Lecture 4(A): Unifying n-tuples, Sequences, Functions, and Subsets

U of Arizona: n-tuples and sequences are shown to be functions. We also describe the set of all functions from one set into another set, and the number of such functions. Continued in Lecture 4(B) for subsets and the power set, and some examples.

[angelmendez-rivera]

The alternative, set-exponent notation for R^2 and R^n respectively is much less awkward if, rather indexing ordered pairs with the set {1, 2}, you index them with the set {0, 1}, and rather than indexing n-tuples with the set {m : 1 =< m =< n}, you index them with the set {m : 0 =< m =< n – 1}. In fact, this built into the set-theoretic axiomatization and construction of the natural numbers, by using the Von-Neumann construction as a model of the Peano axioms. This is also why I decided to remark in the earliest video of the series that, indeed, you should be using the symbol N to refer to the set-theoretic natural numbers, which include 0, and not the primary-school natural numbers, which for some illogical reason, are taught as not including 0. Anyway, the point is that, in the Von-Neumann construction, 2 := {0, 1} is actually a definition, so the equality R^2 = R^{0, 1} is not a mere heuristic identification, but a valid set-theoretic equality. This also justifies why the notation for the set of real sequences, R^N, is not so awkward. Since N is infinite, there is not much of noticeable visual difference if you decide to start with an index other than 0. However, including 0 in N generalizes the notation with more consistency.

Also, I think one problem this video suffers is that, it is included in the playlist under a title which does not follow chronologically with how videos are published in this channel. These definitions only work if you have a reasonable set-theoretic definition of a function, which does not exist in the videos that precede this one in the playlist. My proposal is that a function f from the set X to the set Y is that f is the set of all {{x}, {x, y}}, such that x is an element of X, and y is an element of Y, such that for all x in X, {{x}, {x, y}} is an element of f, and if {{x}, {x, y}} and {{x}, {x, z}} are both elements of f, then y = z, for any y and z in Y. This would solve the problem, and it would make the idea of the empty function work much better.