Questions about Gödel's incompleteness theorems and related topics.

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

Can you explain it in a fathomable way at high school level?
63
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7answers
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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 ...
56
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 ...
45
votes
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 ...
39
votes
7answers
4k 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 ...
22
votes
7answers
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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 ...
22
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3answers
2k views

Why is Gödel's Second Incompleteness Theorem important?

Given that the consistency of a system can be proven outside of the given formal system, Gödel says, It must be noted that proposition XI... represents no contradiction to the formalities ...
21
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6answers
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Does Gödel's Incompleteness Theorem really say anything about the limitations of theoretical physics?

Stephen Hawking believes that Gödel's Incompleteness Theorem makes the search for a 'Theory of Everything' impossible. He reasons that because there exist mathematical results that cannot be proven, ...
21
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2answers
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What is the prerequisite knowledge for learning Godel's incompleteness theorem

I am very interested in learning the incompleteness theorem and its proof. But first I must know what things I need to learn first. My current knowledge consists of basic high school education and the ...
19
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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 ...
15
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1answer
765 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 ...
14
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6answers
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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 ...
14
votes
2answers
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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 ...
13
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
789 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 ...
12
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1answer
459 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 ...
11
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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?
11
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5answers
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Is there a proof of Gödel's Incompleteness Theorem without self-referential statements?

For the proof of Gödel's Incompleteness Theorem, most versions of proof use basically self-referential statements. My question is, what if one argues that Gödel's Incompleteness Theorem only matters ...
11
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4answers
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How does Gödel Completeness fail in second-order logic?

So a while ago I saw a proof of the Completeness Theorem, and the hard part of it (all logically valid formulae have a proof) went thusly: Take a theory $K$ as your base theory. Suppose $\varphi$ ...
11
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2answers
698 views

Is Gödel's incompleteness theorem provable without any model-theoretic notion?

The entry on Gödel's incompletenss theorem in Wikipedia says: Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, ...
11
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4answers
475 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 ...
11
votes
2answers
411 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 ...
11
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1answer
305 views

Problem with completeness theorem and $\mathsf{Con(ZFC)}$

Using the reflection theorem we can prove in $\mathsf{ZFC}$ that for any finite $\Lambda\subseteq\mathsf{ZFC}$ there is some limit ordinal $\gamma$ such that $R(\gamma)\models\Lambda$. Then by the ...
11
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1answer
179 views

Is there a quicker argument from the HBL derivability conditions to the equivalence of fixed points of $\neg\Box$ to $\mathsf{Con}$?

I'm just about to send off the final, final corrected PDF of the second edition of my Gödel book, and the following (neurotic?!?) question occurs to me. In discussing matters around and about the ...
11
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1answer
381 views

Gödel's Second Incompleteness Theorem and Arithmetically Non-Definable Theories

My recursion theory knowledge has become a bit rusty, so I will appreciate any corrections for misstatements. Gödel's incompleteness theorem is often exploited by philosophical discussions which ...
11
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2answers
138 views

Proportion of true statements that are provable

In an axiomatic system, there are true statements, some but not all of which are provable. Is there any sense in which we can quantify the proportion of those statements that are true? Here's an ...
11
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1answer
150 views

Decidability of equality of two set-theoretical terms constructed without replacement or specification

Define the set of NS-terms (NS is for "no schemes") to be the smallest set of terms satisfying the following rules : $\emptyset,\omega$ are NS-terms. if $x$ and $y$ are NS-terms, then so are $x\cup ...
10
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2answers
611 views

Are the Gödel's incompleteness theorems valid for both classical and intuitionistic logic?

I am studying an undergraduate text about math logic. The proofs of the two Gödel's incompleteness theorems are not completely formal: they are admittedly simpler that the real proofs. For what I ...
10
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2answers
735 views

Is there an algorithm that when given a set of axioms, will generate a statement independent of those axioms?

Is there any mechanism or algorithm where one can generate mathematical statements/problems that are undecidable, i.e their proof is independent, from a certain set of axioms?
10
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1answer
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Gödel's way of teaching non-standard models to Takeuti.

In Memoirs of a Proof Theorist, Gaisi Takeuti relates how Gödel taught him about nonstandard models in an "interesting" way: It went as follows. Let T be a theory with a nonstandard model. By ...
10
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1answer
215 views

What was the planned topic of Gödel's second paper on incompleteness?

Gödel's incompleteness theorems first appeared together in a paper titled (translated to English) "On formally undecidable propositions of Principia Mathematica and related systems I," with the Roman ...
10
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1answer
341 views

Freeman Dyson's example of an unprovable truth

Freeman Dyson has claimed that $$\nexists m,n \operatorname{Reversed}(2^n) = m ^ 5 $$ (where $\operatorname{Reversed}(l)$ just is the reverse of the digits of $l$ in base 10), is probably an ...
10
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0answers
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Elementary references on Robinson Arithmetic, Prim. Recursive fns etc.

I'm in the middle of revising my freely available and much-downloaded introductory notes Gödel Without (Too Many) Tears. (They are a sort of cut down version of part of my Gödel book, and I'm ...
9
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2answers
2k 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 consistency....
9
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3answers
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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 ...
9
votes
2answers
293 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 ...
9
votes
4answers
535 views

“The set of all true statements of first order logic”

In one of my lectures, the lecturer put a bunch of examples of sets on the board, stuff like the set of all humans, set of all well typed programs in some programming language, the set of all true ...
9
votes
2answers
319 views

Why is $\omega$-consistency needed in Gödel's original Incompleteness proof?

I don't see why the original version of Gödel's first incompleteness theorem (before Rosser's improvement, I mean) had to include the assumption of $\omega$-consistency in order to show that $F \...
9
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1answer
229 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 ...
9
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2answers
636 views

Is there a mistake in the SEP article about Godel's Incompleteness theorems?

Update: The mistake referred to in this question has now been corrected. The below refers to a previous version of the article: The second supplement to the Stanford Encyclopaedia of Philosophy ...
8
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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 numbers....
8
votes
3answers
363 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 ...
8
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3answers
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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 ...
8
votes
2answers
311 views

How is the standard model of number theory specified, and why can't we use that specification to prove any number theoretical sentence of interest?

According to Gödel's incompleteness theorem, there exists a sentence $G$ in the vocabulary of number theory ($N$) which is not provable from any (recursively enumerable) consistent set of axioms $T$, ...
8
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2answers
313 views

Axiomatic system and Hilbert's 2nd problem

Hilbert's second problem asks if the axioms of arithmetic are consistent. Has this problem been resolved? Shouldn't an axiomatic system ideally be consistent and complete(given that we have the ...
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2answers
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Gödel's (in)completeness theorems and the axiomatization of Euclidean geometry

In David Hilbert's 1899 Grundlagen der Geometrie, Hilbert gives a rigorous axiomatization of Euclidean geometry. As I understand it, some of Hilbert's axioms must be expressed in second order logic (...
8
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2answers
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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, ...
8
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1answer
397 views

Gödelian incompleteness; Smullyan's Puzzle

I am currently doing exercises on the Gödelian theorems; and we are confronted with the introductory puzzle of R. Smullyan's book, which is as follows: Suppose we have a machine which prints strings ...
8
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1answer
351 views

How did Kurt Gödel's Incompleteness Theorem affect the mathematical world?

Hi I am looking not to understand the Incompleteness Theorem, but to find out more about how and what this has effected the mathematics world. I am in high school, in Honors Algebra II, and I am ...
8
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0answers
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Fixed points in computability and logic

I asked this question on CS.SE, too: http://cstheory.stackexchange.com/questions/27322/fixed-points-in-computability-and-logic I would like to understand better the relation between fixed point ...