Godel's 2nd Incompleteness Theorem Explained in Words of One Syllable

[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...

[crntaylor]

In case anyone is interested, the non-witty answer would go like this: Clearly, 1+1=2, which is not 3, so ¬(1+1=3) (by usual rules of addition). But also 1+1=3 (by assumption). Since 1+1=3, then either I am the pope, or 1+1=3. But we know that ¬(1+1=3), which means that I must be the pope.

[btilly]

That is a clever professor.

See http://en.wikipedia.org/wiki/Principle_of_explosion for the usual lines of reasoning that get used.

[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.

[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

[nessus42]

This sentence is true. True or false?

I took two classes by George Boolos at MIT. They were a lot of fun, and Prof. Boolos was a rather strange fellow.

The Department of Linguistics and Philosophy has a memorial display case filled with some of Boolos's favorite puzzles. You should check it out if you're in the area.

[qntm]

The axioms of Zermelo-Fraenkel set theory don't admit the construction of a "set of all sets that don't contain themelves".

[mck-]

Consider superposition, where it is both true and false at the same time (and has been proven)

[chris_wot]

Russell's Paradox :-)

[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

[dmd]

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

[tnicola]

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

[Jach]

Or Python evil. (https://gist.github.com/1208215)

[FreeFull]

Or small values of 5.

[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!

[batista]

>because it can be proven that it can't be proven that two plus two equals five

Really? What does that even mean? In what number system it can it be proven? Does it hold for all?

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

If people get "turned away from math and philosophy" then it's the peoples' problem, not math and philosophy's problem.