[Date Prev][Date Next] [Thread Prev][Thread Next] [Date Index] [Thread Index]

Re: [OFFTOPIC] Goedel and the mathematic (was: Re: Copyright from the lcs-projekt!? [dwarf@polaris.net: Re: First cut at testing and validation]



Marcus Brinkmann <Marcus.Brinkmann@ruhr-uni-bochum.de> writes:

> [warning, this is getting off topic fast]

> On Sat, Aug 15, 1998 at 10:42:14AM +0100, Philip Hands wrote:

> > Are you familiar with Godel's Theorem ?

> >   When defining a mathematical system, you eventually get down to some 
> >   definitions that are neither provable, or disprovable within the system

> You are confusing two things there, the axioms and the theorems. axioms are
> not proofed by definition, they are defined and good is. The you have some
> rules how to draw conclusions from the axioms. If you apply them, you get
> theorems.

> Goedels theorem itself says, that every axiomatic constructions of number
> theory that is free of contradictions contains theorems where you can't
> decide if they are true or false.

> The difference from what you said above is subtle, but
> meaningful. First, the system needs to contain a significant amount
> of complexity, and it is necessary that it does not contain
> contradictions. And, more important, it is not a "definition" that
> is neither provable nor disprovable (definitions are "true" by
> definition), but theorems. This is surprising, as the mathemticians
> had the hope that every theorem could be either proved or
> disproved. That this is not true is the essence of Goedels theorem.

> There is a whole in the mathematic, and you can't repair it. But it is hard
> to give examples of such theorems. In fact, Goedel proofed his theorem by
> constructing such a number theory and a theorem that can't be poved or
> disproved.

> >   (``1 = 1'' in arithmetic for example).

> It's a pity that it is not so easy. Instead Goedels theorem is pretty
> weired, and you can't give any easy example of such a theorem.

IIRC, Goedel used the old "this statement is false" trick.

> maybe an example is the theorem, that there is no set with a
> cardinality between the cardinality of the set of real numbers and
> the set of all subsets of the set of real numbers (is somebody still
> reading here :). At least it was proved that this theorem can
> neither be proved nor disproved.

It was proved that this axiom was independent of the rest of the
axioms of set theory (by Cohen circa 1926, I believe - I skimmed the
article a few years ago).  IIRC, the idea here was to construct two
different set theories, one of which contained the normal axioms plus
that continium hypothesis and the other which contained the normal
axioms plus the negation of the continium hypothesis.  The
construction is done inside set theory without the additional
hypothesis.


(Some details could be incorrect.  The main focus of my study was
Group Theory and Algebra, not set theory or logic.  Anyways, this
stuff, while profoundly interesting to me, is off topic, and perhaps a
bit dry.)


Steve
dunham@cps.msu.edu


Reply to: