Re: Zuhair's ordered pairs

On Sat, 16 Feb 2008 07:12:37 -0800 (PST), Zaljohar@xxxxxxxxx
<Zaljohar@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. 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}} }

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}} }

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

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}}

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.

.