Re: Zuhair's ordered pairs

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.
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
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.

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

NO NO NO, you are mistaken

Why you only read this definition, 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 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.

Second the above definition does't work like Kuratowski, it works
because
{y} is never equal to 0 , that what makes it work, See the proof of it
to understand it.




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.

also works in ZF, so the additional level of nesting seems (?)
needlessly (if only slighly more) complex. The proof is simple: Suppose
{x,{x,y}} = {z,{z,w}}. Then we must have x=z and hence y=w. For if
~x=z, then we have x={z,w} and z={x,y}, in which case x={{x,y},w},
contradicting foundation. However, when Kuratowski formulated his
definition in the early 1920s, foundation was not yet considered a
standard part of set theory. The axiom was first studied explicitly by
Mirimanoff around 1917 but didn't gain wide acceptability until the late
1920s due mostly to the influence of von Neumann's work. I believe
Zermelo himself did not explicitly assume it until his 1930 paper on the
cumulative hierarchy. So if indeed K. tried the definition above, it is
reasonable to think he'd have rejected it and sought a definition that
is independent of foundation.

This is just speculation, of course. For all I know, K. never even
considered the definition in question. Or perhaps he did but simply
found it aesthetically more pleasing to have both elements in the
definiens nested to the same degree.

.

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