

Godel's 2nd Incompleteness Theorem Explained in Words of One Syllable - daviddaviddavid
http://www2.kenyon.edu/Depts/Math/Milnikel/boolos-godel.pdf

======
lucisferre
A math professor was giving a talk and stated that "given a false premise one
can prove anything is true". A member of the audience then interrupted with
"Ok then, 1 + 1 = 3, now prove that you are the pope."

The professor thought for a moment and began, "If 1 + 1 = 3 then 1 = 2 and
since the pope and I are two then the pope and I are one."

~~~
pella
Bertrand Russell is the Pope

"The story goes that Bertrand Russell, in a lecture on logic, mentioned that
in the sense of material implication, a false proposition implies any
proposition. ..."

[http://www.nku.edu/~longa/classes/mat385_resources/docs/russ...](http://www.nku.edu/~longa/classes/mat385_resources/docs/russellpope.html)

~~~
chris_wot
Russell's paradox?

~~~
derleth
Cute, but that's something different again.

~~~
chris_wot
My bad... This is new to me. What is the difference?

~~~
derleth
Russell's Paradox is the statement "Does the set of all sets that don't
contain themselves contain itself?" (Parse that as "The set of (all sets that
don't contain themselves)".)

Now, obviously, the set of all sets contains itself, and in naïve set theory
that isn't a problem: In naïve set theory, a set is just an unordered
collection of any objects you can define; a set that contains itself poses no
logical problems in an of itself.

However, once you define the set of "all sets that don't contain themselves",
you run into an immediate problem: If it does not contain itself, it must, in
which case it cannot, and so on, in a neat infinite recursion.

This is a _real_ paradox, also known as a falsidical paradox, and a _real_
paradox indicates a _deep_ problem with the set of axioms it was derived from.
(Banach-Tarski is _not_ a real paradox by my definition; it is merely called a
paradox because it is a counter-intuitive result.) Russell's Paradox blew
naïve set theory out of the water; later set theories, such as Zermelo-
Fraenkel (more commonly just called ZF), were very careful to not allow sets
to contain themselves.

------
OmegaHN
Eh, this doesn't really explain Godel's 2nd Incompleteness theorem; it only
describes it in a roundabout way. The entire piece could be shortened down to:
in math, all false statements are possible (i.e. their impossibility cannot be
proved), and it doesn't go into any reasoning behind it.

I'm not sure about Godel's 2nd, but his 1st theorem can be described and
explained with one simple sentence: this statement cannot be proven. If it is
proven, then a false statement is proven. If it cannot be proven, then the
proving system is flawed.

~~~
NHQ
I like George Boolos' explanation because it plays with my paradoxic
sensebrain, with something like poetry. Why not call it a poem?

But I like your explanations too.

However, I think that the Goedel card is counter-played well by the
Schrodinger one. "This statement cannot be proven" is only a false statement
because you inspect it with your system. It might otherwise be completely
true.

~~~
md224
Ah, but "true" and "proven true" are two different things!

I'm sensing an epistemology thread developing...

~~~
NHQ
Not so fast!

I am taking issue with the so-called "falsity" of the "this statement is
false". If you imagine that statement, and imagine that what you imagine is
probable in the extreme, outside of your imagination, free, and unmolested by
your probings, then "this statement is false" is always, only, incidental.

And don't come back saying saying "false" and "proven false" are different. I
can compare strings!

~~~
md224
That's the thing though... I interpreted it as being a true statement, but one
that cannot be "proven" to be true. I merely possess the strong belief that it
is true.

Is your argument that the statement could possibly be false? Otherwise we're
already in agreement... Nobody is suggesting the statement is false (I think).

Edit: to clarify, the statement in question is "this statement cannot be
proven," not "this statement is false."

~~~
NHQ
You are right about the statement in question. I got mixed up, but I think the
principal of "same difference" applies :D

He says, "If it [the statement] is proven, then a _false_ statement is
proven." (Proven true or false?)

I am not interpreting the statement "this statement cannot be proven" as true,
as you do, or false.

What I argue is that one's interpretation has an effect, without which the
statement is neither true nor false, but, as I claim, incidental. We can
imagine the statement, but what we imagine is not the statement, even if it
looks and smells like it.

Thus, the Schrodinger Card trumps the Goedel. The trick here is that I am
standing outside of "the system", not permitting myself to enter paradox land.
If you so much as look at me, I'll get sucked in.

------
Xcelerate
This statement is false. True or false?

Does the set of all sets that don't contain themselves contain itself?

~~~
dmvaldman
What's even crazier to me is that there are statements that aren't self-
referential that are both not true and not false at the same time. For
instance the continuum hypothesis [1], or anything to do with the axiom of
choice, like the trippy Banach-Tarski paradox [2].

[1] <http://en.wikipedia.org/wiki/Continuum_hypothesis>

[2] <http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox>

~~~
jerf
The continuum hypothesis sort of fits your description, but Banach-Tarski does
not. If you take the axiom of choice in your system, Banach-Tarski is simply
true (in that the described construction will create two spheres out of one).
If you do not take the axiom of choice, it is simply false (in that it can not
be defined). (And I'm sure somebody somewhere has worked out some sort of in-
between state there, but let's keep it simple.)

While it's a very interesting thing to think about, there is one sense in
which the paradox is trivially resolvable; while the axiom of choice may make
sense in mathematics, it almost certainly is never used by the real universe,
and you _certainly_ can not construct two real spherical shells made of atoms
of the same size as one original spherical shell in the real world. (Whether
the universe is continuous is an open problem, but we _know_ matter isn't.) A
great deal of the reason why the mind rebels at the Banach-Tarski paradox is
that it is built on a thoroughly aphysical axiom; that the aphysical axiom
permits aphysical (and therefore counterintuitive) results is not _that_
surprising.

By no means is that a criticism of BT, it is simply its nature. My personal
position on constructivist vs platonic vs. blah blah blah is that in math, as
long as you specify which axiom set you are using up front, there is nothing
to be emotional about. (Except that I will reserve a special place of interest
for whatever mathematics it turns out to be that precisely represents the real
world. Alas, this is still a work in progress, though we can point to at least
some characteristics of it.)

~~~
dmvaldman
"If you take the axiom of choice in your system, Banach-Tarski is simply true"

\- The exact same is true if you take the continuum hypothesis as an axiom.
Both the axiom of choice and the continuum hypothesis are independent of the
other standard ZF axioms

"If you do not take the axiom of choice, it is simply false"

\- That is incorrect. It is only independent. To say it is false, another
axiom(s) would need to contradict it.

"while the axiom of choice may make sense in mathematics, it almost certainly
is never used by the real universe"

\- neither are the real numbers. would you like to claim that pi, or the
square root of 2, for that matter, also don't exist? How about the number 1?

------
chris_wot
So...

    
    
      If it can be proved that it can't be proved that two plus two
      is five, then it can be proved as well that two plus two is
      five, and math is a lot of bunk.
    

Then:

    
    
      p: it can't be proved that 2 + 2 = 5
      q: it can be proved that it can't be proved that 2 + 2 = 5
    
      q → ¬p
    

Thus if it can be proved that it can't be proved that 2 + 2 = 5 then it can be
proved that 2 + 2 = 5. (i.e. when q is true, p cannot be true)

Sorry, had to do this for myself because I'm just starting a course in
discrete mathematics!

------
marshray
I like the way this explanation considers it a first class possibility that
"math is a lot of bunk".

~~~
dbaupp
This is a bit ingenious, it should actually be "the theory in which we are
working is a lot of bunk" (theory refers to a set of axioms), and maths is a
lot broader than one specific theory.

That is, if one proves Peano arithmetic[1] is inconsistent (which would really
suck), this would not effect the consistency or otherwise of other independent
theories, like, for example, Presburger arithmetic[2] which has actually been
proved to be consistent.

[1]: <https://en.wikipedia.org/wiki/Peano_arithmetic> [2]:
<https://en.wikipedia.org/wiki/Presburger_arithmetic>

~~~
cwzwarich
Peano arithmetic has also been proved consistent (by Gödel himself, Gentzen,
and others) but all of the proofs necessarily use methods slightly beyond
Peano arithmetic. Gentzen proved the consistency of PA from primitive
recursive arithmetic (a much weaker theory than PA, lacking quantifiers)
combined with transfinite induction up to a particular ordinal. Gentzen also
proved that this is more-or-less the best possible.

Similarly, since Presburger arithmetic can't even express its own consistency,
never mind attempt to prove it, any proof of the consistency of Presburger
arithmetic must use methods beyond Presburger arithmetic, which will probably
subsume Presburger Arithmetic itself.

While formal consistency results have some meaningful technical consequences,
from afar they often give off an aire of preaching to the converted. Anyone
who accepts transfinite induction up to the ordinal in Gentzen's proof will
probably have no problem accepting PA itself.

------
tnicola
2 + 2 = 5; for large values of 2.

~~~
Jach
Or Python evil. (<https://gist.github.com/1208215>)

~~~
dbaupp
Or Haskell.

    
    
      let 2+2=5 in 2 + 2
    

(Significantly less mind-blowing though.)

~~~
NHQ
interesting and funny though for somebody who doesn't know Haskell

------
dmd
Reminds me of this classic: <http://www.lel.ed.ac.uk/~gpullum/loopsnoop.html>

------
kmfrk
I suddenly have a very strong urge to read Alice in Wonderland.

------
ThomPete
Ahh but can it be proved that language can prove anything?

~~~
D_Alex
Well... the article attempts to "explain" rather than to "prove".

------
javert
Unfortunately, this write-up is a load of irresponsible crap, because it can
be proven that it can't be proven that two plus two equals five.

This is a load of ivory-tower hogwash, precisely why people get turned away
from math and philosophy.

~~~
chris_wot
Can you supply us the proof that it can't be proven that two plus two equals
five? Even a citation would be fine. Thanks!

~~~
debacle
The Math 101 'proof':

Assume it can be proven that 2 + 2 = 5.

Then 2 + 2 = 5.

It can be proven that 2 + 2 = 4.

The closure axiom of addition states that the expression 2 + 2 is a unique
number.

4 != 5.

Therefore it can't be proven that 2 + 2 = 5.

\-----

I'm sure it can't be that simple, but logically I can't see a reason why not.

~~~
sklipo
You haven't proven that it can't be proven that 2 + 2 = 5. All you have proven
is that if 2 + 2 = 5, then mathematics are inconsistent. Which they could be;
you can't just assume math and logic are consistent.

~~~
debacle
> you can't just assume math and logic are consistent.

What a bunch of bollocks.

~~~
javert
Precisely my point. Wow, that guy really stepped in it.

Claiming that logic isn't valid is worse than a logical contradiction, because
if logic isn't valid, you can't claim _anything_.

~~~
chris_wot
Nobody ever said logic isn't valid. We're talking about consistency.

~~~
javert
If logic isn't consistent, then logic isn't valid.

I'm not assuming specialized definitions of "consistency" and "validity" here,
just your normal everyday layman's usage.

~~~
chris_wot
Well, you were responding to someone who was using the correct formal terms.
You, after all, don't dispute the second theorem - but given it uses the term
"consistent" you can hardly complain when someone makes their argument using
the same terminology as the theorem you agree with!

Interested in what you define "consistent" and "valid" as in laymans terms
though.

