
What Made Gödel’s Incompleteness Theorem Hard to Prove - williamkuszmaul
https://algorithmsoup.wordpress.com/2018/10/01/what-made-godels-incompleteness-theorem-hard-to-prove-its-about-how-you-say-it-not-just-what-you-say/
======
edanm
From the article:

> Here’s another example (known as the Berry paradox):

> Define {x} to be the smallest positive integer that cannot be defined in
> under {100} words.

> This might look like a valid mathematical definition. But again it’s
> nonsensical. And, importantly for the sanity of mathematics, no analogous
> statement can be made mathematically formal.

I thought it was worth mentioning that the Berry Paradox _does_ show up in a
mathematically formal way, and it was fascinating to me. It can be used to
prove that a general algorithm to compute the Kolmogorov complexity of a
string is uncomputable.

[https://en.wikipedia.org/wiki/Berry_paradox#Formal_analogues](https://en.wikipedia.org/wiki/Berry_paradox#Formal_analogues)

------
orcs
I felt like the article was just getting going, then it finished.

~~~
nielsbot
i was ready to roll up my sleeves, expecting “and here’s a walkthrough”....
and that’s how i felt.

~~~
dajohnson89
article had no teeth whatsoever.

------
dandare
A layperson here, I always wanted to ask this question: Godel proves that not
ALL statements that are true can be proven, because here is this one example.
I get it, it's a valid mathematical proof. But does it really say anything
about other statements, except this one? Does it says anything about
statements that are not self-referential? Is it possible that the
incompleteness theorem really applies only to self-referential statements and
says nothing about non-self-referential statements?

~~~
icen
The proof of the incompleteness theorem only shows you that recursively-
enumerable arithmetic systems are always missing _something_ , because the
construction always holds. It doesn't make any grand statements about the
structure or nature of things that are true and without proof.

The second incompleteness theorem gives an example of something undecidable
that's probably more interesting that the uselessly paradoxical statement in
the first: it says that no system can consistently prove its own consistency.

------
vyodaiken
What made the theorem hard is not the details, which are simple, it's the
conceptual leap to thinking about mathematical statements as mathematical
objects - something that is much easier for us than for mathematicians of
Godel's time because we are familiar with programs and computers. Much of the
text is taken up with proving things that were surprising to mathematicians of
the era, but are now commonplace: e.g. that we can treat numbers as encoding
mathematical propositions so that for every proposition P expressible in some
formal logic L, there is a number #P which encodes P in some unambiguous way -
as long as L is expressive enough to formalize some basic arithmetic. Then we
can start making trouble by asking if we can construct a proposition Q so that
Q(#P) is true iff the proposition #P encodes, P, is provable in our system. If
so, we should be able to prove Q(#P) or NOT Q(#P) if L is complete, but things
don't work out well because it's possible to encode paradoxical statements. In
short.

~~~
thrmsforbfast
_> thinking about mathematical statements as mathematical objects..._

Lock yourself away in a room with nothing but pen and paper (no internet, not
books) and try to reproduce Goedel's original proof. Then post the attempt
here :)

When I tried this a few years ago I made a mistake in the definition of
numbering, got through half the proof, ran into trouble, and gave up before I
figured out exactly where the mistake was. And I had already committed the
overall outline and the tactical steps for each part to memory...

Even if you remember the entire outline, it'll be really hard to get the
details right....

And, more importantly, the outline itself is actually _not_ obvious.

 _>... much easier for us than for mathematicians of Godel's time because we
are familiar with programs and computers_

Again, it seems unlikely that this was the sticking point.

E.g. Hilbert understood mathematics as a symbolic game, thought about this
exactly question, and didn't come up with Goedel's proof. And Hilbert was
definitely no dullard.

~~~
repsilat
It's super easy to prove if you assume the Halting Problem is undecidable.
Basically,

1\. Assume every either T or ~T is provable for all theorems T, and that our
logic is consistent. (The opposite of the incompleteness theorem.)

2\. For every Turing machine, there is a proof that it halts or a proof that
it doesn't halt.

3\. A machine that enumerates and validates proofs will find one of those
proofs, so the Halting Problem is decidable -- a contradiction.

Easy peasy. (The proof of the Halting Problem undecidability from the
Incompleteness Theorem is similarly straightforward.)

Godel did his work first though, so he didn't have that machinery (hmm)
available.

~~~
IngoBlechschmid
This proof using Turing machines is indeed very slick! Its elegance is
balanced by its nonconstructiveness: Gödel's proof yields an explicit example
for an unprovable statement (namely a formal rendering of "this statement is
not provable"), while the Turing machine proof doesn't.

Also let me add a remark on a certain subtlety: Step 3 only works if we assume
that, if a claim about the halting behaviour of Turing machines has a proof in
the studied formal system, then it's actually true.

This assumption is believed to be warranted for the standard formal systems of
mathematics such as Peano Arithmetic (PA) or Zermelo–Fraenkel set theory (ZF),
but Gödel's proof also applies to formal systems for which this assumption
doesn't hold, or for which the assumption does hold but cannot be proven in a
weak metatheory.

These systems are typically not very useful as vehicles to carry out
formalized mathematics, but they are nevertheless very interesting and
fundamental to the study of logic. An example for such an anti-real formal
system is PA adjoined by an axiom expressing that PA is inconsistent.

~~~
carlehewitt
As Wittgenstein correctly pointed out, Gödel's proposition "I'm unprovable" is
not a proposition in Russell's system because it leads to inconsistency.
Consequently, Gödel's incompleteness results are not valid for Russell's
system. The proposition "I'm unprovable" _cannot_ be constructed as a fixed
point of P |-> ~|-P because ~|-P has order one greater than the order of P.

~~~
mietek
Could you give a reference for this claim of Wittgenstein’s, please?

~~~
carlehewitt
A reference is:

Ludwig Wittgenstein. 1956. Remarks on the Foundations of Mathematics, Revised
Edition Basil Blackwell. 1978.

There is a discussion here:

[https://hal.archives-ouvertes.fr/hal-01566393](https://hal.archives-
ouvertes.fr/hal-01566393)

------
arithma
Godel Incompleteness Theorem and Turing Halting Problem are two faces to the
same coin, I intuited. It is deeper than I thought, especially that I forgot
about the two godel's incompleteness theorems (not just one.)

Scott Aaranson "popularizes" Kleene's textbook proof of Godel's theorems using
Turing machines in his blog:

[https://www.scottaaronson.com/blog/?p=710](https://www.scottaaronson.com/blog/?p=710)

~~~
scythe
I would say that's a pretty inaccurate analogy. The incompleteness theorem can
be considered a consequence of the halting problem, but not the other way
around. The incompleteness theorem depends heavily on the properties of Peano
arithmetic, as defined by the presence of both addition and multiplication. If
you construct a set of numbers with one defined arithmetic operation
(Presburger arithmetic), the incompleteness theorem does not hold and all
statements can be confirmed or disconfirmed by an algorithm (albeit not
efficiently). The halting problem, by contrast, is much more general, and
recurs in many settings where self-reference is possible.

~~~
johncolanduoni
Godel's completeness theorem rests on a system's ability to _represent_
addition and multiplication, not whether they define it. Basically, it recurs
where reasoning about addition and multiplication is possible within the
internal theory. This is why it applies to e.g. ZF set theory, even though
there is no mention of addition, multiplication, or even numbers in ZF's
definition.

The role of Peano arithmetic is somewhat analogous to the role of the
particular definition of a Turing machine in the proof of the halting problem:
you can easily swap it out with something commensurate and get the same
result.

~~~
scythe
If you can represent something, it's defined. There's no real distinction
there. In _weaker_ systems, this is not possible. ZF proves all of the Peano
axioms, for example.

(Stop pretending to explain things to me that I already know.)

~~~
johncolanduoni
Then what do you mean by "depends heavily on the properties of Peano
arithmetic"? There are lots of axiomatizations of arithmetic that can
represent addition and multiplication that are not equiconsistent with Peano
arithmetic, just like there are lots of definitions of computability that are
equivalent to Turing machines (e.g. the partial recursive functions).

Also when talking about a formal theory distinguishing the definition (i.e.
the axioms) and the theorems is pretty important.

EDIT: I realize now that a specific example would be helpful. Goedel's theorem
can be applied to primitive recursive arithmetic[1], which is neither weaker
nor stronger than Peano arithmetic. Interestingly enough PRA with a small
addition (of broader transfinite induction) can actually prove Peano
arithmetic[2].

[1]:
[https://en.wikipedia.org/wiki/Primitive_recursive_arithmetic](https://en.wikipedia.org/wiki/Primitive_recursive_arithmetic)

[2]:
[https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof](https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof)

~~~
scythe
I mean this:

>represent addition and multiplication

It's the simplest possible explanation. Stop being a pretentious jerk. Yes, I
know about primitive recursive arithmetic. I first learned about it about ten
years ago.

EDIT: Wait a second. Look, I said this:

>depends heavily on the properties of Peano arithmetic, _as defined by
addition and multiplication_

I said it _right there_. How could you not know this? When I said "the
properties of Peano arithmetic", I meant _the presence of addition and
multiplication_.

Let me explain it another way: Goedel's theorem is a theorem about _math_.
Math as it was understood in the 18th century. That's what makes it
interesting. The halting problem is proven with an abstract machine that was
invented, mostly, to use as a basis to form analogies with _other_ machines.
So it's general by construction. (If the integers were not already
interesting, Goedel's theorem would be like proving a variant of the halting
problem for some weird computational structure that has no relevance to
anything and is absurdly cumbersome to prove equivalent to other systems. But
for the integers, the analogy is _why_ it's interesting. Nobody expected the
Goedel numbering.)

Yes, Goedel's theorem applies to all interesting versions of the integers, but
it does not apply to all interesting mathematical systems. IIRC people usually
cite the theory of "real closed fields" or something like that.

~~~
johncolanduoni
Just because I mention something doesn't mean I assume _you_ don't know it.
This is a public forum and although it's Hacker News I still doubt everyone
has heard of primitive recursive arithmetic. You're the only one making
personal statements here. Also I don't know what sort of code I'm supposed to
use to make my arguments without offending you, if PRA is the example I think
illuminates my point I kind of have to write its name whether you know of it
or not.

> I said it right there. How could you not know this? When I said "the
> properties of Peano arithmetic", I meant the presence of addition and
> multiplication.

That's a curious definition of " _heavily_ dependent" on the properties of
Peano arithmetic (when the condition is in fact "any integer arithmetic with
addition and multiplication"), and I'd argue it would be hazardously confusing
to the uninitiated, but that's just semantics. Back to the original point
(that the analogy is poor) I don't see how that breaks the analogy at all.
Just like there are weaker arithmetics that escape Goedel's theorems (I will
not mention their names since that seems to just cause issues), there are
weaker models of computation that dodge the halting problem (again I'm sure
you know of them, the one I'm thinking of off the top of my head rhymes with
schmimative shmecursive shmunctions).

> Let me explain it another way: Goedel's theorem is a theorem about math.
> Math as it was understood in the 18th century. That's what makes it
> interesting.

I think what makes it interesting has nothing to do with the particulars of
what people thought math was in the 18th century. I doubt there are any
mathematicians from ancient Greece to a thousand years from now that would
seriously consider using a logical system incapable of modeling addition and
multiplication as a ground-level theory for the majority of mathematics. They
probably _would_ be fine with one that only killed off Peano arithmetic, just
like we'd be fine with a form of computation that wasn't as limiting as PRFs
but didn't happen to be susceptible to Turing's reasoning. It seems just as
general by construction, and I doubt it is an accident that Goedel using
something so basic.

Basically what I'm saying is just like the fact that the Church-Turing thesis
has held up and all interesting forms of computation and equivalent to Turing
machines, there is an implicit Addition-And-Multiplication-Are-Basic thesis
that claims that systems unable to model addition and multiplication in some
form are fun to visit but not to live in when you're doing mathematics.

------
joker3
While the Berry paradox isn't strictly speaking mathematically sensible, there
are similar ideas that lead to new and less well-known impossibility results.
[https://arxiv.org/pdf/chao-dyn/9406002.pdf](https://arxiv.org/pdf/chao-
dyn/9406002.pdf) has a transcript of an overview of the idea and main results
by the researcher who came up with them, as well as a fairly extensive
bibliography.

There's also a version of Russell's paradox lurking in the standard proof of
Cantor's theorem
([https://en.wikipedia.org/wiki/Cantor%27s_theorem](https://en.wikipedia.org/wiki/Cantor%27s_theorem)).

------
undershirt
I couldn't make it through GEB, but I read a good concise explanation in a
book called Godel's Proof: [https://www.amazon.com/G%C3%B6dels-Proof-Ernest-
Nagel/dp/081...](https://www.amazon.com/G%C3%B6dels-Proof-Ernest-
Nagel/dp/0814758371)

------
carlehewitt
See the following for history and limitations of Godel's results:

[https://hal.archives-ouvertes.fr/hal-01566393](https://hal.archives-
ouvertes.fr/hal-01566393)

------
misiti3780
I just read "I Am a Strange Loop" and he the author a pretty nice job of
explaining this -- Unfortunately, I could never get through GEB so I do not
know if the explanation in there.

~~~
fredgrott
its there, about the middle of the book..

------
technocratius
For those interested, Hoffstadter's Godel Escher Bach gives a great
introduction into formal systems and Godels incompleteness theorem. Highly
recommended.

~~~
afthonos
I read it cover to cover out of a sense of obligation but came away very
disappointed. The Achilles-Turtle-Random Other Creature allegories got very
tiresome, and the content took a very long time to get to the point. For
anyone interested in the proof, I recommend Gödel’s Proof by Nagel instead. It
covers the same material in about a tenth the space.

(GEB has other philosophical discourses that are interesting, if only as an
example of how people can very reasonably mispredict the future. In
Hofstader’s case, he posited that chess would only be played at a human level
by an AGI.)

~~~
et1337
Absolutely agree. GEB constantly tries to prove the cleverness of the author,
complete with winks and nudges. It stacks so many unintuitive analogies on top
of each other that I spent most of the time trying to recall which real-world
concepts they all mapped to.

Also, the book is about 98% Gödel and 2% Escher and Bach. Any time it steps
outside its wheelhouse of math and into the realm of philosophy or art, you
get the distinct sense that it has no idea what it's talking about.

Do yourself a favor and just take discrete math 1 and read a few Wikipedia
articles instead.

~~~
mhneu
_Do yourself a favor and just take discrete math 1 and read a few Wikipedia
articles instead._

+100, exactly. I have some familiarity with mathematical logic, and with Bach
(though admittedly I'm less familiar with Escher) and I found GEB to be
unintuitive at best, and misleading at worst.

To be honest, there may not be anything really magic about Goedel's
incompleteness theorem once you grasp the core concept, which is: If you
encode all proofs into a formal language, you can make a statement that is
true but unprovable. (Which is, effectively: "This statement has no proof").
Most of Goedel's work on the incompleteness theorem was on the systematization
of proofs, which can be tedious to work through.

I found at the time Godel's completeness theorem to be more interesting.
Though I never got into model theory or more advanced mathematical logic so,
grain of salt.

------
7dare
I feel like the further you go in maths, the less attention is given to
details, which are "left as exercise to the reader".

~~~
dan-robertson
This depends a lot on the branch of maths: eg topology has much more of a
reputation for being wishy-washy than analysis.

It also depends what the context is. A paper is not meant to prove things
sufficiently for a layperson but for the author’s peers and so an omitted
proof means one of “proven elsewhere;” “follows in a straightforward way from
the definitions;” “follows in a straightforward way from an earlier
proposition;” or “follows from a well-known pattern in the subfield.” And here
“follows in a straightforward way” typically means something between “just
look at;” “at each step there is only one reasonable thing to do so just do
that;” and “apply all the standard tricks from the subfield and see what
sticks.”

It is not true that advanced mathematics is less rigorous than more elementary
maths, it’s just that materials for advanced topics require more domain
knowledge, and provide less hand-holding.

~~~
SolarNet
And I feel like this is where better hyperlinking of papers could be used.
Like not just citations, but the ability to click through to the "proven
elsewhere" perhaps with a short paragraph of how the given citation proves it.

~~~
abdullahkhalids
Paper writing is hard. Adding more material - even supplementary material just
slows down the work of good people.

On the other hand, I do feel like there should be websites where papers can be
annotated by readers, who can provide missing details etc. Though this does
require a lot more papers to be open access.

~~~
mitchty
Basically something like rapgenius but for papers?

~~~
roywiggins
[https://fermatslibrary.com](https://fermatslibrary.com)

