Questions about Gödel's incompleteness theorems and related topics.

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11
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2answers
134 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 ...
11
votes
1answer
148 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 ...
3
votes
1answer
80 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 ...
1
vote
1answer
55 views

understanding gödel's 1931 paper - number theoritical formulae

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 : ...
1
vote
1answer
53 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: ...
1
vote
1answer
88 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 ...
0
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1answer
90 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 : ...
0
votes
1answer
64 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 : ...
10
votes
0answers
163 views

Elementary references on Robinson Arithmetic, Prim. Recursive fns etc.

I'm in the middle of revising my freely available and much-downloaded introductory notes Gödel Without (Too Many) Tears. (They are a sort of cut down version of part of my Gödel book, and I'm ...
8
votes
0answers
239 views

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 ...
5
votes
0answers
59 views

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 ...
5
votes
0answers
64 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 ...
5
votes
0answers
332 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 ...
4
votes
0answers
120 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 ...
4
votes
0answers
91 views

An “internal” condition on $T$ so that for the standard provability predicate, $T$ proves $\text{Pf}(\underline S)$ implies $T$ proves $S$?

This is probably quite basic, but I'd like to make sure I got this right. Regarding the proof of Goedel's first incompleteness theorem, say that we have $T$ containing $PA$ effectively axiomatizable ...
3
votes
0answers
95 views

Maths, esp. Godel, and poetry

At the risk of being an interloper: I'm a poet with a bit of mathematical training. Right now I've got a grant from the Arts Council of Northern Ireland to write a collection (loosely) based on the ...
2
votes
0answers
43 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 ...
2
votes
0answers
93 views

Minimal number of variables in a ZFC-undecidable sentence?

Let $\phi$ be a sentence of set theory. In Prenex form, $\phi$ can be written $$ {\bf Q}_1 x_1 {\bf Q}_2 x_2 \ldots {\bf Q}_n x_n \ \ \psi(x_1,x_2, \ldots ,x_n) $$ where each ${\mathbf Q}_i$ is ...
1
vote
0answers
27 views

Can we prove undefinability theorem first (using techniques that are different from Godel's) and then deduce Incompleteness from it?

I asked a related question about the matter here in philosophy platform where it was suggested to ask a modified version of the question on Math.se My question is, Is there any known way to prove ...
1
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0answers
23 views

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 ...
1
vote
0answers
55 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 ...
1
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0answers
26 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 ...
1
vote
0answers
75 views

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 ...
1
vote
0answers
112 views

Do we know that if $\pi$ is normal then there is a proof of it?

We do not know whether $\pi$ is normal or it is not and many other weaker statements, e.g. (*) $\pi$ contains infinitely many $0$s. Inspired by the Godel's incompleteness theorem that there are some ...
1
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0answers
67 views

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 ...
0
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0answers
35 views

Rosser Sentences and Theories

Let T be any theory extending Q. Let R be the Rosser sentence of T. Let T0 be T + {R} and T1 be T + {∼ R}. Show that T0 and T1 are both consistent. Show that T0 ∪T1 is inconsistent. Show that for each ...
0
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0answers
26 views

Why is it impossible to prove absolute consistency of a theory falling prey to Godel's theorems?

Why is it impossible to prove absolute consistency of a theory T falling prey to Godel's theorems? I understand that a theory falling prey to Godel's second incompleteness theorem cannot prove its own ...
0
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0answers
18 views

Can there be true but unprovable statements about object other than numbers?

In ZFC, everything is a pure set, and because the necessary amount of arithmetic for the Gödel's incompleteness theorems to go through is interpretable within ZFC, there are undecidable statements ...
0
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0answers
13 views

Exercise Henkin Theory

I'm doing some excercises from the book "The Incompleteness Phenomenom" from Goldstern and Judah. I have to do this excercise: Assume that Γ is a theory satisfying the following: Γ is a Henkin ...
0
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0answers
15 views

Same number of provable as unprovable statements?

Building on this question: Infinitely many nonequivalent unprovable statements in ZFC because of Gödel's incompleteness theorem? is the infinity of provable statements the same infinity of ...
0
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0answers
40 views

Does Gödel sentence depend on numbering?

If we change Gödel's numbering definition of the $Prov$ predicate will change as well, but the meaning won't. How is that going to affect $G$? It seems to me like it will change as it is actually ...
0
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0answers
40 views

Arithmoquine function in Gödel's proof

Could someone explain as detailed as possible how the Arithmoquine{a,a'} function works or is the defined in Gödel's proof of the incompleteness theorem? To describe my question better... In this ...
0
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0answers
37 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 ...
0
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0answers
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 ...
0
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0answers
74 views

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: ...