Re: Kuratowski Ordered Pair

G. Frege wrote:
Han de Bruijn wrote:
Hero wrote:

> So it is impossible, to define an ordered pair with set-theory only.

.......

Hamilton introduced this concept of ordered pairs into math:
Theory of conjugate functions, or algebraic couples; with a
preliminary and elementary essay on algebra as the science of pure
time
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Couples.html

So we have to lift the ban of time in math.

Sir William Rowan Hamilton (1805-1865), theory 1834
Georg Ferdinand Ludwig Philipp Cantor (1845 - 1918)

So ordered pairs were _defined_ before Cantor and set theory were even
born.


No, they weren't.

Read Hamilton.A little bit later Grassman developed tuples for what we
now call vector-spaces. And Hamilton introduced the Quaternions of
four-tuples, with a scalar component, as one can not embedd the real
number-line into 3D and keeping the multiplication intact.

But -of course- ordered pairs can be introduced as
primitives.

Hamilton has done this.

"primitive : ....
2: a mathematical expression from which another expression is
derived"
http://primitive.dict.die.net


Actually,

"Norbert Wiener proposed the first set theoretical definition
of the ordered pair in 1914:

(x,y) := { { {x}, {} }, { {y} } }."

(http://en.wikipedia.org/wiki/Ordered_pair)


In the notation of set theory one can not write:
x = } a, b } or
y = } d, b, f {
So one has already a concept of an ordered pair of brackets
beforehand.

But why Wiener felt the need to alter Hamilton's approach to ordered
pairs?

With friendly greetings
Hero
.