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An important truth before the third truth


Using other words we can say that it will be licit to build a subset B in The from a property A (x). We write: B = {x Î The: A (x)}, and read "B is the subset of The formed by the sets x that belong to The and that satisfy the property THE”.

When the Singer Set Theory, called the Intuitive Set Theory, came up, there was the idea that “any property” could be considered so that the objects that satisfied it formed a set. Thus, there would always be the set of sets that satisfied the property THE, whatever the property THE. So mathematician and philosopher Bertrand Russell made the following reasoning: consider the set of sets x that do not belong to themselves. For Russell the property THE it was "x does not belong to x" It seemed clear that this set existed, for, for example, the set of natural numbers is not a natural number and therefore the set of natural numbers does not belong to itself. Many mathematical objects form sets that are not one of these objects. To give an example outside of mathematics (which is meaningless because out of mathematics we lose the possibility of speaking only “strict truths”) just to illustrate a bit more, the horse set is not a horse, just as a set of men It is not a man.

Let's do as Russell and call M the set of all sets. Consider the subset of M formed by sets that do not belong to themselves. Russell asked if M belongs to M. If it is true that M belongs to M, then M satisfies the property that determines the sets of M, ie M does not belong to M! We come to a contradiction, for a set M cannot belong to itself and at the same time not belong to itself. Well, as the ancient Greek philosophers assumed, we cannot have a true and false statement at the same time, at least in logic that they imagined to be the correct one. Thus we are forced to conclude that M does not belong to M. But then M satisfies the property that determines the sets of M. Therefore, M belongs to M! Contradiction again. This dichotomy, that is, this statement that is true if, and only if, is false, caused a scandal in Cantor's Set Theory. That was the reason why mathematician Ernst Zermelo (1871-1956) “create”The second truth of Set Theory, which is the Axiom ZF (2) above.

The Axiom ZF (2), that is, the second truth of the Zermelo-Fraenkel Set Theory prevents us from building the antinomy discovered by Russell. Assuming this axiom to be true, Russell's reasoning presented above is no longer possible. The reason is that we can no longer build set M of all sets. Simply because a property alone no longer determines a set. We need to have a prior set The, that is, that there is already a set The, to consider a subset of you of sets that satisfy a certain property. Therefore, we can no longer simply consider the set of sets that do not belong to themselves. Is that the “set of x that do not belong to themselves”Is not joint. As mathematician Paul Halmos said, “nothing contains everything" We leave you a challenge (think simply and calmly…): show that it stems from the second truth, that is, from the axiom ZF (2), what there is no set of all sets. It is interesting to note that although we still have no reason for sets to exist, we can already demonstrate that there is no set of all sets. Write it down: From the theory we have so far, we still don't know if it is true that there is any set, but it is already true that the set of all sets does not exist!

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