Questions about Gödel's incompleteness theorems and related topics.

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1answer
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Gödel's Incompleteness Theorem - Diagonal Lemma

In proving Gödel's incompleteness theorem, why does he needed the Diagonal Lemma or the Fixed Point Theorem for building a formula $\phi$ that spoke about itself? Can't this formula be built this way: ...
0
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1answer
471 views

Gödel Incompleteness Theorem - Primitive Recursive Functions

I'm currently studying Gödel's Incompleteness Theorem and I am in doubt about his use of primitive recursive functions. To study it, i'm in the point of view of a system of predicate logic with the ...
1
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1answer
181 views

Consequences of Incompleteness.

Suppose that I'm a working mathematician that has just proved a Theorem, say, in Number Theory. Does Gödel's Incompleteness Theorem imply that I can't know for sure if there exists a proof of the ...
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1answer
339 views

Consistent but incomplete formal axiomatic systems

Is there any known consistent but incomplete formal axiomatic system apart of and simpler than one "capable of doing arithmetic"? Is it even possible? Even if this capability of arithmetic were a ...
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3answers
149 views

If $T$ proves $\operatorname{Con}(ZFC)$, is $T$ at least as strong as set theory?

I am looking for either a proof of counterexample of this: Lemma: Let $\pi$ be a faithful interpretation of $PA$ into $ZFC$, and let $PA'$ be the image of $PA$ under $\pi$. If there is a $T$ with ...
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0answers
83 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 ...
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2answers
204 views

Is this incompleteness result easier to get than incompleteness of PA?

Gödel's theorem for Peano Arithmetic shows that (under consistency hypothesis on PA) there is a statement which cannot be proved or disproved within PA that is true under the standard model ...
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2answers
161 views

How can you add 'not G' to a formal system without introducing omega inconsistency?

In any formal system S that is susceptible to Godel's proof, we can make a formula G which is undecidable. That should mean that we can add either $G$ or $\neg G$ as an axiom to S and still end up ...
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6answers
2k views

What philosophical consequence of Goedel's incompleteness theorems?

I want to write a philosophical essay centered about Goedel's incompleteness theorem. However I cannot find any real philosophical consequences that I can write more than half a page about. I read the ...
2
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1answer
165 views

Differences between between concepts related to Gödel's Incompleteness theorems: self-referencing, diagonalization and fixed point theorem?

I am studying the proof of Gödel's first Incompleteness theorem at the moment and I don't understand the differences between self-referencing, diagonalization and fixed point related to Gödel's proof. ...
0
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1answer
132 views

Uniqueness of super godel numbers of $\varphi$ and $\neg \varphi$

Let $e_{0},e_{1},...,e_{n}$ be a sequence of wffs or other expressions. Code each $e_{i}$ by a regular godel number $g_{i}$, to yield a sequence of numbers $g_{0},g_{1},...,g_{n}$. Then encode this ...
6
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2answers
240 views

“Completeness modulo Godel sentences”?

So this has been bugging me for roughly four years. When I was an undergraduate, I attended a colloquium in which the speaker was a 'cheerleader' for AD (the axiom of determinacy- an alternative to ...
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1answer
111 views

Relationship between $\Sigma_{1}$ and $\Pi_{1}$ functions (Logic)

I am working on the following homework problem for a logic class on Godel's incompleteness theorems and the following question is asked. Is the converse of Theorem $13.1$ true? Explain. Theorem ...
5
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1answer
164 views

Is there any recursive definition, using only addition, of the set of values of $x^2+y^2$?

There is a recursive definition of the set of squares which uses only addition: $CS(x,y) := IS(x) \wedge IS(y) \wedge x \lt y \wedge \forall z: (x \lt z) \wedge (z \lt y)⇒\neg IS(z)$ $IS(x)⇔ x=0 ...
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3answers
316 views

Can ZFC decide number theory?

Among the versions of the Incompleteness Theorem that I've seen are the following: Assuming the Peano axioms are consistent (which they are, if we accept the existence of the natural numbers), there ...
2
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2answers
371 views

How it is posible that $\omega$-inconsistency does not lead to inconsistency

After wikipedia: Theory is $\omega$-inconsistent if, for some property P of natural numbers, T proves P(0), P(1), P(2), and so on (that is, for every standard natural number n, T proves that P(n) ...
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4answers
817 views

Consistency VS Inconsistency , semantics and syntactics

What does it mean when we say that a set of formulas , Sigma , is Consistent , or Inconsistent ? Is ...
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5answers
6k views

What is the difference between Completeness and Soundness in first order logic?

Completeness is defined as if given $\Sigma\models\Phi$ then $\Sigma\vdash\Phi$. Meaning if for every truth placement $Z$ in $\Sigma$ we would get $T$, then $\Phi$ also would get $T$. If the previous ...
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1answer
296 views

About Godel Incompleteness and Multiplication

Godel proved that every system strong enough to include standard arithmetic with multiplication is incomplete. But I've read that systems that do not include multiplication are complete. But ...
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1answer
254 views

Use of undecidability

Suppose someone proved that the Goldbach conjecture was undecidable in an axiomatic system that is consistent as far as we know. Then in some sense we know that Goldbach conjecture must be "true", ...
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1answer
284 views

Completeness of Real Number Arithmetic?

I've recently read that, although Godel Incompleteness holds for the theory of natural numbers, the theory of the real numbers is actually complete. So, why is Godel's Theorem still considered ...
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2answers
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Gödel's incompleteness theorem can't be proven?

I have a very simple question, that I still haven't found an answer to yet: Gödel is said to have proven that Peano Arithmetic (or any system capable of expressing it) can't prove its own ...
2
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1answer
155 views

Why does the existence of independent statements not prove consistency?

I've read before that, by the Principle of Explosion, if a theory is inconsistent, then absolutely any statement can be proven within it. Obviously, there are statements which are independent of ZFC ...
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3answers
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Why Euclidean geometry cannot be proved incomplete by Gödel's incompleteness theorems?

Wikipedia mentioned the limitation of Gödel's theorems. According to it, Euclidean geometry doesn't satisfy the hypotheses of Gödel's incompleteness theorems, that is, it cannot define natural ...
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1answer
159 views

Does robinson arithmetic satisfy modal logic's “axiom 4”?

Does Robinson arithmetic prove the theorem "if sigma is provable then 'sigma is provable' is provable' for a fixed sentence sigma? It's clear to me that you can get a primitive recursive function f ...
2
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1answer
228 views

How to express Con(PA) as a first-order statement?

I read from somewhere that Fact 1. PA, which refers to the first-order version, is not finitely axiomatizable. At the same time, the second incompleteness theorem says that there is no proof in ...
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1answer
164 views

Quantitative version of Godel's incompleteness theorem

Let $A$ be a list of axioms which we assume to be sound (for example, PA or ZFC). Godel's incompleteness theorems imply that if we add only finitely many (true) axioms to $A$, the new list $B$ will ...
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7answers
2k views

True vs. Provable

Gödel's first incompleteness theorem states that "...For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system". What ...
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2answers
1k views

Explanation of proof of Gödel's Second Incompleteness Theorem

I am looking for a simple explanation/outline of the proof of Gödel's Second Incompleteness Theorem, and I haven't yet been able to find anything that is within my grasp. I'm looking for something ...
4
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3answers
578 views

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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1answer
221 views

Best known theory for proving statements about natural numbers

Gödel's incompleteness theorems imply that there is no consistent theory that can be effectively generated and contains all true statements about the natural numbers. Well, what known consistent ...
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6answers
3k views

Why bother with Mathematics, if Gödel's Incompleteness Theorem is true?

OK, maybe the title is exaggerated, but is it true that the rest of math is just "good enough", or a good approximation of absolute truth - like Newtonian physics compared to general relativity? How ...
6
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3answers
979 views

Prove Gödel's incompleteness theorem using halting problem

How can you prove Gödel's incompleteness theorem from the halting problem? Is it really possible to prove the full theorem? If so, what are the differences between original proof and proof by ...
6
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4answers
670 views

A naive inquiry of Godel's incompleteness--or why does mathematics need proofs of unprovability?

My question stems from reading Swetz, 1994 (mostly excerpts from the journal Mathematics Teacher) and Berlinski, 2005 (a popular book on 10 most important mathematical breakthroughs in history). 1) ...
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1answer
181 views

Construction of a sequence of theorems with increasing and unbounded “difficulty”?

Let's define the "difficulty" of a theorem as the logarithm of the size of its shortest proof divided by the logarithm of the size of the theorem itself. For example, if a theorem has difficulty less ...
9
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2answers
263 views

How to show the existence of an infinite set of independent undecidable sentences?

How to show the existence of an infinite set of independent undecidable sentences? By "independent" I mean that no implication between any two elements is provable. A finite set that satisfies the ...
5
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3answers
628 views

P vs NP and Gödel

I apologize for the, perhaps, silly question. My impression, as a layman, is that Gödel Incompleteness Theorem should rule out the possibility that P=NP. Is that true or there are deeper technical ...
12
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1answer
424 views

What's the theory in which incompleteness of PA is proved?

Maybe this is a dumb question, but I have to admit that it is not really clear to me what the theory is, in which incompleteness of PA and stronger theories is proved. The texts I did study so far are ...
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1answer
296 views

Why is Godel's first theorem not a proof from the inside?

Why is Godel's first theorem not a proof for the truth of the so called undecidable proposition? You may say it's a proof from the outside, but if not all proofs from the outside be formalized inside ...
35
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5answers
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Understanding Gödel's Incompleteness Theorem

I am trying very hard to understand Gödel's Incompleteness Theorem. I am really interested in what it says about axiomatic languages, but I have some questions: Gödel's theorem is proved based on ...
3
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1answer
434 views

Is Robinson Arithmetic complete and not-complete?

Is Robinson Arithmetic complete in the sense of Gödels completeness theorem? And is Robinson Arithmetic incomplete in the sense of Gödels first incompleteness theorem? If RA is both it would be a good ...
7
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1answer
389 views

Can Robinson's Q prove Presburger arithmetic consistent?

I made an assertion in What are some examples of theories stronger than Presburger Arithmetic but weaker than Peano Arithmetic? that Q has higher consistency strength than Pres, Presburger arithmetic; ...