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The orderly tender


With the third truth of the Zermelo-Fraenkel Set Theory, we have been able to define the ordered pair which is of great importance in the matter of order throughout mathematics. It should be noted that Norbert Wiener's definition of ordered pair (a, b) = {{a}, {a, b}} may seem shocking at first glance. No wonder the story that Norbert Wiener visiting Italy once gave a lecture to mathematicians and scientists, attended by the famous Italian mathematician Giuseppe Peano, and in which Wiener showed his definition of pair. salary. Peano would have been visibly shocked that such a precise, rigorous, and simple definition was not discovered by himself.

Readers may be bothered by this setting for other reasons. It is useful to dwell a little on this notion of ordered pair. You should keep in mind that (a, b) = {{a}, {a, b}} can also be written as (a, b) = {{a, b}, {a}}, ie not No matter what “order” you write sets {a, b} and {a}, it does not change the order of a and b in the ordered pair (a, b). But of course if you move the order in (a, b), then you have another set: (b, a) = {{b}, {a, b}}.

You should also note that the definition of ordered pair has nothing to do with other concepts such as Euclidean plane, axes of reference, etc. Wiener's definition is very lean, it depends on nothing but the notion of assembly. It is an elegant example of the power of mathematics to produce clear, accurate and concise information.

We leave, in the last column, a “Challenge for you to decipher in a week: why (a, b) ≠ {a, b} Surely you must have already thought of the right solution. Just to make sure everyone understands, let's give a demo here. The set {a, b} is not the ordered pair (a, b) simply because there are sets in {a, b}, for example, b, which do not belong to the set (a, b).

We will still leave a

Challenge: why B does not belong to (a, b)?

To further reinforce the notion of ordered pair, let's define the ordered suit: (a, b, c) = {{a}, {a, b}, {a, b, c}}. Now we leave you with the challenge of writing all the ordered suits we can get with the {a, b, c} set.

Again the notion of ordered suit has nothing to do with the notion of three-axis coordinate referential. Of course we can use the notion of ordered suit to study geometry and / or physics, in particular, to indicate the position of a particle in three-dimensional space. But the definition of ordered suit only requires the notion of set and set formation, that is, one of the axioms of the Zermelo-Fraenkel Set Theory.

The concept of order is fundamental in mathematics. Therefore, it is important that we are able to conceive as simply as possible the ordered pairs, the ordered suits, the ordered quadruples, etc. In the future we will talk more about order in mathematics.

Remember again that we do not yet know if there is anything in our universe of sets. But we already know that if there are two sets a and b, then there will be sets called “ordered pairs”, “ordered suits”, ordered quadruples, etc. We still have a lot to walk. But the path is interesting and surprising. It is not possible to predict which mathematical objects, or which "things" will be discovered in the universe of mathematics. But you can be sure it will remain a fascinating path.

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