Questions about Gödel's incompleteness theorems and related topics.

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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
34 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
51 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
68 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
40 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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0answers
34 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
35 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
166 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
48 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
55 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 ...
9
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2answers
540 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
50 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 ...
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3answers
235 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 ...
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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
135 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
103 views

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
52 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
90 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
27 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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0answers
135 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 ...
4
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2answers
158 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
43 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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0answers
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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: ...
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1answer
27 views

How can Goodstein's theorem be expressed in PA

I understand Goodstein's Theorem and its proof. I'm trying to understand the proof of why Goodstein's Theorem cannot be proved in PA. However, it's not immediately clear to me that Goodstein's Theorem ...
2
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1answer
134 views

True but unprovable?

I would like to ask a question about Gödel's Incompleteness Theorems which I've had in the back of my head for some time. Since I'm a student working in a completely different area of maths (my usual ...
4
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1answer
54 views

About Gödel's Incompleteness Theorem

A short time ago, I've been thinking about what statements in Number Theory are true but not provable. I've seen the proof of the Incompleteness Theorem (in the Gödel's works) and he gave an example ...
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1answer
58 views

Question about the incompleteness proof (Theorem V)

Question in short: Where do I find a complete proof of Theorem V from Gödels incompleteness proof? If it does not exists, can someone provide it? Question in detail: I am trying to understand ...
2
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1answer
143 views

Can unprovability unprovable? Is there an $\omega$-fold unprovability?

I was just thinking about unprovability. I just wanted to know if it is possible to make a concrete boundary between provable problems and unprovable problems in a certain axiomatic system. We know ...
3
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1answer
200 views

Problems with nesting proof predicates in first order logic.

Whenever I start nesting proof predicates, I always seems to run into these bizarre situations. I was wondering if anyone knows about this and could shed some light on it or provide me with some ...
4
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1answer
168 views

Proof of “If $ZFC$ proves there is an inaccessible cardinal, then $ZFC$ is inconsistent”.

Let $I$ be the statement "there is an inaccessible cardinal". I'm aware of two proofs of "If $ZFC\vdash I$ then $ZFC$ is inconsistent". One proof uses the Second Incompleteness Theorem, which I ...
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1answer
71 views

set theory, Incompleteness and axiomatic systems

Is the number of theorems that can be proved (decidable) within a certain set of axioms (for instance ZFC) is finite or infinite ? in other words, are we going to fully exhaust that set of axioms ...
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1answer
45 views

Boolean Algebra and Godel

Can anyone give an example of a theorem in Boolean Algebra that isn't immediately obvious to someone with a computer that can construct a truth table? Clearly no propisition that can be proved using ...
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3answers
126 views

In Godel's first incompleteness theorem the Godel sentence G is true otherwise it contradicts itself, however its truth implies it is not provable .

How can this be? I understand there are two basic definitions of truth in mathematics, one being the formalist definition which includes excluded middle and the second form being the intuitionist in ...
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0answers
100 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 ...
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2answers
656 views

Godel Incompleteness Theorem

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
97 views

How can I imagine a model of $\mathsf{(ZF-Inf)} + \neg \mathsf{Con(ZF-Inf)}$?

Gödel's second incompleteness theorem states that if $\mathsf{ZF-Inf}$ is consistent, then $\mathsf{ZF-Inf} \nvdash \mathsf{Con(ZF-Inf)}$. Moreover, if $\mathsf{(ZF-Inf)} + \neg \mathsf{Con(ZF-Inf)}$ ...
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1answer
79 views

Godel's completeness theorem and formula that states consistency of ZF

Godel's completeness theorem, in original formulation, says that every logically valid statement/formula has finite deduction of a formula. Now then there is Godel's incompleteness theorem. Would this ...
2
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1answer
92 views

Expressibility of Gödel's Incompleteness Theorem

Can Gödel's Incompleteness Theorem be expressed as a formal sentence in ZFC and be proven formally or is it inherently meta-mathematical? (Note: I am referring to the theorem itself, not the ...
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2answers
46 views

Can two distinct formulae (or series of formulae) have the same Gödel number?

As I am studying Gödel's incompleteness theorem I am wondering if two distinct formulae or series of formulae can have the same Gödel number? Or the function mapping each formula or series of formulae ...
2
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1answer
79 views

Gödel's proof method and fundamental theorem of arithmetic

I am a novice to Gödel's proof (the theorem that consistency contradicts completeness), and, as it seems to me, he uses the fundamental theorem of arithmetic to uniquely number any formula. My ...
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1answer
334 views

Could it be that Goldbach conjecture is undecidable?

The result closest to Goldbach conjecture is Chen's theorem [Sci. Sinica 16 157–176], the proposition ``1+2''. It is natural to ask if it is likely that under our arithmetic axioms the Goldbach ...
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1answer
78 views

If a statement holds for all standard models of PA, then does it hold for all models?

Suppose that $\varphi$ is a consequence of every standard model of PA. Then is it provable from PA?
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1answer
201 views

Gödel's way of teaching non-standard models to Takeuti.

In Memoirs of a Proof Theorist, Gaisi Takeuti relates how Gödel taught him about nonstandard models in an "interesting" way: It went as follows. Let T be a theory with a nonstandard model. By ...
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1answer
57 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 ...
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1answer
119 views

Criticism on truth of Gödel sentence in standard interpretation

Mendelson in his book mentioned the Gödel sentence and argued that in standard interpretation it is true. But Peter Milne in his article (On Godel Sentences and What They Say) criticized: "But we know ...
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1answer
69 views

Truth of Godel's sentence in standard interpretation

It is siad that the Godel's sentence: g is true in the standard interpretation. But I have problem in truth of g in the standard interpretation. We proved that if theory K is consistent g is not ...
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2answers
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Gödel incompleteness theorem [closed]

Gödel incompleteness theorem states that any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.So what are some Gödel sentences about ...
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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 ...
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2answers
162 views

Undecidable sentence in Godel's incompleteness theorem

In Godel's incompleteness theorem, the undecidable sentence is g: I am not provable. Ok. I accepted it and realized that in satandard interpretation it is true. So we found a true sentence which ...
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2answers
98 views

Scapegoat theory and PA

A theory T is scapegoat if for every formula A with only one free variable there exist a closed term s such that T proves: (∃x(¬A(x)))⇒¬A(s) I think it is an expectable property for each theory since ...