Can you explain it in a fathomable way at high school level?

5Describe, sure. Give a genuine sketch of a proof, no. – André Nicolas Jul 27 '13 at 15:48

Can you please describe it leaving the proof part? – Hyperbola Jul 27 '13 at 15:51

58Hilbert: "We must know, we will know." Godel: "We won't know, we can't know." – RghtHndSd Jul 27 '13 at 15:51

8There are some books by Smullyan that do a good job of this, http://en.wikipedia.org/wiki/Raymond_Smullyan#Logic_puzzles – Will Jagy Jul 27 '13 at 15:53

2I once attempted a [blog post](http://andrescaicedo.wordpress.com/2010/08/23/toinfinityandbeyondthestruggletosavearithmetic/) on a series of closely related topics, mostly as a consequence of an article by Richard Elwes on recent work of Harvey Friedman. You may find it interesting. – Andrés E. Caicedo Jul 27 '13 at 15:55

I suggest reading the book: Uncle Petros and Goldbach's Conjecture. It is a novel and great mathematical read. Gödel's Incompleteness Theorem comes up in it and you will better understand it after you finish the book. – zerosofthezeta Jul 27 '13 at 16:31

1Spelling note: Either "Goedel" or "Gödel" is correct, and these two ways of writing it are essentially the same. "Godel", on the other hand, is different, and would make the pronunciation different. I think the same some speakers of English whose pronunciations are nonrhotic pronounce "girdle" the same way that Gödel pronounced his name. A rhotic American pronunciation of "girdle" is probably as close as you can get within rhotic American pronunciations. – Michael Hardy Jul 27 '13 at 17:44

My answer emphasizes the connection with _computability_. I think that's often not been emphasized enough. – Michael Hardy Jul 27 '13 at 17:45

Many of the answers talk about "arithmetic"; they are specifically referring to *integer* arithmetic (or an equivalent); e.g. that you can formulate statements like "$2x=1$ doesn't have a solution that is an integer". This is important because, e.g., the firstorder theory of real number arithmetic (called "real closed fields") cannot express such a statement, and so Godel's incompleteness theorems do not apply to it. And, in fact, the theory of real closed fields *is* complete! – Jul 27 '13 at 18:41

There is a fantastic, understandable but technical introduction to incompleteness in "The New Turing Omnibus", which is a book worth reading anyway. – N. McA. Jul 27 '13 at 20:57

13I recommend to read the ["Gödel, Escher, Bach: An Eternal Golden Braid"](https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach) by Douglas Hofstadter, I found it very, *very* interesting and, also, related to the topic. Surprised no one have recommend it yet... – m0nhawk Jul 28 '13 at 07:02

I'd recommend reading [Forever Undecided: A Puzzle Guide to Godel](http://www.amazon.com/ForeverUndecidedPuzzleOxfordpaperbacks/dp/0192821962) by [Raymond Smullyan](http://en.wikipedia.org/wiki/Raymond_Smullyan). Easy to read made sense to me even when I was still in high school. – Warren Hill Jul 28 '13 at 18:28

1[Here](http://math.stackexchange.com/q/16358/462) is an excellent discussion. – Andrés E. Caicedo Aug 21 '13 at 03:23

The fact that nobody has been able to provide a "high school level" explanation to your question is a fairly sufficient (and unironically quite hilarious) description of Godel's Incompleteness Theorems in itself. – Martin Erlic Feb 14 '18 at 23:48
13 Answers
The problem with Gödel's incompleteness is that it is so open for exploitations and problems once you don't do it completely right. You can prove and disprove the existence of god using this theorem, as well the correctness of religion and its incorrectness against the correctness of science. The number of horrible arguments carried out in the name of Gödel's incompleteness theorem is so large that we can't even count them all.
But if I were to give the theorem in a nutshell I would say that if we have a list of axioms which we can enumerate with a computer, and these axioms are sufficient to develop the basic laws of arithmetics, then our list of axioms cannot be both consistent and complete.
In other words, if our axioms are consistent, then in every model of the axioms there is a statement which is true but not provable.
Caveats:
 Enumerate with a computer means that we can write a program (on an ideal computer) which goes over the possible strings in our alphabet and identify the axioms on our list. The fact the computer is "ideal" and not bound by physical things like electricity, etc. is important. Moreover if you are talking to high school kids which are less familiar with theoretical concepts like that, it might be better to talk about them at first. Or find a way to avoid that.
 By basic laws of arithmetics I mean the existence of successor function, addition and multiplication. The usual symbols may or may not be in the language over which our axioms are written in. This is another fine point which a lot of people skip over, because it's usually clear to mathematician (although not always), but for high school level crowd it might be worth pointing out that we need to be able to define these operations and prove they behave normally to some extent, but we don't have to have them in our language.
 It's very important to talk about what is "provable" and what is "true", the latter being within the context of a specific structure. That's something that professional mathematician can make mistakes with, especially those less trained with the distinction of provable and true (while still doing classical mathematics, rather some intuitionistic or constructive mathematics).
Now, I would probably try to keep it simple. VERY simple. I'd talk about the integers in their standard model, and I would explain the incompleteness theorem within the context of this particular model, with a very specified language including all the required symbols.
In this case we can simply say that if $T$ is a list of axioms in our language which we can enumerate by an ideal computer, and all the axioms of $T$ are true in $\Bbb N$, then there is some statement $\varphi$ which is true in $\Bbb N$ but there is no proof from $T$ to the statement $\varphi$.
(One can even simplify things better by requiring $T$ to be closed under logical consequences, in which case we really just say that $\varphi\notin T$.)
 2,047
 11
 12
 370,314
 41
 552
 949

2

4Truth is relative to a structure. Proof is relative to a theory (and an underlying logic). "$\sqrt2$ exists" is true in the context of $\Bbb R$ or $\Bbb C$, but not in the context of $\Bbb Q$ (of course in all cases, the context is a field). – Asaf Karagila Sep 09 '18 at 13:39

Can you give example of a statement which is known to be true in N but which can not be proved starting from, say, Peano axioms ? – user10001 Dec 09 '18 at 09:38

2@user10001: The simplest one would be something like "PA is consistent" which we can turn in a fairly robust way into a statement like that. Other examples also include some strong Ramsey theorems (see the Paris–Harrington theorem), Goodstein's theorem, Kruskal's tree theorem, and there are others as well. – Asaf Karagila Dec 09 '18 at 09:42

@Mark: Try to avoid bumping questions from aeons ago if your edits are bordering on trivial. – Asaf Karagila Nov 22 '20 at 09:25
${\underline{First}}$: The statement of Gödel's theorem(s) requires some knowledge of what axioms and proofs are. To really get a good feeling for what this is, I believe that one would need to work with it. So it is difficult to give a simple description of Gödel's theorems here in any short way. One will almost certainly say something that isn't completely (pun intended) true.
I can recommend the book Gödel's Proof by Nagel and Newman. I can also recommend the book Gödel's Theorem: An Incomplete Guide to Its Use and Abuse by Torkel Franzén. Both of these books aren't too expensive and they are fairly easy to read. (Disclaimer: I haven't actually read all of Franzén's book).
${\underline{Second}}: $That said, this is how I would attempt to explain the whole thing. In high school and, hopefully, earlier you learn for example how to add, subtract, multiply, and divide numbers. You might have learned that numbers are good for counting things. If you have a business, you use numbers to keep track of sales and for that we need to be able to "use math" with the numbers.
Now, you have probably also by now learnt that there are rules in math. We for example now that $4 + 7 = 7 + 4$. The order doesn't matter when we add numbers. One can't, for example, divide by zero. And so it is not hard to understand how math is built up by all the rules for how you are allowed to manipulate numbers.
Now, mathematicians (and other people) have tried to figure out if it is possible to write down a list of axioms and some rules of logic that would "give us all" of mathematics. We in particular want the axioms to "contain" "basic arithmetic" (think of arithmetic as doing math with the natural numbers). An axiom is something that we take for granted. It is a bit like a definition in that we don't have to prove it. An axiom is true because we say so.
With the axioms we have (want) rules for how to deduce new things. From the axioms, we might be able to prove that something is true or false. (If you want to see something concrete, you could look up the "axioms" of what a group is. With these "axioms" one can write down a proof that the socalled identity is unique. So here you have an example where we start with a definition and prove that a statement is true).
Part of what we require is that a statement is either true or false. If I write the statement that $\pi = 22/7$, then that is a false statement. If I write that $\sqrt{2}$ is irrational (as a real number), then that is a true statement. It might not be as obvious. So how would I prove that this is true. Well, using the rules of logic I can prove that from the axioms the statement follows.
Now a question: Since we are saying that all statements are either true or false, can we, given a statement, always arrive at a proof? That is, can we prove that any given statement is either true or false? If we (in theory) can either prove or disprove any given statement, we say that the system is complete. We would of course like a complete system.
Another question is: Can we prove that our system of axioms will never lead to a contradiction? That is, is it possible that we might have a statement that we can prove is true and false at the same time? If this does not happen we say that the system is consistent. Having a consistent system is of course essential. If one statement existed that is both true and false, then you can prove that all statements are true and false. That would be bad.
And with these two questions is problem located.
The answer is: It is impossible to write down a axiomatic system that while being consistent is also complete.
So if, in the "design" of the axiomatic system we want to make sure that we can't have theorems that are true and false at the same time, then we will have statements that we can't write down a proof of. We will have statements/theorems where we can't decide whether the statement is true or false. In fact we will have statements that we can prove that we can't prove i.e. continuum hypothesis

6I can only hope that what you wrote is correct, because I found it to be highly accessible to me (as somebody with a less than impressive math education). The only thing which slightly trips me up: you (mathematicians in general, I guess) talk about 'proving', but what you are *really* doing is deducing implications, right? Like, you are only proving that *if* your axioms are true *then* some statement is also true (or false). – Frerich Raabe Jul 28 '13 at 21:43

...hm, now that I think about it, that's probably what people do in real life when "proving" stuff as well: explaining how things come to be based on things everybody agrees on. – Frerich Raabe Jul 28 '13 at 21:43

4@FrerichRaabe: Thanks. Take what I have written with a grain of salt. It is simplified. As I wrote in the beginning, it is difficult to give a simple explanation. That said, I hope that what I have written will give the right idea of what is going on. And, you are basically right. At the root, mathematicians deduce statements from axioms. We don't talk about *if* the axioms are true, we take(define) them as true. It might be more of a philosophical question, but I would say that we are not necessarily claiming that reality fits the axioms (even though the give a good description of it). – Thomas Jul 29 '13 at 00:51

2

1Given you have recommended some books on the abuse of the theorems, aren't you doing the same by applying them to all axiomatic systems? Isn't the incompleteness theorem only apply to theories where: 1) axioms are recursively enumerable. 2) The theory capable of supporting elementary arithmetic? – Izhaki Dec 28 '15 at 19:22

It would be much more useful to include an example here. E.g. I can't imagine a thing using multiplication/addition/substraction, and stilli being unprovable in those. – HiAngel Jul 26 '16 at 09:12

3@HiAngel: The simplest examples are actually very long sentences. One has to go through the details of the Godel incompleteness theorem to obtain an understanding of a concrete sentence. However, it may be helpful to think about the following. Consider PA Minus (https://en.wikipedia.org/wiki/Peano_axioms#Equivalent_axiomatizations), which more or less captures all arithmetic properties of the natural numbers. It is unable to prove even the very basic fact that every natural number is a multiple of two or one more than a multiple of two (http://math.stackexchange.com/a/1402688/21820). – user21820 Aug 02 '16 at 08:46

1

The Peano axioms are enough to give us arithmetic, right? So they are complete? Then that would mean that they are not consistent, right? – Ovi Jan 17 '18 at 08:50

@Ovi Late to the party, but for posterity: there are *two* things called "the Peano axioms." The *firstorder* version of Peano arithmetic is subject to Godel and is incomplete, although it is powerful enough to (in practice so far) prove the vast majority of the things about the natural numbers we care about. Meanwhile, the *secondorder* version is complete, and indeed pins down $\mathbb{N}$ up to isomorphism (which even the full firstorder theory of $\mathbb{N}$ can't do by the compactness theorem), but is **not** subject to Godel since it's a secondorder as opposed to firstorder theory. – Noah Schweber May 31 '21 at 05:50
I'm not sure it's worth adding yet another discussion, but the following presentation of Gödel's theorem, due to Raymond Smullyan, is extremely simple. Suppose that you have some sort of machine for printing out statements about things. The machine might be some mathematical system, and instead of "printing out" statements it contains methods for "proving" statements. Or it could be a literal machine that actually prints statements on a long roll of paper. It could even be a silk top hat containing slips of paper with claims written on them. Whatever it is, we'll call it $\def\M{\mathcal M}\M$, for "mathematics" or "machine". (Or "mathematician"!)
Now suppose that among the statements that $\M$ might, or might not, print, assert, prove, or contain, are statements in the following forms, with the following interpretations:
$$ \begin{array}{rl} \mathtt{P*}x & \text{“The machine $\M$ will print $x$”} \\ \mathtt{NP*}x & \text{“The machine $\M$ will not print $x$”} \\ \mathtt{PR*}x & \text{“The machine $\M$ will print $xx$”} \\ \mathtt{NPR*}x & \text{“The machine $\M$ will not print $xx$”} \end{array} $$
For example, $\mathtt{P*FOOFOO}$ means that $\M$ will at some point print $\mathtt{FOOFOO}$.
$\mathtt{PR*FOO}$ also means that $\M$ will print $\mathtt{FOOFOO}$.
We would like to find some implementation of $\M$ that prints all possible true statements. (Or, if it's a hat, it should contain slips of paper with all the true statements.) Unfortunately, it is the case that any $\M$ that prints all true statements must also print some false statements as well. Or conversely, any $\M$ that prints only true statements must fail to print some true statements. This is not hard to see: just ask whether $\M$ prints $\mathtt{NPR*NPR*}$. This is an assertion that $\M$ does not print $\mathtt{NPR*NPR*}$. If $\M$ does in fact print $\mathtt{NPR*NPR*}$, it has printed a false assertion. On the other hand, if $\M$ fails to print $\mathtt{NPR*NPR*}$, then $\mathtt{NPR*NPR*}$ is true and $\M$ has failed to print the true assertion $\mathtt{NPR*NPR*}$.
So far everything is fairly simple. The stroke of genius (and the difficult details) in Gödel's theorem is that arithmetic is sufficiently expressive to be able to express $\mathtt{NPR*}x$. This means that any machine (or system) $\M$ that is capable of expressing certain simple arithmetic properties is also able to express the property $\mathtt{NPR*}x$, which asserts that $xx$ will not be produced by $\M$ itself; it can then express a complex statement of arithmetic which amounts to asserting $\mathtt{NPR*NPR*}$. If it prints this complex statement $\mathtt{NPR*NPR*}$ it has printed a falsity; if it fails to print the complex statement, it has failed to print a truth.
Therefore, there is no machine that can print all the true statements of arithmetic unless it also prints some false statements of arithmetic (because one such statement is $\mathtt{NPR*NPR*}$, which says "this machine never prints $\mathtt{NPR*NPR*}$"); there is no logical system that can prove all the true statements of arithmetic unless it also proves some false statements of arithmetic (because one such statement is $\mathtt{NPR*NPR*}$, which means "this system cannot prove $\mathtt{NPR*NPR*}$"); there is no silk top hat that contains slips of paper with all the truths of arithmetic unless it also contains some slips with false statements (because one such statement is $\mathtt{NPR*NPR*}$, which means "the hat does not contain a slip that says $\mathtt{NPR*NPR*}$"), and so on.
It's not at all obvious that such an arithmetic formula could exist, which somehow expresses $\mathtt{NPR*NPR*}$ for a particular machine $\M$, but that is the technical part of the proof, and it's why Gödel is considered a genius.
 467
 2
 9
 62,206
 36
 276
 489

8Oh yes, it *is* worth adding this! And for anyone who wants to follow this up, start with the first chapter of Smullyan's simply wonderful book *Gödel's Incompleteness Theorems*, and keeping reading ... – Peter Smith Jul 28 '13 at 21:28

3I get stuck at $\mathtt{NPR*NPR*}$ : what's the double $\mathtt{*}$? Previous invocations of $\mathtt{*}$ appear to have a variable bound after the last $\mathtt{*}$. Also, in "does *not* print $\mathtt{NPR*NPR*}$."  is there an 'N' too much? – Frank Meulenaar Aug 01 '13 at 09:41


1I do agree with @FrankMeulenaar, you lost me on $\mathtt{NPR*NPR*}$, it has quite little meaning: either you missed *x*, in which case it is hard to make any sense — the left $\mathtt{NPR*}$ would either accept nothing *(the right doesn't return a value)*, or *(which is also the case if you omitted x on purpose)* it considers the left part of expression as symbols, and doesn't print them — then there's no contradiction. – HiAngel Jul 21 '16 at 11:11

1


Arithmetic is a system that can express statements about addition, multiplication, and equality of the natural numbers, such as “For all numbers $x$ and $y$, $(x+y)·(x + y) = x·x + 2·x·y + y·y$” or “$2 + 2 = 5$”. – MJD Aug 28 '19 at 05:45
Asaf, as always sensible, gives a good answer on some basics: for some amplification and further related ideas, you can download the first chapter of my book on Gödel's Incompleteness Theorems at http://www.logicmatters.net/igt/
This is certainly intended to be accessible without significant background knowledge (you just need to know that the negation sign '$\neg$' can be read 'It is not the case that'), and it tells you in informal terms what the Theorems say. So give it a go ... this opening chapter is only seven pages long!
 49,532
 3
 58
 138

11It is nice that you provide your book, but, in the interest of selfcontainment, it would be best if you could bring the relevant ideas here. – robjohn Jul 28 '13 at 10:30

14That's a rather silly comment. There would have been precisely zero point, wouldn't there, in replicating Asaf's perfectly sensible and perfectly mainstream quick remarks at similar length. The link is evidently for those who would welcome a somewhat more expansive treatment along similar lines. – Peter Smith Jul 28 '13 at 11:59

2"Book" might be misleading here, the [freely available chapter](http://www.logicmatters.net/resources/pdfs/godelbook/SmithGoedelChap1.pdf) the OP is referring to is 7 pages long. – Did Jul 28 '13 at 12:33

14If that is the case, this might have been better as a comment to Asaf's answer. – robjohn Jul 28 '13 at 12:35

1[Chapter 1](http://www.logicmatters.net/resources/pdfs/godelbook/SmithGoedelChap1.pdf) is a hearty and fulfilling read. @robjohn is right, some excerpt belongs here, such as the first paragraph of section 1.4. I now proffer my own conjecture, true and unprovable, that the more you read about Godel the more you feel stoned. Emotions rise, logic feels foggy and defeated. – Bob Stein Jul 28 '13 at 14:31

2I don't know about the rest of a book, but the available 7 pages are disappointing. It is full of marketing praise of the theorem, but doesn't show even a single example. – HiAngel Jul 26 '16 at 09:17
In terms that I'm sure have been put on a tshirt:
 Complete
 Consistent
 Nontrivial
Choose Two
This is more of a wave in the direction of the meaning, but its whatI generally remember, knowing that I can look up the exact definition as required.
Complete: A system is complete if anything that can be stated, can be proved (within the system).
Consistent: A system is consistent if you can never prove a statement and its opposite.
Nontrivial: A system is nontrivial is it can be used for arithmetic calculations.
For example First Order Logic (aka Propositional Logic), is trivial, this means it can be both complete and consistent.
 1,390
 1
 11
 23

6That's a horrible way of explaining Godel's incompleteness, which will assure people will not understand it completely, and its necessarily requirement. Then you get questions (here, and elsewhere) "We know that there are no complete and consistent theories ...". No, that's definitely not a good way of explaining. – Asaf Karagila Jul 28 '13 at 11:50

I admit this. You are correct, but it is still how I would explain it to a layman. – Frames Catherine White Jul 29 '13 at 05:30

3Then please don't. How does the fact that the theory of real closed fields is complete, consistent, and nontrivial? Oh yeah, it is also recursively enumerable. – Asaf Karagila Jul 29 '13 at 10:05

3+1 Good answer, and appropriate for the layman who is looking for a short explanation of this size upon which to ponder. – Matt Sep 05 '13 at 01:15

1@oxinabox, Very good answer for layman. It's absolutely understandable and please continue to go further posting such things. :) – IDOK Mar 13 '14 at 05:44

@AsafKaragila So the statement which I've always assumed to be correct, namely "No consistent formalized theory of arithmetic can be complete" is actually untrue because of the existence of the theory of real closed fields? – Ray Toal Mar 16 '14 at 18:43

@Ray: It depends on what you call "theory of arithmetic". Just any theory of "numbers"? Sure, then it's incorrect. I think that the colloquial definition would be something which interprets "enough" of the theory of the natural numbers, in which case the theory of realclosed fields is not a theory of arithmetic. – Asaf Karagila Mar 16 '14 at 18:47

1In most formulations of Gödel's theorems one uses zero, successor, addition, and multiplication only, so I guess that would be called "Successor, addition, and multiplication of the natural numbers." Do real closed fields operate on a theory much larger than this or much smaller than this? At any rate the answer to the OP's question definitely has to state _exactly_ what is meant by the formal system in question. – Ray Toal Mar 16 '14 at 21:09

There's too much that's wrong here. First, as Asaf pointed out a key condition is missing: that the theory in question be sufficiently simple (= recursively axiomatizable)  the version you've written is false. As a more minor point the statement "First Order Logic (aka Propositional Logic)" is blatantly false: propositional logic is *quite* different from first order logic. Also it's misleading to describe FOL as "trivial:" due to the existence of finitely axiomatizable theories to which Godel applies, FOL is undecidable! This answer unfortunately entrenches common confusions  sorry, but 1. – Noah Schweber May 22 '21 at 18:30
In layman's terms: Any axiomatic system which is capable of arithmetic is either incomplete or inconsistent. An incomplete system is the one in which there are theorems which may be true but cannot be proven. An inconsistent system, on the other hand, is the one in which there are contradictions. However, if we do not want contradictions, every axiomatic system which is capable of arithmetic must be incomplete. Moreover, a "stronger" result is due to Turing, who showed that the process of finding theorems that cannot be proven is undecidable. In general, a problem is undecidable if there exists no algorithm that, in finite time, will lead to a correct yesorno answer. In this case, the problem is undecidable because there is no algorithm that, in finite time, will tell you whether the theorem you want to prove can be proven.
By the way, the proof of Godel's Theorem is essentially a sophisticated Liar's Paradox, and the proof of Turing's Theorem uses Cantor's diagonal argument.
 9,670
 2
 30
 52

3You probably want to say a bit more. The point of having arithmetic is that we can talk about recursive sequences (such as factorials, towers of exponentials, Fibonacci sequences, etc), and any computer program or formal system is essentially producing recursive sequences, so we can use this to talk about proofs, via "coding". This representation allows us to "diagonalize" and so produce statements that are unprovable. – Andrés E. Caicedo Jul 27 '13 at 15:59

2(The point is that it is not really arithmetic that we care about, but the possibility of discussing/mimicking a formal system within the system itself.) – Andrés E. Caicedo Jul 27 '13 at 16:00
One thing not sufficiently emphasized in many accounts (although this may have improved in recent years) is the connection with computability.
In mathematics, the correctness of a proposed proof can be checked by very efficient algorithms.
Actually finding a proof is nowhere near as easy as checking its correctness after it's found.
This is essential because if one didn't insist on algorithmic proofchecking, then one could decide every question as follows: Let every true statement in arithmetic be an axiom. Then every true statement's proof could say "This is an axiom. Qvod erat demonstrandvm. ${}\ \ \blacksquare\ {}$"
The "language of arithmetic" can express things like "Every even number greater than $2$ is a sum of two prime numbers."
Gödel's theorem says you can't have a system of proof for which there is a proofchecking algorithm, and that proves only statements that are true of natural numbers, unless there are some statements in the language of arithmetic that cannot be proved or disproved within that system.
 1
 30
 276
 565

5You might want to replace "Qvod erat demonstrandvm. ${}\ \ \blacksquare\ {}$" by either "Qvod erat demonstrandvm." or "${}\ \ \blacksquare\ {}$". – Did Jul 28 '13 at 07:37

3I think this doesn't quite hit the mark. Contrary to what you imply, there *is* a simple algorithm to find a proof of any provable statement $S$, namely enumerate every possible proof until you find the proof of $S$. The crucial point is that this algorithm doesn't help you decide whether or not *S* is provable in the first place. The content of Gödel's theorem is precisely that you can't decide if $S$ is true by searching in parallel for a proof of either $S$ or $\lnot S$, because there may not be a proof of either one. – MJD Aug 05 '14 at 15:52

@MJD : The difference between what your comment says and what I wrote escapes me. Yes, ever provable statement is provable, and one can find the prove in the way you suggest. But how is that relevant to whether my answer hits the mark? Read my last sentence: "Gödel's theorem says you can't have a system of proof for which there is a proofchecking algorithm, and that proves only statements that are true of natural numbers, unless there are some statements in the language of arithmetic that cannot be proved or disproved within that system." – Michael Hardy Aug 05 '14 at 16:02
An informal but meaningful presentation would be to present Gödel's incompleteness theorems (and JB Rosser's trick) as rigorous versions of Richard's paradox for integers.
Create a list of English definitions of properties of natural numbers, ordered first by length of phrase and then (within phrases of equal length) by dictionary order (lexicographically). Number this list by natural numbers, starting with 1.
A natural number $n$ is said to be Richardian if it does not have the property defined by the $n$th phrase on the list. Since being Richardian is a property of natural numbers, it must appear somewhere on the list. For the $n$ corresponding to this definition, one asks if that natural number $n$ is Richardian. A paradox ensues. If this $n$ were Richardian, then it would not have the property of being Richardian. But as well, if this $n$ were not Richardian, it would mean that it was. Contradiction.
Because the English language is not formal (we have just introduced a "new" word Richardian to mean something that presumbably would not be known to any significant fraction of native English speakers), the paradox just outlined is open to a number of criticisms. However the version Gödel gives involves a numbering of proofs in a formal theory of arithmetic, and so far no one has produced an invalidating criticism of his argument. It quickly became an accepted result in the foundations of mathematics.
 35,235
 19
 69
 133
Perhaps not an actual answer, unless you stretch the term "layman" a bit. These are some excerpts from one of Scott Aaronson's lectures from his Quantum Computing course.
The Incompleteness Theorem says that, given any consistent, computable set of axioms, there's a true statement about the integers that can never be proved from those axioms. Here consistent means that you can't derive a contradiction, while computable means that either there are finitely many axioms, or else if there are infinitely many, at least there's an algorithm to generate all the axioms.
(I think it can qualify as high school level, providing that you did not skip math classes)
The proof is actually not that hard, if we cheat and use some (semi)modern advances in mathematics:
First, though, let's see how the Incompleteness Theorem is proved. People always say, "the proof of the Incompleteness Theorem was a technical tour de force, it took 30 pages, it requires an elaborate construction involving prime numbers," etc. Unbelievably, 80 years after Gödel, that's still how the proof is presented in math classes!
Alright, should I let you in on a secret? The proof of the Incompleteness Theorem is about two lines. The caveat is that, to give the twoline proof, you first need the concept of a computer.
...
As I said, once you have Turing's results, Gödel's results fall out for free as a bonus. Why? Well, suppose the Incompleteness Theorem was false — that is, there existed a consistent, computable proof system $F$ from which any statement about integers could be either proved or disproved. Then given a computer program, we could simply search through every possible proof in $F$, until we found either a proof that the program halts or a proof that it doesn't halt. (This is possible because the statement that a particular computer program halts is ultimately just a statement about integers.) But this would give us an algorithm to solve the halting problem, which we already know is impossible. Therefore $F$ can't exist.
I'm just going to be honest with you. Hopefully you don't find this reply too brash.
If you want a true appreciation for Gödel's, you have to do the intellectual legwork and learn propositional and predicate logic. With that under your belt, you can read a book like Nagel, Newman and Hofstadter's "Gödel's Proof" to really get to grips with it. Gödel's Incompleteness Theorems just aren't simple enough to capture their essence in a Layman's explanation. As always, there is no royal road to geometry (or number theory!).
If you want a gentle introduction to the ideas behind Gödel's Incompleteness Theorems, there's a wealth of explanations written "for the layman" such as Logicomix, various websites, popular mathematics books, etc. I don't feel these materials generally convey the heart of the matter. It's like the difference between a postage stamp depicting the Sistine Chapel, and the Chapel itself.
Alright, all that being said, the most compressed explanation I can come up with is this: Right around the turn of the 19th century into the 20th, mathematicians such as Bertrand Russel, Gottlob Frege, Guissepe Peano and others were enamoured with the idea that logic and mathematics were, at the most basic level, the same thing. That is, all true mathematical statements could be reduced down to and derived from the laws of logic. The subtleties of the laws of logic are often difficult for people to grasp on first contact. This is why it's so important to know logic before you can proceed.
Now, skipping over a whole lot of stuff, Gödel showed that this notion was mistaken in every important sense. There are statements in mathematical logic which are of the form "I am not a provable statement"; which if false lead straight into paradox, and if true represent a dislocation between logic and mathematics  a true mathematical statement that cannot be derived from the laws of logic alone. These statements are close cousins of the ordinary language "liar paradox" which is the sentence "this sentence is false". If it's true, then it's false, and if it's false, then it's true. Similarly, if the statement "I am not provable" is true, then there's a hole in mathematics that logic can't fill. If it's false, then it's true that it's IS provable, implying it must also be unprovable, etc.
Hope that helps.
 110
 6
The following helped me with the First Theorem. I'll leave it to somebody else to add the Second.
We can use math to talk about the real world. Even though we don't find perfect points, lines, etc. outside of math class, geometry is still useful because large parts of it crossapply to the real world. My favorite example is conic sections and planetary motion, but you can find plenty of other instances.
Turns out, we can "go meta" with this. One specific part of the real world is "mathematicians doing mathematics", and we can use math to talk about that.
Gödel's First Incompleteness Theorem is sort of the opposite of the infinitemonkeys scenario. An infinite number of mathematicians working for an infinite amount of time will never resolve all the questions you can put in front of them. Try as they might, as long as they play by the rules of math (and are working on a minimally complicated part of it) you can always put a question in front of them that they just cannot prove or disprove.
At this point, remember that there are many diverging views on the significance of that fact. I personally don't think you can judge between those views without diving into the relevant literature.
 119
 1
There are lots of sets of rules: school dress codes, traffic laws, the rules of Calvinball, etc. These are all valid sets of rules, but they're arbitrary  new rules can be made up that completely ignore old ones. Godel studied sets of rules where every new rule is a combination of older rules (like math where you use basic definitions to prove new rules), and he proved two theorems about them.
Godel's first theorem says that one of the following two things must be True about every set of rules that meet his conditions:
 There must be rules that you can never prove
 There must be rules that you can never express
Which just says that if you can prove everything you can write, then you must not be able to write everything. That doesn't seem so bad  there are things we can't write in English (e.g., what colors look like, or what emotions feel like), so why should math be any different?
You can also turn that around to say that if you can write everything, then you won't be able to prove some of the things you can write. That doesn't seem so bad either, there are lots of things we can write that we can't prove (e.g., The Liar's Paradox).
His second theorem says that if you try to define a set of rules that includes the statement:
Every rule can be proven True or False.
then that rule must be False. His second theorem just means that you can't cheat your way out of the first theorem.
 69
 6
 127
 2
I can't, but Professor Eric C.R. Hehner did, in his article "Beautifying Gödel", chapter 18 in Feijen, van Gasteren, Gries, Misra (eds.): Beauty is our Business, SpringerVerlag silver series, New York, 1990, p.163172 (PDF via his Publications List).
 5,499
 3
 29
 58