🚀 go-pugleaf

On Feb 23, 8:11 pm, Zuhair <zaljo...@gmail.com> wrote:
> On Feb 23, 3:16 am, Chris Menzel <cmen...@remove-this.tamu.edu> wrote:
>
> > On Tue, 21 Feb 2012 07:48:03 -0500, Shmuel Metz
> > <spamt...@library.lspace.org.invalid> said:
>
> > > While I find (a,b)={{a},{a,b}} to be more natural for ordered pairs,
> > > it does not generalize well to m-tuples.
>
> > But only in regard to "naturalness", right?  I mean, in general, for
> > n>1, we have <a1,...,an,an+1> = <<a1,...,an>,an+1> = {{<a1,...,an>},
> > {<a1,...,an>},{an+1}}, which is perfectly well defined under K's
> > definition.
>
> Ok Chris examine this definition of pairs which can generalize to n-
> tuples
> in a nice manner so that one can compare and see
>
> Def.)  (a,b)= { {1,{a}} , {2,{b}} }
>
> where 1 and 2 are the {{}} and {{},{{}}}  as in standard notation.
>
> Now you can call this pair  Hausdorff's - Zuhair pair.
>
> It is definitely less natural than Hausdorff's { {1,a} , {2,b} }
> which is the most natural of all possible pairs because
> it uses tagging with 1 and 2 to determine the order of
> projections by simply pairing them in unordered manner
> which is the most natural one, although this naturalness comes
> at a high expense of complexity of defining projections.
>
> Now my modification both ease the definition of projections
> of that pair and not only that it is more powerful in the sense
> of making it extending to any n-tuple for any ordinal n whether
> finite or not!!! and this extension happens in a very natural manner!
>
> so for example (a,b,c)= { {1,{a}} , {2,{b}} , {3,{c}} }
>
> You see extending the pair here follows the same strategy
> so (a,b,c,d,e)={ {1,{a}} , {2,{b}} , {3,{c}} , {4,{d}} , {5,{e}} }
>
> This is much nicer and more elegant the Kuratowski nesting
> and even more elegant than n-tuples defined as sequences
> of length n depending on Kuratowski based ordered pairs.
>
> Actually this method allows simpler definition of sequences
> also, so instead of defining a sequence S of elements of set A
> of length n as: S:N-->A, where S is a collection of Kuratowski
> ordered pairs, one can similar define S:N--> P_1(A) where
> S is a collection {n,{a}} pairs where neN and a e A.
>
> Cheers
>
> Zuhair


None of these work!

You are confusing  { (1, a), (2, b), (3, c) }  with  { {1, a},  {2,
b}, {3, c} }

What is the order here?

{ {4, 1}, {1, 2}, {2, 3}, {3, 4} }

<2 3 4 1> or <4 1 2 3> ?

***********************************************************

You have to nest the braces outside of the ordered pair.

 {   {a,{}} ,   {{b,{}}} ,   {{{c,{}}}} ,   {{{{d,{}}}}}   }

Now (a, a) = { {a,{}}, {{a,{}}} }

has 2 full elements, no singleton!

since my design is based on your idea, how about Herc-Zuhair Set
Ordering?

Herc