The ordered pair

The third axiom allowed us to form the "even" set: {The, B}. Do you remember that this set only exists if The and B already existed previously. We should not confuse "pair" with the "ordered pair" set. "Order" in mathematics is of fundamental importance. We are talking here about 'relationship of order'. In order to build future relationships of order that will be very useful to us, we need to take a first step. The first step, as the astronaut who first stepped on the moon proved, can have great significance. Something similar happened with the "discovery" of the "ordered pair." It was Norbert Wiener who first "saw" correctly what an ordered pair is. He had the happy intuition that the ordered pair is nothing more than the whole {{The}, {The, B}}. At this point we can naturally consider three problems: the first is whether the whole {The} exist; the second is whether the whole {The, B} exists and the third is whether ordered pair exists. We must not forget that our hypothesis is that The and B they are given sets. We can say the following: supposing that The and B are existing sets because there would also be sets

{The}, {The, B} and {{The}, {The, B}}?

You may have already noticed that the second problem was solved by the third axiom, that is, the ZF (3). Now note that if {The} exist then again at ZF (3), we immediately conclude that the third problem is solved, that is, that the ordered even set exists. It remains for us, therefore, to justify that {The} exist. We do not know how to work miracles with the Zermelo-Fraenkel set theory, so the only chance we can solve our problem is to resort to the axioms already assumed to be true or their consequences already deduced. This is how part of mathematical research works. But which of the three axioms is what we need? Or do we need all three and some more truths already deduced?

It is very common in mathematics to discover simple and withering arguments for the demonstration of truths. This is our case, because to see that the whole {The} exist, just argue that {The, The} exists by the axiom ZF (3), since we are assuming that The exist!

We really only need one detail: why {The} = {The, The}? Do you remember the first truth? Let's remind her:

ZF (1) Extension Axiom:

if The and B are sets and if for all xThe if and only if x B, then The = B.

The first truth of the Zermelo-Fraenkel theory means that two sets are equal if, and only in this case, the relevance of x one of them is equivalent to the relevance of x to each other. Now it is not clear now that {The} = {The, The}?

It costs nothing to emphasize this point because you may be just beginning your mathematical experience and not yet very familiar with the rigor and subtlety of mathematics: these two sets are the same because all x which belongs to one of them belongs to the other. Note that the word all may give the impression of many, but here there is only one set (the set The) playing the role of x. The existence of the ordered pair is thus established. {{a}, {a, b}} which we will indicate by (a, b). Remember that we still don't know if exist some set in the universe of Zermelo-Fraenkel theory. We just demonstrate that if there is any set, then there will also be some ordered pair. Challenge for you to decipher in a week: why (a, b) ≠ {a, b} ?

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