The set that contains absolutely everything, it is interesting to think about something like that, can you imagine it? It is not unreasonable to believe that something like this exists, considering that our notion of set is that it is a collection of objects that have certain characteristics in common.However, the reality is completely different. In fact, the formal definition of a set does not exist. It is similar, in fact quite similar to what happens with the geometric point. It is something that everyone recognizes, but that cannot be formally defined without falling into some contradiction.And similar to the geometric point, in order to understand the sets, it is necessary to give certain rules that they must satisfy with, these rules are known as axioms. These are rules that must be followed to the letter within this world that we know as set theory.In the universe of mathematics, each area is a world, and each of these worlds has its rules and its language, with which we can build or move on this world, always under the shadow of those rules.When any of these rules are broken, we are actually moving out of the world.With this idea in mind, let's think about how we are moving in a place governed by what is known as the Zermelo-Fraenkel axioms.In particular, there is one of them that is vital to the purpose of this article. Said axiom says the following:For all set A and a property P, there exists another set R such that x is in R if and only if x is in A and furthermore x satisfies the property P.But what does it mean? Basically, this axiom gives us a recipe to build new sets from others and from some property, that is, this axiom tells us about the possibility that certain sets are as we believed.Do you remember? It tells us that, in effect, there are some sets that are collections of objects that have some characteristic in common. However, it is only one of the axioms, this means that there are sets that are not of this style.Let's try to understand this axiom better. We all know that the set of even numbers is a set that is just like this, that is, a collection of objects, in this case natural numbers, that have a common characteristic, which is, that they must be multiples of 2.In the language of this axiom, we can rewrite the above as follows.For the set A, the set of natural numbers, and the property: "must be a multiple of 2", there must be another set R, such that x is in R if and only if x is a natural number and is also a multiple of 2. And indeed, we all know that this set is the set of even numbers.Okay, okay, but what does this have to do with the set that contains everything?
Well, let's get to that.
Imagine for a moment that in this strange world of set theory, there exists a set that contains everything. In particular, such a set should contain itself.
Can you imagine something like that? What does a set that is an element of itself look like? It would be similar to thinking that you are one of your hair. Pretty crazy idea.
Basically this idea of the non-existence of a set that can have itself as one of its elements is the one that contradicts the existence of the set that contains everything.
Let's try to understand this in depth.Since this idea of the set that has itself as an element is quite crazy. Let's think about the opposite, that is in those sets that do not have themselves as elements.This is a property, which is why it serves as an ingredient in the axiom we mentioned earlier. The other ingredient we need is a set. Ah! But we are imagining that the set that contains everything exists, so under this assumption, it can serve as the missing ingredient.So, for the set that contains everything, let's call it U (it is funny, it is called U, like universe) and for the property: "it does not have itself as an element", there must be another set R, such that x is in R if and only if x is in U and furthermore x does not have itself as an element.This R is another set for which it must live in U (since U contains everything). There are only two possibilities 1) R contains itself as an element 2) R does not contain itself as an elementIn the first case, if R has itself as an element, then R is in R, so it must satisfy the property that says: ... Do you remember? ... It says that R should not have itself as an element, that is, R should not be in R.That? !!! R is in R but not in R? Exactly, this is completely absurd.On the other hand, in the second case, since R does not have itself as an element, then it satisfies the property and is also in U, so R must be in R. Again, what? !!!!, Exactly, we have the same absurd result again.Ok, ok calm down, so if we imagine that there is a set that contains everything, this leads us to that there should be another set that has itself as an element, but at the same time, does not have itself as an element. This is all absurd, right? For all this, it cannot be true that there is a set that contains everything, at least within this world called set theory.
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