Questions about Gödel's incompleteness theorems and related topics.

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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
61 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
20 views

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
129 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
95 views

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
50 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
48 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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2answers
45 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 ...
4
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1answer
88 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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0answers
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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 ...
3
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1answer
37 views

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
78 views

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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2answers
103 views

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 ...
3
votes
3answers
96 views

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 ...
4
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1answer
105 views

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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0answers
23 views

Closed term conditions in PA

The situation I have to transfer statements from the "recursive world" into the "$\color{red}{\text{syntactical world}}$", in the context of binumerability of primitive recursive predicates into the ...
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1answer
40 views

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: ...
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0answers
42 views

Is there a difference between induction in Peano Arithmetic and Presburger Arithmetic

Following this question I still do not get clearly the difference between defining exponentiation in PA but impossiblity of recursively define multiplication in Presburger Arithmetics I was thinking ...
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1answer
73 views

Explanation of $\mathrm{ZFC} + \lnot \mathrm{Con(ZFC)}$

I read in JDH's answer to this mathoverflow question that $\mathrm{ZFC}$ is equiconsistent with $\mathrm{ZFC} + \lnot \mathrm{Con(ZFC)}$. But this second statement is a bit weird. What does ...
2
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1answer
66 views

diagonalization about incompleteness

In the Enderton's logic book(page 186-187), he writes 'diagonalization approach' for proving undefinability of $\#Th\mathfrak{N}=\{\#\tau |\vDash_{\mathfrak{N}}\tau \}$ where $\#\tau$ is the godel ...
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3answers
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Why is Gödel's Second Incompleteness Theorem important?

Given that the consistency of a system can be proven outside of the given formal system, Gödel says, It must be noted that proposition XI... represents no contradiction to the formalities ...
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1answer
21 views

Proving divergence of a sequence in a normed linear space

My aim is to prove that the space $(C_\mathbb{R}[0,1],||•||)$, where $||•||=\int_{0}^{1} t(1-t)|f(t)|dt$, is not complete. I have already proved that the sequence $(f_n)$ defined by ...
4
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0answers
140 views

Gödel's Incompleteness Theorem in “Gödel, Escher, Bach”

Ok, so I'm reading the chapter on Gödel's Incompleteness Theorem in "Gödel, Escher, Bach" and I want to make sure I'm getting this right: the idea of the book's proof is to form the sentence "There ...
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0answers
38 views

Is the probabilitistic distribution of the digits in the Chaitin's constant computable?

The Chaitin constant can in principle be computed with exponential effort on each sucessive digit by brute forcing all programs of a given length and simply proving special theorems on each case that ...
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1answer
94 views

What is the mathematical meaning of this statement made by Gödel (see details)? [closed]

It appears as Proposition VI, in Gödel's 1931 paper "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I" : "To every $\omega$-consistent recursive class $\kappa$ ...
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1answer
229 views

Gödel's Second Incompleteness Theorem and Arithmetically Non-Definable Theories

My recursion theory knowledge has become a bit rusty, so I will appreciate any corrections for misstatements. Gödel's incompleteness theorem is often exploited by philosophical discussions which ...
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0answers
91 views

Incompleteness theorem

Correct me if I am wrong at any point! Godel's incompleteness theorem allows us to express "PA is consistent" in the language of Peano arithmetic, and shows that this is not provable in PA. Let's ...
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votes
1answer
101 views

How can you come to the truth of a statement without proving it?

I was reading a bit about Gödel's incompleteness theorems. I haven't took the time to really study it, but I'm very curious about statements like these: In other words, if our axioms are ...
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1answer
38 views

Why is the set of all true first-order statements about non-negative integers in the language with only equality, $+$ and $\times$ undecidable?

Apparently Tarski and Mostowski proved this, but intuitively I'm not seeing the difference between statements in a language of non-negative integers with equality, addition, and multiplication vs ...
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0answers
63 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 ...
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1answer
73 views

Trouble Reading “On Formally Undecidable Propositions”

I've been working my way through Godel's original paper of the incompleteness theorem in my spare time, and I'm stuck with something stupidly simple. I'm looking at the list of 45 definitions of ...
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1answer
56 views

Lindenbaum's Lemma

I am working on the proof of the Lindenbaum's lemma and there are some passages which are not very clear for me. Here is the statement: Let $\mathbb{L}$ a countable signature, $T$ a consistent set of ...
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1answer
61 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
40 views

The existence of concatenation functions in Godel Numbering?

I know that there are many schema of Gödel Numbering, and each has its own method of Concatenation, n★m. But is there a general proof that shows 'For every Gödel Numbering scheme there exists a ...
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2answers
212 views

How to prove that Gödel's Incompleteness Theorems apply to ZFC?

Let us denote Robinson Arithmetic as Q and Primitive Recursive Arithmetic as PRA. Let $T$ be a formal theory formulated in the language of arithmetic. According to this page on the Stanford ...
2
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1answer
57 views

What does not having a first-order frame imply for models?

There are formulas in modal logic which which do not have a first-order frame condition, as stated here (Non-Sahlqvist formulas, Wikipedia). An example is the McKinsey formula for $p$: ...
2
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1answer
71 views

$\omega$-consistent in Gödel I

In a very accessible form one could state the first incompleteness as follows: Incompleteness Theorem I Assume that $\textbf{PA}$ is consistent. Then there is a sentence $\phi$ such that ...
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2answers
584 views

Is there a mistake in the SEP article about Godel's Incompleteness theorems?

Update: The mistake referred to in this question has now been corrected. The below refers to a previous version of the article: The second supplement to the Stanford Encyclopaedia of Philosophy ...
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0answers
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Importance of Gödel Numbering System

How important is Gödel numbering to his incompleteness proofs, set theory, logic theory in general and proofs employing ZFC? Can we use some other numbering or 'meta' programming? How about if one ...
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3answers
276 views

Can we prove that axioms do not contradict?

We construct many structures by chosing a set of axioms and deriving everything else from them. As far as I remember we never proved in our lectures that those axioms do not contradict. So: Is it ...
48
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12answers
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What is a simple example of an unprovable statement?

Most of the systems mathematicians are interested in are consistent, which means, by Godel's incompleteness theorems, that there must be unprovable statements. I've seen a simple natural language ...
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1answer
142 views

Problem in Kunen - suitable representation of ZF proves the consistency of ZF?

I tried to prove the exercise problem in Kunen (Chapter IV, problem 36.) Problem. Show that there is a formula $\chi(x)$, such that $\chi$ represents ZF; i.e.,$$\phi\in \mathsf{ZF}\to ...
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2answers
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Is it possible to prove that some point belongs to Mandelbrot set? Is this an example of Gödel's theorem?

Everybody knows about Mandelbrot set drawing computer programs. Program takes some point, builds sequence from it, and if found that sequence goes out of circle with 2 radius, then knows that this ...
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1answer
65 views

What does First Godel's Incompleteness theorem mean?

I am terribly confused what really "Incomplete" mean in terms of the Godel's theorem. Does it mean there are some theorems that are definable in First order theory of natural numbers and true but ...
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1answer
94 views

Uncountable reals in the theory

The Question I'm looking for a possibility to somehow proof the "essence" of Cantor's diagonal argument within a recursive first-order theory which is satisfied by the reals (better: within a theory ...
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0answers
28 views

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

Consider pairs $(\Phi,n)$ where $\Phi$ is a finite set of statements in Peano arithmetic and $n$ is an integer. Say that $p'=(\Phi',n')$ is an elementary intuitionistic extension of $p=(\Phi,n)$ iff ...
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Fixed points in computability and logic

I asked this question on CS.SE, too: http://cstheory.stackexchange.com/questions/27322/fixed-points-in-computability-and-logic I would like to understand better the relation between fixed point ...
4
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2answers
206 views

Is it possible to deduce Godel's first incompleteness theorem from Chaitin's incompleteness theorem?

I want to ask if it is possible to deduce Godel's first incompleteness theorem from Chaitin's incompleteness theorem. I am reading the following AMS-Notice article. The authors claim that: The ...
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1answer
49 views

Show that the metric $d_1$ on $C([0,1])$ does not give rise to a complete metric space. [duplicate]

Show that the metric $d_1$ on $C([0,1])$ does not give rise to a complete metric space. $$ d_1(f,g) = \int_0^1 |f(s)−g(s)| \, ds $$
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Questions about godel's first incompleteness theorem

I'd rather not get into the formal proof of godel's first incompleteness theorem. But I have 2 general questions. Looking at the statement from wikipedia: ...