A little more about ordered pair

If The and B are sets, so we define “ordered pair” as the set (a b) = {{The}, {a b}}. This is the famous definition first given by Norbert Wiener. We had left you a challenge: why (a b) ≠ {a b} ? We argue that the set {a b} not equal to ordered pair (a b) simply because there are sets in {a b}, for example, B, which do not belong to the set (a b). But we had still left a second challenge for you: why B does not belong to (a b)? In fact, we could also say that The does not belong to (a b). Come on: if The belong to (a b) = {{The}, {a b}}then or The = {The} or The = {a b}. In either case, we concluded that The belongs to The. We could then write The = {The,… } = {{The,… },… } = {{{The,… },… },… } =… .

Of course we need to make some comments now. First, we need to explain what we want to conclude. Well, we want to conclude nonsense! This is because we are reasoning according to a very old model of demonstration, which was already known to the Greeks before Christ. It is the famous “Reductio ad absurdum”. This demonstration strategy consists in supposing that the conclusion we want is false and, from this hypothesis, deducing an absurd proposition, that is, not valid according to our logical conventions. So we assumed that the ordered pair (a, b) was equal to the set {a b}. This hypothesis is the negation of what we want, that is, the negation of the claim that these two sets are different. We then deduce that The belongs to The. Now then we can indicate this fact by writing The = {The,… }, precisely because The is one of the sets that belong to The. But we can think about it again and write: a = {{The,… },… }. Note that we are already using two key pairs for the set. The. If we think again about the fact that The belongs to Thewe can once again write a = {{{The,… },… },… }. Now, you realize that this process does not stop anymore.

You could question our procedure by saying: look, you have not yet defined what you are "three points"so why are you writing “… ”? It would be a withering criticism. But let's say that we are just trying to indicate our reasoning, and that after intuitively reaching our goal we would put everything on clean plates, that is, we would make a deduction respecting all of our already established logical conventions, definitions and axioms. If you grant us this truce, then we can say that we are facing a fact that is at least uncomfortable and probably undesirable. This story of "infinite" keys scares us, and it doesn't seem like a good thing! Moreover, the idea of ​​a set belonging to itself is not intuitive, at least for us (and for you?).

What to do? To study mathematics one can draw on her story, trying to learn how other mathematicians faced the same questions we discovered for ourselves. Sometimes we discover new questions, and then it may be that the story doesn't help us. But in the case of the aberration of infinite keys, we are lucky: the mathematician D. Mirimanoff discovered in 1917 that such infinite decreasing sequences of pertinence may exist. In other words, they may exist mathematically. But he seems to have disliked them very much and proposed an axiom that forbids them. It will be our ninth axiom. Thus, as the ninth and final axiom will prohibit this kind of “freak of belonging,” we conclude that we have come to absurdity because we have assumed that the ordered pair (a, b) were equal to the set {a b}. We now use the principles of non-contradiction and the excluded third to conclude that (a b) ≠ {a b}.We owe you an explanation of our suggestion that “B does not belong to (a b)”In the last column for the demonstration that (a b) ≠ {a b}.

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