Questions about Gödel's incompleteness theorems and related topics.

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1
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2answers
156 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
1k 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 ...
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1answer
161 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. ...
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1answer
131 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
237 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 ...
1
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1answer
108 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 ...
4
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1answer
159 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
311 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
354 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
736 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
288 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 ...
1
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1answer
250 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", ...
5
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1answer
270 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
1k views

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
140 views

Why does the existence of independent statements not prove completeness?

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
1k views

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 ...
4
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1answer
158 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
votes
1answer
222 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
163 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 ...
13
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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
540 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 ...
6
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1answer
220 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
948 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
votes
4answers
654 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) ...
2
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1answer
178 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
votes
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
votes
3answers
610 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
423 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
292 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 ...
33
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5answers
6k views

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
428 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
385 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; ...