AtD pg 675

[Lawrence Bryan]

I no longer have the beginning of this thread and am too lazy to go  
look it up. However in dimension theory it has long been established  
when dealing with Euclidean spaces - the kind of everyday spaces in  
which our intuition tells us we live - that there exist n-dimensional  
objects which can only be embedded in a Euclidean space of dimension 2n 
+1. For example if one takes two linked circles - a circle is a one  
dimensional object -  that do not intersect - have no points in common  
- then that object can only exist in a 3 dimensional space. Perhaps it  
is this notion that is being discussed.

It's an interesting challenge to come up with a two dimensional object  
which can only exist in a 5th dimensional space. Makes my head hurt to  
do that so I'll refrain.

Lawrence

On Jan 7, 2008, at 5:40 AM, Michael J. Hußmann wrote:

> Lawrence Bryan (lebryan at speakeasy.net) wrote:
>
>> The only thing I would add is that a simple consequence of the  
>> theorem
>> that might be easier to understand is that Godel showed that there
>> exist legitimate statements (by legitimate I mean the lexicon and
>> syntax is valid within that system) in any system that's at least as
>> rich as arithmetic whose truth cannot be determined from the axioms  
>> of
>> that system and even if the axiom system is extended there will still
>> be such unprovable statements.
>
> I still don't see how the analogy between Gödel's theorem and Kit's
> remark about n+1 dimensions being necessary would work out. After all,
> you can add up vectors within an n-dimensional space just fine. This
> would correspond to proving statements within a given system, which  
> you
> cannot, in the general case. Kit introduces an observer who needs to
> travel in an additional dimension so he can "see" the resultant,  
> rather
> than arriving at the result by calculation. But Gödel's theorem isn't
> about "seeing" a proof one could also arrive at within the system.
>
> - Michael
>
>
> Michael J. Hußmann
>
> E-mail: michael at michael-hussmann.de
> WWW (personal): http://michael-hussmann.de
> WWW (professional): http://digicam-experts.de
>
>
>