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 Maarten Maartensz:    Philosophical Dictionary | Filosofisch Woordenboek                      

 T - Tuple

 

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 (the empty set). This is one way to generate - a set-theoretical equivalent of - the natural numbers.

Thus with "#(.)" = "the number of":

#()               = 0
#({})            = 1
#({,{}})      = 2
#({,{,{}}) = 3  (etc.)

Those seeking even more refinement can use {} = . 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.

 


See also: Pair, Cartesian Product, Structure, Set Theory


Literatuur:

Cech, Halmos, Quine,

 Original: Aug 28, 2004                                                Last edited: 12 December 2011.   Top