godel'...
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1answer
31 views

What is the diagonalization of ‘∀x¬Gdl(x,y)'?

Correct me if I am wrong here.‘∀x¬Gdl(x,y)' simply states that There does not exist godel number for a given number y, right? So if we say that there exist a diagonalization of ‘∀x¬Gdl(x,y)', then we ...
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1answer
104 views

How do inconsistent arithmetics get around Gödel incompleteness?

Gödel theorems imply that arithmetic can not be complete under recursive axiomatization and consistency assumptions. Unexpectedly, consistency can be dropped in a meaningful way by switching to a ...
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1answer
110 views

Proving that the existence of strongly inaccessible cardinals is independent from ZFC?

ZFC can't prove that strongly inaccessible cardinals exist, or else it would prove that a model of itself exists and hence $Con(ZFC)$. So this leaves us with two options: ZFC proves there are no ...
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Vaught's essentially undecidable set theory

Today I was reading this paper, which includes discussion of essential undecidability of various weak theories. On page 24, I was surprised to find out that Vaught has showed that set theory with the ...
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1answer
79 views

Concerning the canonical example for Gödel's first incompleteness theorem

Concerning the canonical example for Gödel's first incompleteness theorem: G cannot be proved within the theory T If G were provable under the axioms and rules of inference of T, then T would ...
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2answers
63 views

Gödel's theorems and nonstandard model of $PA$

According to the Second incompleteness theorem $Con(PA)$ is independent of $PA$. So if $PA$ is consistent $PA + \neg Con(PA)$ is also consistent which means that there exist a number $t$ which codes a ...
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1answer
91 views

understanding gödel's 1931 paper - elementary formulae

I am trying to understand Gödel's first incompleteness theorem from his original 1931 paper. Here is a translation i am using for my studies : http://www.research.ibm.com/people/h/hirzel/papers/...
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1answer
70 views

understanding gödel's 1931 paper - proof of theorem / proposition V 5

I am trying to fully understand gödel's proof of the first incompleteness theorem from it's original 1931 paper. Here is the document I am using : http://www.research.ibm.com/people/h/hirzel/papers/...
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0answers
39 views

Standard model of the Meta theory in the proof of arithmetic incompleteness

I have a question about the incompleteness of arithmetic. I wonder in which meta theory we are reasoning. For example, if we are in ZFC, then there are some non standard models of ZFC. As far as I ...
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Completeness of the space $C_{2\pi} \left ( \mathbb{R} \right )$ [duplicate]

Denote $C_{2\pi} \left ( \mathbb{R} \right )$ is the set of complex-value, continuous on $\mathbb{R}$, periodic functions with period $2\pi$. Fix $p \in \left [1, \infty \right )$. For each $x \in C_{...
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1answer
146 views

understanding gödel's 1931 paper - the undecidability theorem

I am reading a translation of Gödel's original paper about in completeness theorem and there are a couple things i don't understand. Here is the document i am using primarily : http://www.research....
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1answer
69 views

Infinitely many nonequivalent unprovable statements in ZFC because of Gödel's incompleteness theorem?

Am I right in thinking that there are countably infinitely many nonequivalent unprovable statements in ZFC due to the Gödel's first incompleteness theorem? If not, why?
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2answers
138 views

Proportion of true statements that are provable

In an axiomatic system, there are true statements, some but not all of which are provable. Is there any sense in which we can quantify the proportion of those statements that are true? Here's an ...
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3answers
186 views

[Paradox]How can Godel prove that Godel sentence is unprovable but true, if such proof itself proves that Godel sentence is true?

Isn't the proof that Godel sentence is unprovable but true a proof itself that Godel sentence is true? Godel in the preface of his proof remarked: “From the remark that [the unprovable statement] ...
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2answers
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understanding gödel's 1931 paper - primitive recursive functions - projection and equality

I am reading a translation of gödel's original, and i'm a bit confused about the primitive recursive functions. Everywhere on the internet (or some resources like courses/classes i could put my hands ...
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2answers
126 views

understanding gödel's 1931 paper - gödel numbers

I am a little confused about gödel numbers and what numbers exactly we are manipulating. are the numbers "real" natural numbers (than we obviously represent as) 1, 2, ... or are we always dealing ...
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1answer
42 views

Godel's Incompleteness Theorem and Algorithms [closed]

According to Godel's incompleteness theorem, not every problem can be solved using algorithms. How do we know if a problem can be solved using algorithm? How do we know that NP problems are ...
0
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1answer
58 views

understanding gödel's 1931 paper - primitive recursive functions “FR(x)” and “n Gl x”

I am trying very hard to fully understand gödel's paper on the incompleteness theorem. I have a slight technical question about one of the 45 primitive recursive functions introduced to build the ...
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1answer
64 views

What's wrong with this proof that ZFC is consistent

If we take ZFC minus all it's axioms, we can easily prove that ZFC's first axiom is independent (because all statements are independent in a formal system without any axioms). We can then prove that ...
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1answer
32 views

Will assuming an undecidable statement result in a consistent system?

If you assume that an undecidable statement in a consistent axiomatic system is true (or false), will that new system also be consistent? For example, does $\mathsf{ZF}$ being consistent imply that ...
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1answer
191 views

What does a Godel sentence actually look like?

My understanding is that Godel's first theorem says that there is a sentence that is in a sense true in a formal system F but cannot be derived in that system. I then hear that Godel actually goes on ...
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2answers
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$L$ models set theory and so does $V_\kappa$ for $\kappa$ inaccessible

Background: I have been reading the 1980 edition of Kunen. Theorem VI.2.1 states it is provable from ZF that $\mathbf L$ (Kunen writes classes in bold) is a model of ZF. Also, it is a well-known ...
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4answers
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How does Gödel Completeness fail in second-order logic?

So a while ago I saw a proof of the Completeness Theorem, and the hard part of it (all logically valid formulae have a proof) went thusly: Take a theory $K$ as your base theory. Suppose $\varphi$ ...
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1answer
61 views

Does the Law of the Excluded Middle imply syntactical completeness?

The Law of the Excluded Middle (LEM) states that for any proposition $p$, we have $\vdash p \lor \lnot p $. Syntactic completeness (a.k.a negation completeness) states that for any proposition $p$, ...
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1answer
77 views

$\mathrm{NBG}$ proves its own consistency

In $\mathrm{NBG}$, one can construct a model of $\mathrm{ZFC}$: $V$, the class containing every set. Thus: $\mathrm{NBG}$ can prove "$\mathrm{ZFC}$ is consistent". Furthermore, the following ...
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1answer
43 views

Consistency hierarchy between models with many urelements

The axiom of extensionality can be weakened, asking only for non-empty sets with the same elements to be equal. Then there can be many "different empty sets" called urelements. I call this theory ZFU (...
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2answers
165 views

Could someone show me a simple example of something being proved unprovable?

Could someone show me a simple example of something being proved unprovable? Pretty much what the title says, I want to understand a proof of some statement being proved unprovable. E: Please read ...
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1answer
54 views

Is there some result that says a theory cannot prove the consistency of any of its extensions?

Is there some result that says a (sufficiently strong) theory cannot prove the consistency of any of its extensions? Or something along these lines?? More generally, is there a result that says a (...
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1answer
31 views

What does it mean to encode some statement in Theory of Natural Numbers $Th(\mathbb{N})$?

For example Godel's Incompleteness Theorem says "Some statement in $Th(\mathbb{N})$ has no proof". Consider sentence "This sentence is not provable". We can encode this sentence in $Th(\mathbb{N})$. ...
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Gödel's theorem, un-decidable propositions and axioms of a formal theory [closed]

There was a similar question of mine some time ago (which is now closed, but should not be). Anyway the question is similar, but simplified to avoid (mostly) "pedantic" objections, which bypass the ...
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1answer
28 views

Is there any relationship between Gödel numbers associated with proofs of undecidable theorems and infinitesimals?

I recall from my reading of the popular book, Gödel, Escher, Bach some years ago that Hofstadter speculated on the value of Gödel numbers of undecidable theorems. If I recall correctly, he suggested ...
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3answers
76 views

Godel's incompletness theorem - proving a statement is false

I have two question regarding Godel's incompletness theorem. The theorem says that every axiomatic system is either incomplete or inconsistent. If it's consistent, then there are true statements that ...
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1answer
70 views

Does the foundations of Arithmetic need to be effective?

I was reading about Godel's incompleteness theorem which is true for any formal theory that satisfies certain properties. One of these properties is the following: "The theory is assumed to be ...
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1answer
83 views

Independence of existence of inaccessible cardinals

Let $I$ be the formula which states that there exists strongly inaccessible cardinals. My question is regarding the proof of $ZFC\nvdash I$ appearing in Jech (part of theorem 12.12). He starts by ...
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23 views

Quantitative results on PA completeness?

Are there any results estimating the number of sentences in PA that are not provable together with their negations, as a function of the sentence length or the depth of the sentence parse tree or ...
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3answers
127 views

Godel's Second Incompleteness and the Assumption of Consistency

As I understand it, Godel's Second Incompleteness Theorem states that given a theory $T$ that is any extension of Robinson Arithmetic, that if that theory is consistent then it cannot prove a given ...
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1answer
150 views

Decidability of equality of two set-theoretical terms constructed without replacement or specification

Define the set of NS-terms (NS is for "no schemes") to be the smallest set of terms satisfying the following rules : $\emptyset,\omega$ are NS-terms. if $x$ and $y$ are NS-terms, then so are $x\cup ...
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2answers
186 views

Why does Gödel's (First) Incompleteness Theorem apply to ZFC?

Okay, so I'm reading Smullyan's book on Gödel's incompleteness theorems, and I've just about finished the part where he shows that Peano arithmetic is incomplete using Tarski's truth set (chapter IV). ...
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1answer
108 views

Absolutely undecidable statements in Peano arithmetic

Let "Undec(x)" be a predicate in Peano Arithmetic that says "x is the Gödel number of a sentence that is neither provable nor refutable" It is easy to see that this predicate is in fact expressible in ...
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Enumerating the reals using a definability hierarchy

(edit : for those perplexed with the meaning of "truth" in the following, let us say we believe in the consistency of ZFC, use the completeness theorem and reason in a fixed model $M$ of ZFC. The ...
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1answer
186 views

Since arithmetic has a model (thus it is consistent) why care if consistency can't be proved?

Since arithmetic has a model, the numbers as we know them, it is consistent. Why do we care if consistency can't be proved within arithmetic? Do I miss something, ie in what we can consider a model?
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1answer
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Does semantic inconsistency guarantee syntactic inconsistency?

I'm wondering about the possibility of circumventing the problem of incompleteness posed by Roger Penrose in his book "Shadows of the Mind". It occurred to me (and, Googling has revealed to me, ...
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3answers
225 views

Gödels incompleteness vs incompleteness

This has been nagging me, and might be an unfit question, but still: I've been taught that completeness of a theory $T$ means that for any sentence $\varphi$ in the language of the theory, we have ...
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1answer
195 views

How to show incompleteness of second order logic?

I'm trying to see/show that second order logic (with full semantics) is incomplete - i.e. that there are sentences that are true in all models of some theory $T$, and yet still can not be proved from $...
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1answer
172 views

What is wrong with this “proof” that there is no $\omega$th inaccessible cardinal?

"Theorem": There is no $\omega$th inaccessible cardinal. "Proof": Assume ZFC. Let $\kappa_n$ be the $n$-th inaccessible cardinal; since $V_\kappa$ is a model of ZFC for inaccessible $\kappa$, $V_{\...
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1answer
114 views

Consistent, complete axiom system that proves its own consistency

Is there a consistent, complete axiom system that proves its own consistency? I know that this question isn't exact and I haven't defined when an axiom system proves its own consistency because that'...
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1answer
66 views

Peano and consistency, how to understand it rightly.

I'm struggling with the notion of consistency, and a few cases : I'm writing in the following $Con(T)$ to denote the arithmetic formula which expresses the consistency of $T$, for $T$ a consistent ...
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2answers
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Why is $\omega$-consistency needed in Gödel's original Incompleteness proof?

I don't see why the original version of Gödel's first incompleteness theorem (before Rosser's improvement, I mean) had to include the assumption of $\omega$-consistency in order to show that $F \...