
World's shortest explanation of Gödel's theorem - MartinMond
http://blog.plover.com/math/Gdl-Smullyan.html
======
idlewords
Here's my attempt:

Gödel figured out how to encode the statement "you can't prove me" as an
arithmetical expression. Minds blew.

~~~
DougBTX
Can you point me to an explanation of the arithmetical expression itself?

~~~
idlewords
[http://books.google.com/books?id=FtclgZ1yYDEC&printsec=f...](http://books.google.com/books?id=FtclgZ1yYDEC&printsec=frontcover&dq=godels+theorem&ei=bwglS8GONoqGzAS07NCZCw&cd=1#v=onepage&q=godels%20theorem&f=false)

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mdemare
Wow, that was a disappointment. I've always thought of Gödel's incompleteness
theorem as something so arcane that I'd never understood it. Now it turns out
to be both trivial and really similar to the Halting problem.

Good explanation though.

~~~
anatoly
This isn't a proof of Goedel's incompleteness theorem. It's a demonstration of
Tarski's undefinability theorem. Goedel's theorem is far stronger and more
difficult to prove.

The confusion between the two is widespread.

Goedel's first incompleteness theorem doesn't talk about "true" or "false"
statements at all. Technically speaking, it is entirely syntactic, not
semantic. It says that in any sufficiently complex axiom system there will be
statements that it can neither prove nor refute, "undecidable" statements. The
notions of "prove", "refute", "axiom", "undecidable" do not require any
definition of "true" or "false".

~~~
joe_the_user
Correct - he's referencing Tarski not Goedel.

Incorrect - Goedel is not much harder to prove or wider-ranging than Tarski.

Goedel is proven as a corollary to Tarski in Manin, Course In Mathematical
Logic (Springer - [http://www.amazon.com/Course-Mathematical-Logic-Graduate-
Mat...](http://www.amazon.com/Course-Mathematical-Logic-Graduate-
Mathematics/dp/0387902430))

Strongly recommend Manin to the mathematically literate interested in a
sophisticated and worldly introduction to logic. Manin gives very
understandable details of the Smullyan proof this guy is taking bits and
pieces of.

\-- And Antoly's "semantic/syntax" distinction is overdone.

~~~
anatoly
Well, let's agree to disagree on the recommendation of Manin's book. I think
it's confusing, rambling and often downright unhelpful.

The reason Godel's theorem is proved as (little other than) a corollary to
Tarski's in Manin's book is that Manin presents a weaker, semantic version of
Godel's theorem that is indeed little other than a corollary to Tarski's. See
Manin's discussion of "Godel's argument" and particularly step (c) on p.255 of
his book. Manin explains that Godel's proof doesn't use the assumption that D
is a subset of T (all provable statements are true), but that he, Manin,
doesn't think it's important to care about that, so he presents the theorem
this way. In the form Godel proved it, the theorem doesn't require any
semantic assumptions.

Manin is entitled to his own spin on what is and isn't important about the
Incompleteness Theorem, but it should be apparent to you that his presentation
uses an assumption that Godel's theorem in its general form doesn't require.
It's interesting (where by interesting I mean "weird, and goes toward
explaining my opinion of Manin's book") that Manin nowhere even deigns to
mention Godel's _Second_ incompleteness theorem, the one about axiomatic
systems being unable to prove their own consistency. Now if you were to take
up a better book than Manin's, say Smullyan's _Godel's Incompleteness
Theorems_ , and study the proof of the second incompleteness theorem, you'd
find out that its proof _requires_ a syntactic proof of the first theorem, the
proof that Manin says isn't important. This is because proving the second
theorem entails formalizing the proof of the first theorem inside the
axiomatic system itself, and you can't do that with a proof that uses semantic
arguments and talks about true and false theorems (why? because of Tarski's
undefinability of truth!). Think of that what you will.

~~~
joe_the_user
Well, I think I'm starting to understand your take on the subject. Thank you
for putting in the effort to explain it (where I agree or not).

It seems like the issue is whether the case of D not being a subset of T is
"interesting" or not.

I think Manin has a point in sense that if we know that D is not a subset of
T, then we are in a rather pathological case.

But I can see your point in the sense that if it's impossible to prove within
an axiomatic system that the system itself is inconsistent, then one has to
keep a situation like D not being a subset of T in mind.

It's very roughly speaking, the computational versus the syntactic view of
logic...

~~~
anatoly
Thank you for the kind words. Understanding is always more valuable than
agreement.

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dbz
Okay. I have a problem maybe someone can help me with.

The article says:

 _"If the machine prints NPRNPR, it has printed a false statement. But if the
machine never prints NPRNPR, then NPRNPR is a true statement that the machine
never prints."_

This part seems logically true:

 _"If the machine prints NPRNPR, it has printed a false statement."_ It seems
logically true, and I wont say why because I view it as obvious =/

This part seems logically false:

 _"if the machine never prints NPRNPR, then NPRNPR is a true statement"_ I
don't understand the logic of "If it is not printed, it is true." I was under
the impression that the only true statement was "If it prints, then it is
true", and of course the contrapositive: "If it is not true, it doesn't
print". The statement taken from the article is NOT the contrapositive. It is
the inverse, and because the converse of a statement is not always true, the
inverse is not always true because the inverse is always equal to the
converse. _"if the machine never prints NPRNPR, then NPRNPR is a true
statement"_ That statement is the inverse of the original if-then statement,
and the inverse/converse is FALSE because it allows opportunities which
violate the original if-then statement.

I logically concluded that the statement in the proof:

 _"if the machine never prints NPRNPR, then NPRNPR is a true statement"_ is a
false statement. And therefore the proof is incorrect.

Okay. If I am wrong, can someone please tell me where?

~~~
evo
The truth of the statement does not derive from being an inverse, converse, or
contrapositive of the statement before it. Rather, it is inferred from what we
defined "NPR" to mean.

Since "NPR" is defined as true if what follows it is never printed twice in
succession, and "NPRNPR" is never printed as a given, then "NPRNPR" as a
statement is true.

~~~
dbz
Ah. Wonderful. I see now. Thanks a lot =]

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rjprins
Gödel's theorem:

If you print only true statements, you can not print all true statements.
Proof:

"I do not print this statement"

~~~
anatoly
This is a version of the Liar paradox, not Goedel's theorem.

~~~
dejb
They are pretty close to a 1:1 mapping

[http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_t...](http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Relation_to_the_liar_paradox)

'The analysis of the truth and provability of G is a formalized version of the
analysis of the truth of the liar sentence.'

~~~
anatoly
This is only approximately true. What really is a straightforward formalized
version of the liar paradox is Tarski's undefinability theorem, which is
what's demonstrated in the blogpost.

Goedel's incompleteness sentence is more like inspired by the liar sentence
rather than being its formalization; and the proof of Goedel's incompleteness
is much more challenging than the simple true->false->true bounce of the liar
paradox.

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statictype
While we're on the topic, something that always puzzled me: doesn't Gödel's
theorem apply only to 'non-trivial' axiomatic systems?

Meaning you could come up with a trivial system to which it doesn't apply.
Which begs the question: How do you define triviality here?

~~~
idlewords
Trivial in this context means a system that's not powerful enough to express
basic arithmetic.

This includes some surprisingly complicated systems, like real (as opposed to
integer) arithmetic.

Bonus pedantipoint: it's not begging the question - that actually means 'to
assume X in your proof of X'. It's just raising the question.

~~~
DoctorToWhom
_This includes some surprisingly complicated systems, like real (as opposed to
integer) arithmetic._

Real arithmetic is not enough to express integer arithmetic?

I suppose it all boils down to how you define "arithmetic", but what is it
that the integers have that the reals lack under the relevant definition? I
would think that real arithmetic by any reasonable definition would give you
enough tools to check whether a number was an integer, and in that case,
doesn't it necessarily contain integer arithmetic as a subset?

~~~
idlewords
Cribbing from this book:

[http://books.google.com/books?id=71pK8Zz9Dd8C&printsec=f...](http://books.google.com/books?id=71pK8Zz9Dd8C&printsec=frontcover&dq=godels+theorem&ei=nhglS777Non4zAS3n62uCw&client=safari&cd=2#v=onepage&q=godels%20theorem&f=false)

"Since the natural numbers form a subset of the real numbers, it may seem odd
that the theory of real numbers can be complete when the theory of the natural
numbers is incomplete. The incompleteness of the theory of the natural numbers
does not carry over to the theory of the real numbers because even though
every natural number is also a real number, we cannot _define_ the natural
numbers as a subset of the real numbers [...]

How would we ordinarily define the natural numbers as a subset of the real
numbers? The real numbers 0 and 1 can be identified with the corresponding
natural numbers, and using addition of real numbers we get the natural numbers
as the subset of the real numbers containing 0, 1, 1+1, 1+1+1, and so on.
However, this "and so on" cannot be expressed in the language of the theory of
real numbers [because it requires the notion of sets, a second-order concept
outside of the theory]."

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swixmix
From the article, "Either the machine prints NPR@NPR@, or it never prints
NPR@NPR@."

This makes sense in that the machine has told itself never to print NPR@NPR@
in the future.

It would be like trying to verify someone doesn't know a secret by asking them
about the secret they shouldn't know.

Zero knowledge proofs come to mind.

And that's probably a good description of my knowledge on this subject.

edit: changed asterisks to @ because HN doesn't display asterisks.

------
RiderOfGiraffes
The article in Wikipedia is pretty good, although stops short of the full
details. In particular it addresses this question of the Liar's Paradox versus
Gödel's theorem "proper".

[http://en.wikipedia.org/wiki/Goedel%27s_incompleteness_theor...](http://en.wikipedia.org/wiki/Goedel%27s_incompleteness_theorems#Relation_to_the_liar_paradox)

