Questions about Gödel's incompleteness theorems and related topics.

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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 ...
4
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3answers
131 views

Does Gödel's incompleteness theorem contradict itself?

I have problems understanding Gödel's incompleteness theorem. I presume I have a misunderstanding of some phrase or I have to look closer at the meaning of some detail. Gödel's second incompleteness ...
3
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1answer
64 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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0answers
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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
92 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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7answers
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Does Gödel's Incompleteness Theorem really say anything about the limitations of theoretical physics?

Stephen Hawking believes that Gödel's Incompleteness Theorem makes the search for a 'Theory of Everything' impossible. He reasons that because there exist mathematical results that cannot be proven, ...
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1answer
136 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
85 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). ...
5
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1answer
66 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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0answers
51 views

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 ...
2
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1answer
63 views

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, ...
5
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1answer
174 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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2answers
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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 ...
3
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1answer
74 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 ...
4
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1answer
118 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 $\kappa$, ...
3
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1answer
87 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 ...
2
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1answer
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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 ...
4
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1answer
134 views

Why a system becomes incomplete once it's capable of doing arithmetic?

For a Formal axiomatic system to obey Godel's incompleteness theorems, It has to be powerful enough to incorporate Peano Axioms. Why It does not apply to say, Presburger arithmetic or the axioms of ...
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1answer
85 views

Algorithm to force decidability of statements using an intuitionistic series of new axioms

Say that a set $\Phi$ is a finite set of statements in Peano arithmetic is meekly consistent if it contains no "inner,immediate" contradiction, i.e. for any statements $\alpha,\beta$, it does not ...
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6answers
983 views

prove that it's not provable

Is it possible to prove that you can't prove some theorem? For example: Prove that you can't prove the Riemann hypothesis. I have a feeling it's not possible but I want to know for sure. Thanks in ...
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2answers
48 views

show the set of valid second-order $\emptyset$-sentences is not R-enumerable

show the set of valid second-order $\emptyset$-sentences is not R-enumerable this would have the empty symbol set i.e. $S = \emptyset$ so it would be sentences that are universally or existentially ...
3
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1answer
76 views

Is it possible to show that a particular theorem or its negation is provable, without knowing which of the two is true?

I've been thinking about this for a while: as far as we know, is it possible that for a particular statement $\sigma$ of $\textsf{ZFC}$, we can prove that $(\textsf{ZFC} \vdash \sigma) \vee ...
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2answers
698 views

Is there an algorithm that when given a set of axioms, will generate a statement independent of those axioms?

Is there any mechanism or algorithm where one can generate mathematical statements/problems that are undecidable, i.e their proof is independent, from a certain set of axioms?
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1answer
52 views

How does PA prove all $\Delta_0$-formulas which are true in the standard model?

Let $\varphi(x_1,\dots,x_n)$ be a $\Delta_0$-formula, i.e. a formula in which every quantifier is bounded. I want to prove that $$ \text{PA}\vdash\varphi(\overline{n_1},\dots,\overline{n_k}) \iff ...
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1answer
88 views

Non-standard model of arithmetic and Gödel's theorem [closed]

This is a cross-post of a question asked on History of Science and Mathematics Stack Exchange. I've read Skolem's paper on his non-standard models of the arithmetic ("Über die ...
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2answers
73 views

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 ...
5
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0answers
55 views

Deductive closure of sentence $\forall x \forall y F(x,y) \stackrel{.}{=} F(y,x)$ in language $\mathcal{L}$ is undecidable.

$\mathcal{L}$ is the language that contains a single binary function symbol $F$. In the earlier parts of this question, we were told to take the $\mathcal{L}$-structure $\mathcal{M}$ with universe ...
3
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2answers
75 views

Lindebaum's Lemma seemingly inconsistent with Gödel's incompleteness theorem?

Lindenbaum's Lemma: Any consistent first order theory $K$ has a consistent complete extension. First Incompleteness Theorem: Any effectively generated theory capable of expressing elementary ...
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1answer
73 views

Why are all computable functions representable in PA?

I'm trying to understand the proof of the first incompleteness theorem, and more specifically, the diagonal lemma. Suppose $GN(x)$ is the Gödel Number of a formula $x$. The first step of the diagonal ...
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1answer
76 views

What does “Fixed-Point Lemma” says intuitively?

The lemma as stated in Enderton's logic says: Fixed-Point Lemma.   Given any formula $\beta$ in which only $v_1$ occurs free, we can find a sentence $\sigma$ such that $$ A_E \vdash ...
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1answer
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Question about Godel's first incompleteness theorem and the theory within which it is proved

Please explain the error in my reasoning with this: Let T be a formal theory within which Godel's first incompleteness theorem can be proved. In other words... suppose when we write the proof of ...
3
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2answers
147 views

Do we know if the consistency of $ \mathsf{ZFC} $ is sufficient to prove that $ \mathsf{ZFC} \nvdash \text{Con}(\mathsf{ZFC}) \to \mathsf{SM} $?

This is basically a sequel to this earlier post. There, I asked: If $ \mathsf{ZFC} $ is consistent, then is $ \mathsf{ZFC} \nvdash \neg \text{Con}(\mathsf{ZFC}) $, i.e., $ \neg ...
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1answer
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Reference request for the independence of $ \text{Con}(\mathsf{ZFC}) $.

Gödel’s Second Incompleteness Theorem says that if $ \mathsf{ZFC} $ is consistent, then $ \mathsf{ZFC} \nvdash \text{Con}(\mathsf{ZFC}) $, i.e., $ \text{Con}(\mathsf{ZFC}) $ is not provable in $ ...
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2answers
54 views

Confusion in Godel's numbering for subscripts

I don't understand how to represent subscripts in Godel's numbering. Suppose I have a formula: $$x_1 + sx_{11} = s(x_1 + x_{11})$$ and an encoding: then what should be the Godel Numbering? Should ...
2
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0answers
66 views

How does Godel Escher Bach support Artificial Intelligence? [closed]

Typically, Godel's Incompleteness theorems have been used to argue against the possibility that the human mind is essentially equivalent to a formal system. However, in Daniel Dennett's book "Darwin's ...
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3answers
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Classifying Types of Paradoxes: Liar's Paradox, Et Alia

The well-known Liar's Paradox "This statement is false" leads to a recursive contradiction: If the statement is interpreted to be true then it is actually false, and if it is interpreted to be false ...
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2answers
53 views

Limit of Cauchy Sequence

Let $(X,d)$ be a metric space and $(x_n)$ be a Cauchy sequence in $X$. Is there a limit $x$ for $(x_n)$ whether $x$ in X or not ? In general, does every Cauchy sequence has a limit, if that limit ...
63
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14answers
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4
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1answer
89 views

Integer induction without infinity

In ZFC minus infinity (let us call this T), one can still define ordinals, and then define integers as ordinals all of whose members are zero or successor ordinals. I am looking for a formula $\psi$ ...
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2answers
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Does “provable” include “proved by reduction to absrdity”?

The incompleteness says that formal logic system (under certain condition) contains non provable TRUE sentence. It seems that "prove" means here "derive". Only TRUE sentence could be proved. If a ...
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Formalisms of Mathematics in Gödel's Incompleteness Theorem

Unfortunately I have been more or less introduced to Gödel's Incompleteness Theorem(s?) via computer science and Turing machines, and we haven't addressed them very rigorously. My professor often says ...
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3answers
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Why can PA + $\neg G_{PA}$ be consistent?

Wikipedia and other sources claim that $PA +\neg G_{PA}$ can be consistent, where $\neg G_{PA}$ is the Gödel statement for PA. So what is the error in my reasoning? $G_{PA}$ = "$G_{PA}$ is ...
3
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1answer
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Incompleteness of formal systems as opposed to completeness of a non-formal theory

I have read that Gödel's incompleteness theorem does not apply to real closed field theory. But the incompleteness theorem applies only to formal systems, that is systems whose alphabet of symbols and ...
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2answers
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Are Gödel's incompleteness theorems really about primitive recursive functions?

Any formulation of Gödel's incompleteness theorems seems to involve arithmetic. Why is arithmetic so fundamental? After thinking about the issue a little bit, I came to the conclusion that the ...
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1answer
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Is there a relationship between Turing's Halting theorem and Gödel Incompleteness

Turing's proof that a Halting oracle is impossible and Gödel's proof that and omega-consistent first order theory of arithmetic must be incomplete are similar in that they use self-referential ...
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1answer
158 views

question about Gödel numbering

I have a question about Gödel numbering, it is trivial but I would like to know how can you know the length of an expression through its Gödel number. ¿? I think you can use a recursive function but ...
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2answers
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How is Goedel's 1st incompleteness theorem related to the Axioms of a theory [closed]

i am thinking of various connections and formulations of Goedel's 1st incompleteness theorem. Apart from connections to Turing's Halting Problem and Algorithmic Complexity Theory, i am looking for ...
6
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2answers
206 views

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 ...
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1answer
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Understanding Rosser's Theorem

Initial Situation For some time now I'm trying to understand a proof for Rosser's Theorem -- the proof given in Smorynski's article "The Incompleteness Theorems" (here is a first entry from google: ...