Questions about Gödel's incompleteness theorems and related topics.

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32
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6answers
2k views

Why is the Continuum Hypothesis (not) true?

I'm making my way through Thomas W Hungerfords's seminal text "Abstract Algebra 2nd Edition w/ Sets, Logics and Categories" where he makes the statement that the Continuum Hypothesis (There does not ...
35
votes
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 ...
9
votes
5answers
1k views

What does it mean for something to be true but not provable in peano arithmetic?

Specifically, the Paris-Harrington theorem. In what sense is it true? True in Peano arithmetic but not provable in Peano arithmetic, or true in some other sense?
51
votes
13answers
8k views

Can someone explain Gödel's incompleteness theorems in layman terms?

Can you explain it in a fathomable way at high school level?
13
votes
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 ...
14
votes
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 ...
13
votes
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 ...
7
votes
1answer
390 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; ...
4
votes
3answers
352 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 ...
0
votes
2answers
128 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 ...
6
votes
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 ...
11
votes
2answers
801 views

Examples of statements which are true but not provable

Speaking informally and working, for example, in Peano Arithmetic (PA), we know that the essence of the Gödel's first incompleteness theorem is that there are true statements (in our model PA), which ...
8
votes
3answers
1k views

Consistency of Peano axioms (Hilbert's second problem)?

(Putting aside for the moment that Wikipedia might not be the best source of knowledge.) I just came across this Wikipedia paragraph on the Peano-Axioms: The vast majority of contemporary ...
8
votes
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 ...
3
votes
2answers
336 views

Proving and understanding the Fixed point lemma (Diagonal Lemma) in Logic - used in proof of Godel's incompleteness theorem

http://en.wikipedia.org/wiki/Diagonal_lemma I am wondering about the proof of the "Fixed-Point Lemma" $\text{Mod } \Sigma$ is the class of all models of $ \Sigma$. $\text{Th Mod } \Sigma$ is the set ...
5
votes
1answer
106 views

Impossibility of theories proving consistency of each other?

By Godel's second incompleteness theorem, a consistent theory (to which the theorem applies) cannot prove itself consistent. I learned that it's also impossible to have a pair of consistent theories ...
5
votes
3answers
505 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 ...
3
votes
1answer
201 views

System with infinite number of axioms

Assume we have a set of axioms $A_0$. There exists a statement that can be formulated with these axioms that cannot be proven to be true with this system. Assume we give such a statement axiomatic ...
2
votes
1answer
113 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 ...
2
votes
1answer
342 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: ...
9
votes
1answer
203 views

Given a φ independent of PA which is true in the standard model, will always (PA+ ¬φ) be ω-inconsistent?

Given a φ independent of PA which is true in the standard model, will always (PA+ ¬φ) be ω-inconsistent? Does it mean that every such ¬φ can be used to prove that a Turing machine halts on a given C ...
6
votes
2answers
143 views

Formalizing proof of Godel sentence's non-provability?

In the question What does it mean for something to be true but not provable in peano arithmetic? Henning Makholm states, "...he [Godel] gave a (formalizable, with a few additional technical ...
5
votes
2answers
146 views

Modern book on Gödel's incompleteness theorems in all technical details

Is there a modern book on Gödel's incompleteness theorems that goes into each and every technical aspect of the proof of them (a classical one, if such exists)? I'm not interested in popular ...
5
votes
4answers
272 views

Proving Undecidability of first order logic without first proving it for arithmetic.

All text I have read prove the Undecidability of first order logic a bit as an afterthought and after having proved the incompleteness and Undecidability of (Peano) Arithmetic. This proof also ...
4
votes
3answers
198 views

Concrete example for diagonal lemma

Diagonal lemma says that in a theory with enough assumption for any formula $A(x)$ there exist a sentence $B$ such that $B$ $\iff$ $A(\#(B))$ is a theorem in that theory, in which $\#(B)$ represents ...
4
votes
3answers
582 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 ...
3
votes
1answer
59 views

The effects of requiring a recursive vs. a recursively enumberable axiomatization in the incompleteness theorem

I believe that the (paraphrased) original statement of Gödels first incompleteness theorem (including Rosser's trick) is If T is a sufficiently strong recursive axiomatization of the natural ...
3
votes
1answer
118 views

Non-self-referential undecidable sentences in arithmetic

Are there any known undecidable sentences for PA are neither "self-referential" (like a sentence equivalent to its own nonprovability) nor imply consistency of PA (like in the Paris Harrington ...
3
votes
1answer
84 views

Looking for counterexamples where the output of a computable function always has a computably checkable property, but PA cannot prove this

Suppose we have a computable function $f$, say over the naturals, and a decidable set $S$ of naturals, such that $f(x) \in S$ for all $x$. In this case, for any specific $x$, there is some specific ...
3
votes
1answer
435 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 ...
2
votes
1answer
182 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 ...
0
votes
2answers
92 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 ...
-4
votes
2answers
77 views

How is Goedel's 1st incompleteness theorem related to the Axioms of a theory

i am thinking of various connections and formulations of Goedel's 1st incompleteness theorem. Apart from connections to Turing's Halting Problem and Algorithmic Complexity Theory, i am looking for ...