The set-theoretical meaning of ordered tuples shows that being an $n$-tuple is a predicate of a logical calculus. Therefore, we can use the axiom of separation to define what today's mathematicians call the Cartesian product. René Descartes (1596 - 1650) invented this concept in the 17th century, long before the set theory was born in its strict form but with our new set-theoretical tools we can provide a foundation for this concept:

# Definition: Cartesian Product

Let $$A_1\ldots A_n$$ be sets. Using the set-theoretical meaning of ordered tuples and the axiom of separation we define the set $A_1\times A_2\times\cdots\times A_n :=\{(x_1,\ldots,x_n)\mid x_i\in A_i, 1\le i\le n\}.$ as the Cartesian product of the sets $$A_1\ldots A_n$$. In other words, the Cartesian product is the set of ordered $$n$$-tuples $$(a_1,a_2,\ldots,a_n)$$, such that $$a_i\in A_i$$ for $i=1,\ldots,n.$1.

If all of the $$A_i$$ denote the same set $$A$$, then we use the short form

$A^n:=\underbrace{A\times A\times\cdots\times A}_{n\text{ times}}.$

Corollaries: 1
Definitions: 2 3 4 5 6 7
Examples: 8
Explanations: 9
Motivations: 10 11
Parts: 12
Proofs: 13 14 15 16 17 18
Propositions: 19 20

Github: #### Footnotes

1. Please note that, in general, the Cartesian product is not commutative, i.e. $$A\times B\neq B\times A$$, because of the order or the elements would change, in contradiction to the definition of ordered tuples.