
An Introduction to Godel's Theorems (Second Edition) [pdf] - furcyd
https://www.logicmatters.net/resources/pdfs/godelbook/GodelBookLM.pdf
======
braindongle
If you're uninitiated and simply want to grok where true-but-unprovable comes
from, I recommend: [https://www.quantamagazine.org/how-godels-incompleteness-
the...](https://www.quantamagazine.org/how-godels-incompleteness-theorems-
work-20200714/)

Wolchover is masterful here. The layers of abstraction keep piling up, and I
had to read the last part more than a couple of times to really get it, but
then you have it.

~~~
dvt
Just going to echo @dwohnitmok here in saying that this is a pretty 'meh'
article. Understanding Godel's First Incompleteness Theorem is actually very
accessible, I wrote about it a few years ago[1]. His second is _much_ more
involved and laymen won't have the required tools to grasp it. In my opinion,
the easiest way to understand it is probably using Löb's Theorem, but that's
neither here nor there. Either way, I'm of the opinion that arithmetic coding
is very confusing and shouldn't be used to introduce people to Godel.

[1] [https://dvt.name/2018/03/12/godels-first-incompleteness-
theo...](https://dvt.name/2018/03/12/godels-first-incompleteness-theorem-
programmers/)

~~~
krick
I'm not sure this is equivalent to Gödel's theorem. Actually, I'm not sure
this is a correct proof of anything at all, merely a sort-of demonstration of
Richard's paradox.

First, for reference, let me quote First Incompleteness Theorem: "Any
consistent formal system F within which a certain amount of elementary
arithmetic can be carried out is incomplete; i.e., there are statements of the
language of F which can neither be proved nor disproved in F." (from
Wikipedia)

Now back to your post.

Obviously, f' is not in T and it is not computable. But neither is T. It is
incomputable merely by being the list of computable functions, which we cannot
construct because of halting problem and stuff like that.

And your definition of f' relies on having T. So we don't in fact have seen
"statement f'". So, the direct connection between Gödel's theorem and your
construct is not obvious to me, because first is about constructable, but
unprovable statements, and second seems to be a faulty (i.e. mathematically
uninterpretable) construct itself.

~~~
dvt
> I'm not sure this is equivalent to Gödel's theorem.

It is. In fact, I've seen it proven this way several times (mostly in CS-y
papers), but it's definitely not common. My post is in fact based (almost
beat-by-beat) on this UC Davis lecture:
[https://www.youtube.com/watch?v=9JeIG_CsgvI](https://www.youtube.com/watch?v=9JeIG_CsgvI)

~~~
krick
Your plain and confident statement "it is" doesn't really address any of my
concerns, and I just explained why, as it seems, it actually isn't.

P.S.

I did some random googling, and while I'm not sure, all this story seems to be
based off of Paul Finsler's work, which he himself thought to be the priority
for an incompleteness theorem (Collected Works Vol. IV., p. 9), but didn't
seem so to Gödel (and, seemingly, the majority of those who had to say
something about it).

If this indeed is the same thing, it would explain to me why this "proof",
which doesn't seem to me like a proof at all is so widespread. But, again, I'm
not sure I'm reading into it correctly, and operating under assumption that if
somebody could provide something more weighty than "it is" statement, I could
be persuaded otherwise.

~~~
dvt
> Your plain and confident statement "it is" doesn't really address any of my
> concerns...

Your concerns aren't really valid (for example, the computability of T is
irrelevant). These are questions I already answered before on both HN and in
the Disqus comments. Further, metalogic was my area of focus in school, so I'm
quite familiar with intricacies here, and don't really feel it's useful to get
_too_ deep into the weeds. But don't take my word for it; Dr. Gusfield's paper
(that the lecture is based on) can be found here:
[https://csiflabs.cs.ucdavis.edu/~gusfield/godelproofreviseda...](https://csiflabs.cs.ucdavis.edu/~gusfield/godelproofrevisedarXiv.pdf)

The ideas in this proof have been well-tread by Scott Aaronson, Peter Smith,
and Michael Sipser. My curt "it is" is also meant to nip in the bud --
_correct_ \-- claims that it's not _exactly_ what Godel proved (the OG variant
is actually slightly stronger). Why I'd want to nip these in the bud is that
in school, you technically learn the weaker variant (albeit starting with a
Henkin construction of Godel's _Completeness_ Theorem). Sort of like this[1].
But the difference between the two is _so_ nuanced, it's not even really worth
bringing up unless we're in a graduate seminar.

[1] [https://hal.archives-
ouvertes.fr/hal-00274564/document](https://hal.archives-
ouvertes.fr/hal-00274564/document)

------
boyobo
Here is a nice, short proof, utilizing the conceptual framework of
"Computation".

[https://www.scottaaronson.com/incompleteness.pdf](https://www.scottaaronson.com/incompleteness.pdf)

See chapter 3.

~~~
rstuart4133
Yes, that Scott Aaronson paper was an eye opener for me.

It wasn't because of the correspondence between Godel's theorems and the
halting problem, although for a computer programmer Aaronson's formulation
turns Godel's inscrutable theorem into something easily accessible. (Oh, and
for the curious, the reason Godel's theorem is so hard to understand is he
needed a computer programming language for his proof but there were none lying
around back then, so he invented his own. I strongly suspect he wasn't
thinking of it as a series of executable steps, let alone a programming
language whose most important property is to be readable. As a consequence
it's bloody terrible.)

No, it was because it Aaronson made it plain both the halting problem and
Godel's theorem are really statements about the difficulty of reasoning about
infinities. The infinities are really well hidden in both approaches, but once
you uncover them it's like seeing the elephant in the room - it all becomes
obvious. And it also becomes evident they aren't particularly relevant to
anything you are likely to encounter here on earth. That's because there are
no infinities in our corner of the universe. And, there there is no halting
problem for finite, bounded systems. For some reason, no one ever tells you
that.

This paper, listed here on HN recently, explains it much better than I can:

[https://arxiv.org/abs/math/0411418](https://arxiv.org/abs/math/0411418)

------
dtornabene
A Couple of comments here on "this isn't an introduction, its a book!!", for
anyone who checked here first, that is the _title_ of the book, and while I
can't speak to Smith's coverage, if you're going to "introduce" someone to the
incompleteness theorems, a book is how you're going to do it. They're two of
the most significant theorems in modern mathematics, maybe all of mathematics.
Logicomix is fine, but I'm not pushing anyone to actually _learn_ these
theorems through a graphic novel or a blogpost. As far as book alternatives,
an interesting and useful way to get your head around what Godel was doing is
to try to read Gottlob Frege, his notation isn't that hard to parse, and it
marks the origin point of modern mathematical logic and the attempt to do that
which Godel proved impossible.

Foundations of Arithmetic is easily readable with only a slight amount of
contextualization for someone with a high school level mathematical education.

------
ntabris
For a shorter (but still fairly thorough) introduction by the same author, I
recommend:

[https://www.logicmatters.net/igt/godel-without-
tears/](https://www.logicmatters.net/igt/godel-without-tears/)

I found this super helpful when I was studying Gödel's theorems in grad
school. (Fwiw, the 3rd edition of Boolos & Jeffrey's _Computability and Logic_
is also great and takes a somewhat atypical approach to the proofs.)

------
Smaug123
I can recommend Peter Smith as an author; I was part of a small reading group
with him while we were all getting to grips with Cambridge's Part III
Introduction to Category Theory, and he's definitely good at identifying the
difficult bits and breaking them down.

------
Kednicma
It's a book! I was expecting a short paper along the lines of [0] or [1] but
this is going to take a while to evaluate. Nonetheless I like the surface-
level presentation and it seems like the author takes the time to carefully
explain each concept in detail.

[0]
[http://tac.mta.ca/tac/reprints/articles/15/tr15.pdf](http://tac.mta.ca/tac/reprints/articles/15/tr15.pdf)

[1] [http://r6.ca/Goedel/goedel1.html](http://r6.ca/Goedel/goedel1.html)

------
bmitc
I need to learn more about Godel and read the books about him. From the
outside, he almost seems like an intellectual rebel, showing how systems are
broken when people assume they aren't. With his completeness theorems and his
Godel metric, he seemed to be excited by logical exotica. I find the Godel
metric and its closed timelike curves to be an interesting case study of
general relativity. This is just me spitballing here, as I have not yet dove
into the many books I have on him.

------
irontinkerer
402 pages is one incredible introduction!!

Godel’s theorems are fun to study though. What other theorems have equally
blown your mind?

------
antman
This is a book, if someone wants to introduce himself to the field and its
actors I suggest Logicomix.

------
hwc
Why does the author use "sound" instead of "consistent"?

~~~
moefh
It's more or less explained in 1.2. A system is "sound" if it's consistent AND
can only prove true things.

To see the difference between sound and consistent, suppose you have a formal
system A which you know is sound. You can then (by Godel's construction) make
a proposition G that says A can't prove or disprove G. We know that G is true
because A can't in fact prove or disprove G, by construction.

Make a new formal system by taking the axioms of A and adding the negation of
G, that is, B=A+not(G). B is consistent but not sound. It's consistent because
A can't prove G, so adding not(G) as an axiom can't possibly lead to a
contradiction. And it's not sound because it can prove a falsehood, namely
not(G) (granted, it's a _very_ easy proof, since it's an axiom).

Scott Aaronson calls systems like B "self-hating" [1], because they "believe"
they're not consistent (even though they are).

[1]
[https://www.scottaaronson.com/blog/?p=710](https://www.scottaaronson.com/blog/?p=710)

------
billman
I read this, but it seems incomplete.

------
sfpoet
How clear the writing!

