Questions about Gödel's incompleteness theorems and related topics.

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5
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3answers
481 views

Is the negation of the Gödel sentence always unprovable too?

The incompleteness theorem says that certain theories+deduction system contain at least one sentence (the Gödel sentence "$G$"), which can't be proven (in the system in which it holds). (i) Is ...
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2answers
219 views

Is there a non-contradictory non-trivial axiomatic system in which Gödel's theorem is undecidable?

Gödel's Incompleteness Theorem states that in any non-contradictory non-trivial axiomatic system there are certain statements or theorems whom cannot be proved in that system, For example in ZFC, ...
0
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2answers
361 views

What is actually “relatively consistent”?

Gödel's incompleteness theorem states that: "if a system is consistent, it is not complete." And it's well known that there are unprovable statements in ZF, e.g. GCH, AC, etc. However, why does this ...
2
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1answer
185 views

A qualitative, yet precise statement of Godel's incompleteness theorem?

I read online a statement to the effect that (I'm paraphrasing): Goedel's incompleteness theorem shows that we cannot even have a complete and consistent theory for the natural numbers. I am ...
4
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5answers
291 views

Gödel says: countable proofs, uncountable conjectures?

I thought I understood Gödel's Incompleteness Theorem to say: Starting from ZF, there only a countable number of proofs you can write The number of possible conjectures is uncountable. Thus, ...
4
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3answers
335 views

Aftermath of the incompletness theorem proof

This is somewhat of a minor point about the incompletness theorem, but I'm always a little unsure: So one proves that there is a formula which is unprovable in the theory of consideration. Okay, at ...
3
votes
1answer
113 views

properties of the provability predicate applied to open formulas

Good day! Let $\mathrm{T}$ be a first-order theory which contains the Peano arithmetic and has a recursively enumerable set of axioms. It is well known that one can construct a predicate ...
5
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3answers
304 views

An ignorant question about the incompleteness theorem

Let me preface this by saying that I have essentially no background in logic, an I apologize in advance if this question is unintelligent. Perhaps the correct answer to my question is "go look it up ...
0
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1answer
142 views

Why isn't GL system of provability logic reflexive?

Formula $\square p \rightarrow p$ (axiom T; corresponding to reflexive modal frames) is interpreted as "if p is provable, then p", or more precisely: for all realizations (all substitutions for $p$), ...
1
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1answer
149 views

Definition and meaning of “Proof Schema”, “Class Sign”

I'm a newbie in advanced mathematics, and I'm trying to understand Godel's theorem. I came across these two words which I couldn't understand clearly. "Proof Schema" and "Class-Sign" Can anybody ...
5
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1answer
377 views

Is there any direct application of Gödel's Theorems outside of logic?

Gödel's incompleteness theorems was a major achievement with ramifications outside the field of mathematics itself. Are there any direct applications of the theorem(s), or any of the methods pioneered ...
2
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1answer
327 views

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
448 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 ...
2
votes
1answer
164 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. ...
6
votes
3answers
148 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 ...
0
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1answer
319 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 ...
4
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0answers
82 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 ...
13
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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 ...
7
votes
3answers
312 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
votes
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
743 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 ...
3
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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 ...
5
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1answer
271 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 ...
8
votes
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
votes
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 ...
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
225 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 ...
12
votes
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 ...
6
votes
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 ...
6
votes
3answers
951 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
657 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 ...
1
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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 ...
3
votes
1answer
430 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
votes
1answer
386 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; ...