Re: Zuhair's ordered pairs

On Feb 17, 4:02 pm, Chris Menzel <cmen...@xxxxxxxxxxxxxxxxxxxx> wrote:
On Sun, 17 Feb 2008 04:02:33 -0800 (PST), Zaljo...@xxxxxxxxx
<Zaljo...@xxxxxxxxx> said:

On Feb 16, 2:15 pm, Chris Menzel <cmen...@xxxxxxxxxxxxxxxxxxxx> wrote:
On Sat, 16 Feb 2008 07:12:37 -0800 (PST), Zaljo...@xxxxxxxxx
<Zaljo...@xxxxxxxxx> said:

The following is my definition of ordered pairs:

Define: <x,y>= { {0,x} , {x,{1,y}} }.

Can't see much point to that.

Yes true , because you don't read all my posts, that's why you get
half the idea.

Well, I like my ideas half-baked. :-)

It is obvious this definition will work
if you know Kuratowski's definition works.  For it is Kuratowski's idea
that is doing all the work.  In particular, 0 and 1 are completely
superfluous; note that the definition

  <x,y>= { {0,x} , {x,{0,y}} }

Yes, this works but it has unnecessary addition that is 0 with the y

Er, well, you missed the point, viz., that neither 0 nor 1 was doing any
heavy lifting for you in your definition above -- though I see now in
light of your response that you wished to avoid ordered pairs being
singletons.  They did have the virtue (?) of performing that function.

See below:
works just as well and so the presence of 0 and 1 in your definition is
obviously playing no essential role in "ordering" the elements of the
pair.  So there is really no significant difference between your
definition and the definition

    <x,y>= { {x} , {x,{y}} }

Let x={y} then <x,y> is singleton. which is something that I already
wanted it to be forbidden.

Ok, but, question: why?

which is just Kuratowski with an additional, and unnecessary, level
of nesting thrown in.

NO NO NO, you are mistaken

Not with respect to the point *I* was making.

Why you only read this definition,

Er, well, because I found the exercise entirely pointless and
uninteresting.  Sorry...

I have posted a lot of posts to this thread and I said in the second
post that I simplified this definition to the following, in addition I
simplified Wiener's ordered pairs, see the whole of my posts they
contain the answers to your question:

First of all I simplified the above definition to the following:

Define: <x,y>= { {x,0} , {x,{y}} }.

Now you think that this definition is Kuratowski , No it is not.

I never said anything about that definition.

I said that Kuratowski ordered pairs has the property of being
Singleton if x=y and we need a pair that do not have this property,
i.e a pair that always have the cardinality of 2 for all x and y.

We *need* it?  Again, why?  I mean, once you *define* ordered pairs to
be sets of any kind, you are playing the representation game.  It seems
to me that all that really matters at that point, in the case of ordered
pairs, anyway, is getting a definition that preserves the desired
properties simply and efficiently.  After all, once you've *got* the
definition and have proved it preserves those properties, you rarely
even have to revisit it, and when you do (e.g., proving some function
exists at some level V_n) Kuratowski's works just great.  There just
seems no point in fixing what ain't broke.

If your motivation is to capture more accurately the "essence" of an
ordered pair, then, rather than introducing yet another set theoretic
representation that perhaps gets you a little closer, you should just
introduce ordered pairs directly into your set theory as objects in
their own right and axiomatize them.  Then you'd be talking about the
real thing.  By your own definition, for example, an ordered pair always
contains the empty set in its transitive closure.  That is certainly no
part of the essence of an ordered pair -- intuitively, they don't even
*have* transitive closures, let alone ones that always contain the empty
set.  Surely the fact that an ordered pair could turn out to be a
singleton is no worse a property for a representation of pairs to have
-- vis-á-vis their "essence" -- than the property of having the empty
set in their transitive closures.


All of what you said above is perfectly right! I never said that
imposing another condition on ordered pairs other than the
characteristic property of ordered pairs is IMPORTANT, my reply was
essentially made to the poster G.Frege who proposed a complicated type
of ordered pair that always has the cardinality of 2.

He objected to the ordered pair

<x,y>:= { {x,0} , {y,1} } by saying that by letting y=0 and x=1 the
pair will be singlton which is a defect in his opinion that it shares
with Kuratowski pairs.

Of interest the following ordered pairs ALL have cardinality 2.

1) Wiener's pair: <x,y>:={ {{x},0} , {{y}} }

2) My simplification of Wiener's pair

<x,y>:= { {x} , {0,{y}} }

3) My ordered pair

<x,y>:= { {x,0} , {x,{y}} }

4) Another version that I defined:

<x,y>:= { {x,0} , {{y},1} }

5) The poster G.Frege's ordered pairs

<x,y>= { (x,1) , (y,2) }

were: for all a , for all b : (a,b) is Kuratowski pair of a and b.

From the above You see that the simplist of all these pairs is my
simplification of Wiener's pair, and the most complicated pair is the
poster G.Frege one. So that was my main message that is there are many
ways to define pairs of cardinality 2 always that are simpler than
5).

Zuhair


However, here's something mildly interesting here that I hadn't
thought about before.  One might wonder why Kuratowski's definition
uses the singleton {x} instead of simply x.  For the definition

  <x,y> = {x, {x,y}}

To get Rid of Regularity , this is well known.

Could be!  You are the expert on this Deep and Critical matter.  I only
said it was something I hadn't noticed and it got me to thinking about
it.

.

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