
Gödel's Incompleteness Theorems (2015) - mutor
https://plato.stanford.edu/entries/goedel-incompleteness/
======
threepipeproblm
I took a full semester course on Godel's Incompleteness Theorems in college
and found it rewarding. It was one of those experiences that convinced me that
hand-wavy familiarity with stuff sometimes pales by contrast to a deep dive.

One of the professors also had a very old newspaper clipping taped up (it was
brown then and that was quite a while ago), about several mathematicians
committing suicide in the wake of those proofs. I've looked for corroboration
of this and have never found it. A bit difficult to comprehend from a modern
perspective, but more plausible with an appreciation of the 19th Century faith
in rationalism / logical positivism that Godel helped overturn.

Not many people are aware that Godel spent much of his life secretly working
on a modern update of an ontological argument for God's existence, and didn't
reveal this until late in life --
[https://en.wikipedia.org/wiki/G%C3%B6del%27s_ontological_pro...](https://en.wikipedia.org/wiki/G%C3%B6del%27s_ontological_proof)

~~~
qwert-e
As far as I know it's not like the proofs caused a wave of suicides, but
mathematicians studying this topic often became mentally unstable, including
Godel (who starved himself in a sanatorium) and others like Georg Cantor.

I'm curious where you studied; I also took a semester course on "logic and
computability" where the main text we read was 'Godel, Escher, Bach'

~~~
dvt
I studied at UCLA and also took several logic and metalogic classes (it was my
AOF). For one of the metalogic classes, this was our text:
[http://www.math.ucla.edu/~dam/135.07w/135notes.pdf](http://www.math.ucla.edu/~dam/135.07w/135notes.pdf)

We briefly talked about Godel's proofs, but they are nontrivial. Henkin's
proof of completeness is hard enough[1]. I don't mean to sound dismissive, but
a class where _Godel, Escher, Bach_ is the text does not seem very rigorous.
Logic is very tricky stuff. And once you get into infinities, it's not even
intuitive.

[1] [https://www.cs.nmsu.edu/historical-
projects/Projects/complet...](https://www.cs.nmsu.edu/historical-
projects/Projects/completeness.pdf)

------
cousin_it
Here's a nice way to make peace with Godel's theorems.

Let's say your beliefs about the integers can be summarized by some formal
theory T. ("Every number has a successor", and so on, until your intuition
runs out of things to say.) Now Godel jumps out of the bushes and says aha,
your T doesn't include Con(T), so you aren't the math genius that you thought
you were!

But let's stop for a moment and consider what it would take for T to include
Con(T). First of all, compared to other sentences about integers that you
believe, Con(T) is an absolutely huge sentence. It must contain an
arithmetization of all of T (encoding the axioms, inference rules, etc. into
integer arithmetic).

Second of all, if Con(T) is included in T while speaking about T, it might
need to include an arithmetization of Con(T) itself! That uses a diagonal
construction ("quine" in computer science terms), making the sentence even
bigger. Now you're looking at some kind of hundred-kilobyte Diophantine
equation, with no intuitive reason to believe it at all.

And third of all, it's easy to see that an inconsistent theory T would easily
prove Con(T) (because it proves any sentence), so having Con(T) inside T
doesn't even give you any positive evidence for trusting T. In fact, we're
lucky to live in a world where Godel's theorems are true, and having Con(T)
inside T is negative evidence instead of none at all!

~~~
Certhas
To some degree Chaitlin's equations [1] make the issue less abstract than that
though.

"We outline our construction of a single equation involving only addition,
multiplication, and exponentiation of non-negative integer constants and
variables with the following remarkable property. One of the variables is
considered to be a parameter. Take the parameter to be 0, 1, 2, ... obtaining
an infinite series of equations from the original one. Consider the question
of whether each of the derived equations has finitely or infinitely many non-
negative integer solutions. The original equation is constructed in such a
manner that the answers to these questions about the derived equations are
independent mathematical facts that cannot be compressed into any finite set
of axioms."

So there are finite (though several hundred pages long) equations for which
the finiteness of the solution set is independent of every finite
axiomatization of arithmetic. So suddenly, even though the formula is
enormous, the type of sentence is perfectly normal. Does f(x) have finitely
many solutions, where f(x) is built from addition, multiplication and
exponentiation.

Much more so than Gödel’s incompleteness this deeply violates my intuition
about what type of statements should be decidable.

[1]
[https://www.cs.auckland.ac.nz/~chaitin/berlin.pdf](https://www.cs.auckland.ac.nz/~chaitin/berlin.pdf)

~~~
cousin_it
There's no single f(x) for which finiteness of the solution set is independent
from all axiom systems. (We can always just add an axiom saying f(x) has
finitely many solutions.) Chaitin shows something different, an f(x,y) for
which finiteness of solution set in x for a given y becomes independent from
more and more axiom systems as y grows (because it encodes larger and larger
instances of the halting problem). That doesn't disagree with my intuition at
all, in fact it seems obvious.

~~~
Certhas
Well given that it was considered a surprising and highly non-trivial result
when it came out less than thirty years ago, I'll say that it's obvious only
in the sense that all statements that have been proven are obvious after the
fact.

And yes, I worded that ambiguously. Let me rephrase, to see if I got it
correct: Pick any finite axiomatization. Then there is a concrete f(x) that we
can write down in about 200 pages, for which finiteness of the solution set is
not decidable.

The fact that there are diphantine equations for which finiteness is
undecidable is really surprising to start with to me.

------
matt_morgan
If you're interested in this and you haven't read Logicomix,

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

get it at your library now.

[http://www.worldcat.org/title/logicomix/oclc/708346776&refer...](http://www.worldcat.org/title/logicomix/oclc/708346776&referer=brief_results)

~~~
osullivj
Great comic book treatment of the Frege - Russell - Godel story. Strongly
recommended.

~~~
Y_Y
They do mention this in the book, but it does take artistic license with some
of the history.

------
curuinor
Carl Hewitt (of Actor model fame) has this thing where he claims that
contradiction alone allows you to defeat Godel Incompleteness, so he shops it
around places and everyone's like, "man, wtf, this isn't how that works". I
saw him shop it to D. Hofstadter and Hofstadter basically smiled and took the
little piece of paper and threw it away when Carl wasn't looking.

This doesn't have anything to do with anything, just a thing that happened

~~~
wfn
I know that the above was written in jest anyway :) but just for completeness
(har), a contradiction would help you defeat _many_ things (i.e., it's not
specific to incompleteness theorems, so to speak), because given it, (at least
in traditional logic) you can prove _anything_. This is trivial and known, but
to spell it out, if we have a contradiction formalized by _p_ and _not p_
(given), we can prove that _q_ :

1\. _p_ and _not p_ (given).

2\. _p_ (from 1).

3\. _not p_ (from 1).

4\. _p_ or _q_ (from 2 (disjunction introduction)).

5\. _q_ (from 3 and 4).

So obviously this would _upset_ quite a few things around :) that said, the
interesting stuff is with Graham Priest's (et al.) paraconsistent logic
systems wherein your system can tolerate a contradiction without exploding in
whole. And (so the story goes) _those_ systems may offer an actual insight
into handling incompleteness (while still being usable). If anyone has looked
into this more, would be interesting to hear about it!

~~~
jesuslop
That behaviour was desribed by the old masters as 'ex falso quodlibet': from
falsity whatever (when else am i having the chance to be this pedantic). In
the uni I was taught that hippie-era AI dealt with that with things called
non-monotonic reasoning, abduction, truth maintenance systems.

~~~
dvt
Also known as the (much cooler-sounding) Principle of Explosion :)

------
dataphyte
I recommend Torkel Franzén's Gödel's Theorem: An Incomplete Guide to Its Use
and Abuse. It is readable without sacrificing rigor.

It is especially good at ensuring the reader doesn't come away infected with
the pseudo-profound BS that plagues many discussions of the Gödel's Theorems.

~~~
theoh
That, and Nagel's book on the proof, have been recommended on HN before (very
recently!). And Rebecca Goldstein's book has been dismissed as too hand-wavy.

For important, wiki-friendly topics like this one, HN is frustratingly not
great for building a readily checkable/cumulative knowledge base, rather than
encouraging a forgetful herd discussion that goes around in circles. It is as
I say frustrating for anyone who wants long-term knowledge rather than flimsy
"news".

~~~
Jun8
Very apt obeservation! I was thinking on how to address that for some time.
Any ideas? Would you like to collaborate on this?

------
psyc
This isn't really about Gödel, but I always wish that people had a little bit
_less_ of a natural instinct towards bastardized Gödelian thinking. So that
there would be fewer infuriating people arguing things like "Infringing on my
right to infringe on the rights of others infringes on my rights." Or, "You
can't be against criticism, because then you'd be criticizing criticism." I've
been rolling my eyes at these arguments for decades, and I wish fewer people
thought they were clever.

------
maskedinvader
I recently read 'Incompleteness: The Proof and Paradox of Kurt Gödel' by
Rebecca Goldstein [1], I highly recommend it for those interested in learning
more about his life and the proof itself.

1.[https://smile.amazon.com/Incompleteness-Proof-
Paradox-G%C3%B...](https://smile.amazon.com/Incompleteness-Proof-
Paradox-G%C3%B6del-Discoveries/dp/0393327604/)

------
okket
A non-mathematical interpretation of Gödels Theorems and their implications

[http://rationalwiki.org/wiki/G%C3%B6del's_incompleteness_the...](http://rationalwiki.org/wiki/G%C3%B6del's_incompleteness_theorems)

------
crypto5
I always was wondering: Godel proved his theorem regarding formal systems
described in Principia Mathematics. In my understanding this is some higher
level logic with recursive functions.

But how this can be extended to the whole universe of all possible formal
systems? Who guarantee that there will be no some new system with quantum-
oracle-operator, which will not be affected by incompleteness theorem, and can
self-proof self-consistency?

Even well known m-recursive functions (which are essentially Turing machines)
are wider class than primitive recursive functions used in the proof..

~~~
chowells
Godel's technique is a constructive proof that depends on two things to work.
One, the logic needs to be at least strong enough to express the Peano axioms.
Two, it needs a finite formalism of the logic to construct the proof out of.

If you have those two things, Godel's technique works. The details vary
greatly based on the formal logic in use, but part of the process is encoding
the formalism in Peano arithmetic.

So no, adding more things doesn't break it, because the construction takes
those additional things into account.

Unless you want your logic to be infinite, of course. In that case, the method
would break down. But it has to be infinite in the sense of having no finite
representation, rather than the much weaker sense of having some infinite
representation.

~~~
crypto5
See my comment about what do I mean:
[https://news.ycombinator.com/item?id=14039104](https://news.ycombinator.com/item?id=14039104)

> Unless you want your logic to be infinite, of course. In that case, the
> method would break down.

Godel numbering is infinite, because it is just natural numbers. Nothing
prevents you to build the same proof for infinite number of axioms, rules,
functions, variables..

~~~
chowells
> Godel numbering is infinite, because it is just natural numbers. Nothing
> prevents you to build the same proof for infinite number of axioms, rules,
> functions, variables..

That's kind of a non-sequitor. Second-order logic, the logic required to
express Peano arithmetic, consists of a finite number of axioms and inference
rules. The number of statements that can be generated with those rules is
irrelevant. The logic is finite.

Godel numbering requires assigning a number to each logical axiom and
inference rule. The mechanism of Godel's numbering implies that every
derivation has a unique number as well, but that number is derived, not
assigned. The difference is important. It means that the information required
to construct the system is finite. If the supply of axioms and inference rules
was infinite, with no finite alternate encoding, it would mean Godel numbering
would fail because the process of assigning numbers to each axiom and rule
would never complete.

But that's all a bit of an aside. To get back to your original question...

Godel's proof provides a mechanism to construct a statement that refers to
itself in any logic that can express Peano arithmetic. Once you can make a
statement refer to itself, the game is over. You've lost.

But Godel's approach is even more clever. Not only can a statement refer to
itself, it can refer to proofs in its own formal system.

So it constructs "This statement has no proof within the formal system it's
expressed in."

That's far more subtle of a statement than you give it credit for. It makes no
claim about other formal systems. But once you start addressing it from a
different formal system (or an informal metalogic) the statement no longer
refers to the logic system in which you are examining it.

So sure, you can say that Godel statement G0 constructed in formal system L0
is provable (or disprovable!) in formal system L1. But that doesn't prove
anything about the completeness or consistency of L1, because G0 is a
statement about L0. And if L1 happens to be strong enough to encode Peano
arithmetic, you can construct a statement a Godel statement G1 which refers to
logic L1.

This is always applicable. Whatever your G0 and L1 are, G0 says nothing about
L1, but the existence of L1 is irrelevant to the statement G0, and very
probably also allows the existence of G1. (I've spent lots of time discussing
when it doesn't, and won't belabor the point further.)

~~~
crypto5
> If the supply of axioms and inference rules was infinite, with no finite
> alternate encoding, it would mean Godel numbering would fail because the
> process of assigning numbers to each axiom and rule would never complete.

I strongly disagree with that. There is no evidence of that.

> And if L1 happens to be strong enough to encode Peano arithmetic, you can
> construct a statement a Godel statement G1 which refers to logic L1.

As I said before, Godel proved that your G1 is unprovable in specific
framework of Principia Mathematics, which is first/second/higher theory/logic
(consist on quantors, functions, predicate, variables, rules).

I don't see evidence that there can be no other framework even with Peano
arithmetic inside which can't deliver consistent theory. You started
speculating about framework with infinite set of axiom, and I disagreed with
that, but there can be other type of infinite framework, say when you allow
proofs of infinite length, then Godel numbering may be impossible, and it can
be example of such framework.

------
rb1
Just chipping in with my interesting reading about the implications of Godel's
incompleteness theorems.

This piece about what the theorem means for developing "deep AI" and the human
mind, was a fascinating eye opener for me, about the far stretching
implications of the theorem.

"The Lucas-Penrose Argument about Gödel's Theorem" \-
[http://www.iep.utm.edu/lp-argue/](http://www.iep.utm.edu/lp-argue/)

~~~
clairity
penrose wrote a whole book walking through his argument around deep AI (and
providing a gentle explanation of gödel's incompleteness theorem in the
process):
[https://en.wikipedia.org/wiki/The_Emperor%27s_New_Mind](https://en.wikipedia.org/wiki/The_Emperor%27s_New_Mind)

------
danharaj
Goedel's results imply some surprising and counterintuitive things about our
formal models of computability:
[https://johncarlosbaez.wordpress.com/2016/04/02/computing-
th...](https://johncarlosbaez.wordpress.com/2016/04/02/computing-the-
uncomputable/)

~~~
pizza
Enjoyed rereading this, thanks

------
laxd
"A common misunderstanding is to interpret Gödel's first theorem as showing
that there are truths that cannot be proved." I've heard it stated that
confusing way so many times in popular media. And then often follows quantum
mechanics, and geniuses going mad, and the world is just your imagination...
forever.

~~~
saguiar
Agreed it has been abused. On the other hand, if you adhere to a computational
theory of mind and see the mind as computation over a formal system, then it
is saying something about the mind and the fact that some (true) sentences
cannot be proven in that system, isn't it? It is a big if though.

(though maybe in a completely irrelevant sense of truth).

~~~
laxd
Just poking fun at the way these things are presented in pop.sci. media.
Anyway, talking over my head, I'm not shure formal systems are a good way of
modelling the mind itself. Formal systems are (usually) considered consistent.
The mind is not. Neither are neural nets. And it's interesting how neural
nets, and other statistical models/algorithms that drop the requirement of
always beeing right, seems more much more capable in certain practical
matters.

------
quizotic
When I rant about it, I claim his theorem says "any system that reference
parts of itself is either incomplete or inconsistent," and that the universe
is such a system, as we're a part of it, and refer to parts of it.

So therefore, the universe is either full of magic (in a system with
inconsistency, anything can happen), or mystery (there are true things that we
can never prove).

My belief is that the quest to find a unified physics that describes
everything is provably impossible due to Godel's theorem.

And on the rare occasions when I look for evidence of God, the fact that one
of the things we can know for sure - is that we can't know everything -
provides about as much comfort as I need.

~~~
petegrif
This sort of application is wildly out of bounds of the true scope of the
theory. The relationship between physics (unified or otherwise) can't be
resolved with reference to Godel's theorems.

~~~
quizotic
Why? I honestly don't see how it's out of scope at all (majored in
mathematics, minored in physics)

~~~
mrbrowning
Well, you're actually making metaphysical claims more than mathematical or
physical ones. One of two assumptions, depending on how literally the
sentiment that the universe is a formal system is intended, is being elided
here:

1\. That the universe _is_ a formal system, rather than being describable in
the language of some formal system. It's not evident what the universe being a
formal system even means, or how it squares with basic intuition regarding
e.g. the fact that physical systems have state.

2\. That, dropping the physical system <=> formal system equivalence and given
some real system R consisting of some fundamental entities whose behaviors can
be described in full in the language of some formal system S, (borrowing a
useful construct from Lucas' anti-mechanism argument, even though I don't buy
that argument) no machine can be constructed in R which computes theorems of
some formal system S' in which all true statements of S are provable, meaning
that no state of the system R can be said to contain a description of S', and
that S' is therefore not describable by any arrangement of the entities in R
(assuming some reasonable predicate over states of R that is true for a state
when some arrangement of a subset of the entities in that state describes S').
Intuitively, this doesn't seem to hold up: by analogy, I can describe a
universal Turing machine with a computer equipped with only finite memory. You
could then attempt to go down the road of claiming that, even if a description
of S' is possible in R, that a mind within R would not be capable of
formulating that description, but then you're heaping on an even larger tangle
of assumptions, unknowns, and things you have to define if you're going to
argue the case rigorously.

The point being that confidence about _any_ hypothesis about the nature of
reality made on the basis of Gödel's incompleteness theorems is not
epistemologically warranted.

------
cantagi
I am halfway through reading Raymond Smullyan: A puzzle guide to Gödel, after
finding out about his recent death here:
[https://news.ycombinator.com/item?id=13626221](https://news.ycombinator.com/item?id=13626221)
. It explains Godel's incompleteness Theorems accurately to a layperson
through a set of logical puzzles.

------
jchassoul
Wittgenstein bitches!

