[FOM] N-tuples, Ordered Pairs and Sequences
Zuhair Abdul Ghafoor Al-Johar
zaljohar at yahoo.com
Fri Feb 24 10:06:01 EST 2012
Dear FOmers.
In a separate forum and some privet correspondence via e-mail I asked the
question: Why Kuratowski ordered pairs are so famous?
What matters is the abstract notion of an ordered pair, the details of any
implementation of that pair are in a sense immaterial as long as the
implementation is copying the properties of the abstract notion.
However the K-pairs are so famous so why? And in a specific manner I asked
that question by stipulating a comparison with one of the oldest set
implementations of ordered pairs that of Felix Hausdorff (1914) which
appears to be a very natural pair to me.
I received many answers most stressing that the issue of choice of the
pair is immaterial and seldom has it made any difference!
But we have a situation here that is the K-pair is almost the official
pair, so why?
A convincing answer was that it is formally the simplest: the formal
definition of pair itself and its projections and the structure of the
pair all are strikingly simple, and there is also some kind of elegance to
it. However I see this somewhat ironic since all of those are actually
technical reasons even the seemingly superficial elegance of the pair is
like a kind of being captured in the fondness of a nice trick rather than
due to its natural proximity to the original abstract notion of ordered
pair, so technical details did have a role in the choice of the
implementation to be used, although at the end it is not essential and
truly immaterial.
I see Hausdorff pair as the most natural version of an ordered pair, it is
the one nearest to the abstract notion of an ordered pair, and it employs a
tagging strategy to discern the order of the relevant projections, which
is something at the core of this concept. I'm sure that if a person is
first exposed to the abstract concept of an ordered pair and then
presented with Kuratowski and Hausdorff constructions and then asked which
one is an implementation of an ordered pair, I'm sure that Hausdorff's
would be the most chosen.
But what is wrong with Hausdorff's, most answers: it involves the use of
arbitrary sets not related to the projections which are the core elements,
it has a rather complex formal definition of projections because of
peculiarities of projections when being equal to 1 or 2, etc... all are
strict technical formal answers, and after all formal detail is
immaterial!
How I see matters is that "naturalness" can sometimes come at a high
cost, it comes with increasing formal complexity, now if the later is
immaterial, naturalness is important, it stimulates further thought and
creative act which is very important. So why sacrifice naturalness for
formal simplicity if the later is immaterial!
Now there is something nice to Hausdorff's pairs, which is actually
related to its tagging strategy which is in my simple own opinion
something at the heart of the concept of ordered pair, and this is
that it is extend-able to n-tuples in a more natural manner than Kuratowski
whose extensions are rather ugly and remain as a pair despite nesting. An
example of how Hausdorff's extend to a triple is (a,b,c)={ {1,a},{{2,b},
{3,c}}}. However this is still not really a genuine extension. During my
search for one I noticed that a simple modification of Hausdorff's pair
actually yields a nice result as follows:
Def.) (a,b) = { {1,{a}} , {2,{b}} }
where 1, 2 are the usual known specific Von Neumann ordinals.
A nice comment that I had is that this is a {position,{data}} pair. This
really gives it an intuitive flavor.
This pair extends to any n-tuple whether n is finite or even infinite.
Examples:
(a,b,c) = { {1,{a}} , {2,{b}} , {3,{c}} }
(a,b,c,e,d) = { {1,{a}} , {2,{b}} , {3,{c}} , {4,{e}} , {5,{d}} }
etc...
where 1,2,3,4,... are the usual specific Von Neumann ordinals (in
particular only 1 is singleton, all the others are not singletons)
All those are genuine extensions of the definition that are related to the
tagging strategy of this definition which is in my opinion the simplest of
copying strategies of the essential core of ordered tuples.
A nice result is that this actually simplifies sequences, a sequence
traditionally defined as a function (a set of ordered pairs) from an
ordinal domain, so it is a set of ordered pairs, this can be simplified to
a set of {position,{data}} unordered pairs! An example:
S= 5 , 3, 8
S= { {{1},{1,5}} , {{2},{2,3}} , {{3},{3,8}} }
S= { {1,{5}} , {2,{3}} , {3,{8}} }
Clearly the third is much simpler and more natural, and not only that it
also admits a simpler definition of its projections. So it is the simplest
formal presentation of sequences and tuples in general.
The projections can actually be defined by a common definition using
ordinal parameters:
x is the i-th projection of t <-> Exist q,z. q e t & z e q & i e q & x e z
& (i=z ->~Exist m. m e i & ~m=x)
I personally think that this nice result (which might be well known) comes
as a reflection from the natural looking and easy to understand pair of
Hausdorff's, so at the end although we began with a rather formally complex
pair albeit very natural yet we reached to both a simple and a natural and
a more general version of definition of pairs, n-tuples and sequences.
So it is not a good habit to sacrifice naturalness for formal simplicity
when the later is immaterial!
Best Regards
Zuhair Al-Johar
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