Foundations of mathematics

A tuple (or n-tuple, for any integer n) is an interpretation of a list of n variables. It is thus a meta-function from a finite meta-set, to the universe. Tuples of a given kind (list of variables with their types) can be added to any theory as a new type of objects, whose variables are meant as abbreviations of packs of n variables (copies of the list with the same old types) x = (x0, · · · , xn−1). In practice, the domain of considered n-tuples will be the (meta-)set Vn of n digits from 0 to n − 1. Set theory can represent its own n-tuples as functions, figuring Vn as a set of objects all named by constants. A 2-tuple is called an oriented pair, a 3-tuple is a triple, a 4-tuple is a quadruple. . . The n-tuple definer is not a binder but an n-ary operator, placing its n arguments in a parenthesis and separated by commas: ( , · · · , ). The evaluator appears (curried by fixing the meta-argument) as a list of n functors called projections: for each i ∈ Vn, the i-th projection πi gives the value πi(x) = xi of each tuple x = (x0, · · · , xn−1) at i (value of the i-th variable inside x). They are subject to the following axioms (where the first sums up the next ones) : for any x0, · · · , xn−1 and any n-tuple x, x = (x0, · · · , xn−1)⇔ (π0(x) = x0 ∧ · · · ∧ πn−1(x) = xn−1) xi = πi((x0, · · · , xn−1)) for each i ∈ Vn x = (π0(x), · · · , πn−1(x))