[FOM] 336:Undecidability/Euclidean geometry/2

[Rob Arthan]

On 30 Apr 2009, at 12:23, pax0 at seznam.cz wrote:

> Harvey Friedman wrote:
>>
>> An integral line system is a finite set of rational lines whose
>> intersection points have integer coordinates.
>>
>>
>> We say that f is an equivalence between integral line systems S,T if
>> and only if f is a bijection from S onto T such that for any
>> L_1,...,L_k in S,
>
> I think, that here is missing some continuity condition on f, as is  
> in the original formulation.
>
> What happens with theorems 1 and 2 if we require moreover that the
> bijection from S onto T maps every line of S _analytically_ onto a
> line of T?


Note that Harvey changed the definition of "integral line system": in  
#335, it was the union of a finite set of lines, i.e., a subset of the  
plane. But in #336 and #337, it is a finite set of lines, i.e. a  
finite set of subsets of the plane.

With the definitions of #335, it won't make any difference if the  
bijections are required to map each line analytically. The problem  
only imposes a finite set of constraints, so if there is any solution  
there will be one where the bijection will restrict to an algebraic  
function on each line.

Like Richard Pollack, I would very much like to see more of the proofs  
for the results announced in Harvey's postings #335-#337.

Regards,

Rob.