Questions about Gödel's incompleteness theorems and related topics.

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38
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7answers
3k 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 ...
45
votes
5answers
8k 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 ...
11
votes
5answers
2k 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?
55
votes
12answers
7k views

What is a simple example of an unprovable statement?

Most of the systems mathematicians are interested in are consistent, which means, by Gödel's incompleteness theorems, that there must be unprovable statements. I've seen a simple natural language ...
22
votes
7answers
4k 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 ...
78
votes
14answers
29k views

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

Can you explain it in a fathomable way at high school level?
11
votes
4answers
462 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 ...
15
votes
1answer
688 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 ...
63
votes
7answers
5k views

Is linear algebra more “fully understood” than other maths disciplines?

In a recent question, it was discussed how LA is a foundation to other branches of mathematics, be they pure or applied. One answer argued that linear problems are fully understood, and hence a ...
14
votes
2answers
2k 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 ...
6
votes
2answers
233 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
3answers
433 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 ...
4
votes
3answers
324 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 ...
19
votes
6answers
4k 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 ...
11
votes
2answers
403 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 ...
8
votes
2answers
352 views

Proof of Proposition/Theorem V in Gödel's 1931 paper?

Proposition V in Gödel's famous 1931 paper is stated as follows: For every recursive relation $ R(x_{1},...,x_{n})$ there is an n-ary "predicate" $r$ (with "free variables" $u_1,...,u_n$) such that, ...
7
votes
1answer
91 views

What's wrong with this proof of ZF being inconsistent?

I'm studying (independently) mathematical logic and in investigating self-referential statements I developed a result which I don't know how to interpret. I'll use the notation from Enderton's "A ...
5
votes
3answers
781 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 ...
4
votes
1answer
399 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 ...
-8
votes
2answers
196 views

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

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 ...
7
votes
1answer
472 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
985 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 ...
12
votes
6answers
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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 ...
12
votes
2answers
780 views

Where, specifically, did Principia Mathematica fail?

I'm very fascinated by the book Principia Mathematica. From what I've learned so far, Principia Mathematica set out to be, essentially, the bible of mathematics and logic, from which all mathematical ...
14
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 ...
9
votes
3answers
2k 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
2k 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 numbers....
2
votes
1answer
384 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 ...
8
votes
3answers
2k 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
3answers
2k views

are there non-standard models of arithmetic in second order arithmetic?

non-standard models of arithmetic in second order arithmetic? Background: According to Godel's theorem, if we have, in a given consistent system S, a non-provable wff. A, then we can extend the ...
4
votes
1answer
186 views

What does a Godel sentence actually look like?

My understanding is that Godel's first theorem says that there is a sentence that is in a sense true in a formal system F but cannot be derived in that system. I then hear that Godel actually goes on ...
3
votes
2answers
906 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 ...
6
votes
5answers
882 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 ...
6
votes
1answer
108 views

Absolutely undecidable statements in Peano arithmetic

Let "Undec(x)" be a predicate in Peano Arithmetic that says "x is the Gödel number of a sentence that is neither provable nor refutable" It is easy to see that this predicate is in fact expressible in ...
1
vote
1answer
120 views

Reference request for the independence of $ \text{Con}(\mathsf{ZFC}) $.

Gödel’s Second Incompleteness Theorem says that if $ \mathsf{ZFC} $ is consistent, then $ \mathsf{ZFC} \nvdash \text{Con}(\mathsf{ZFC}) $, i.e., $ \text{Con}(\mathsf{ZFC}) $ is not provable in $ \...
6
votes
2answers
296 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 ...
5
votes
1answer
134 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 ...
4
votes
1answer
96 views

Integer induction without infinity

In ZFC minus infinity (let us call this T), one can still define ordinals, and then define integers as ordinals all of whose members are zero or successor ordinals. I am looking for a formula $\psi$ ...
4
votes
3answers
158 views

Why can PA + $\neg G_{PA}$ be consistent?

Wikipedia and other sources claim that $PA +\neg G_{PA}$ can be consistent, where $\neg G_{PA}$ is the Gödel statement for PA. So what is the error in my reasoning? $G_{PA}$ = "$G_{PA}$ is ...
3
votes
1answer
113 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
2answers
2k 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 ...
2
votes
1answer
476 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: ...
2
votes
1answer
157 views

Con(PA) implies consistency of $\mathsf{PA}$ + ¬Con($\mathsf{PA}$)

The Wikipedia article for $\omega$-consistency says "Now, assuming PA is really consistent, it follows that $\mathsf{PA}$ + ¬Con($\mathsf{PA}$) is also consistent, for if it were not, then PA would ...
2
votes
2answers
65 views

Trying to understand self-reference as it relates to Godel's Second Incompleteness Theorem

As noted in this post, I'm trying to understand how a sufficiently powerful consistent theory $T$ can prove statements about itself without contradicting Godel's Second Incompleteness Theorem. Let $...
2
votes
5answers
2k views

Are (some) axioms “unprovable truths” of Godel's Incompleteness Theorem?

Like any math newbie, Godel's Incompleteness Theorems are easy to understand in general layman's terms, but difficult to understand beyond the typical "liar's paradox" and "barber's paradox" type ...
2
votes
2answers
247 views

clarify the term “arithmetics” when talking about Gödel's incompleteness theorems

I am not quite sure what really is meant when talking about "arithmetics" in context of Gödel's incompleteness theorems. How I so far understand it: Gödel proved that every sufficiently powerful ...
1
vote
1answer
68 views

Infinitely many nonequivalent unprovable statements in ZFC because of Gödel's incompleteness theorem?

Am I right in thinking that there are countably infinitely many nonequivalent unprovable statements in ZFC due to the Gödel's first incompleteness theorem? If not, why?
9
votes
1answer
228 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
211 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 ...
4
votes
1answer
487 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 ...