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Tuple:
In logic and mathematics:
Element made up of n things, with a first,
second ... nth element, as in "(x1, .., xn)"
Tuples are often named with their number of terms explicated: as in
"n-tuple". A pair is a two-tuple.
It is not difficult to reduce tuples to pairs with suitable definitions
or assumptions, for one can analyse a three-tuple as a two-tuple made up
of an element and a tuple, and similarly for four-tuples (an element and
a three-tuple) etc.
This is
the standard practice in set theory.
Also, as is also common practice in set-heory, one may take the element
to be Ø (the empty set). This is
one way to generate - a set-theoretical equivalent of - the natural
numbers.
Thus with "#(.)" = "the number of":
#(Ø)
= 0
#({Ø})
= 1
#({Ø,{Ø}}) = 2
#({Ø,{Ø,{Ø}}) = 3 (etc.)
Those seeking even more refinement can use {} = Ø. And the percipient
reader will have noticed that set-theory thus enables one to generate
natural numbers from nothing at all.
And also one may define an ordered two-tuple
(xa, xb) using sets:
(xa, xb) =df
{{x}, {x,y}} & {x} = xa & {x,y} = xb
on the principle that in {{x},{x,y}} one
can keep apart the two elements (one an
unit-set, one a pair-set), and
thus as it were pair off first and second element of the tuple involving
xa and xb in that order.
This was independently thought of - in that order - by Wiener and
Kuratowski. It showed that a special logic of relations as conceived in
Principia Mathematica could be avoided, as relations could be
analysed as tuples, and tuples as sets of the above kind. In case one
has proper classes, the above
reduction of tuples to sets doesn't hold for
proper classes. To have pairs of these too, Cech assumes special axioms
for pairs in general, for both sets and proper classes. This then
guarantees arbitrary n-tuples of arbitrary things. |
