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No fun an empty mathematical universe


However imaginative and creative we may be, we cannot deduce the existence of sets of other axioms that no longer contain in themselves the hypothesis of the existence of any set. Or at least, to be more rigorous, we could say that no one to this day has achieved the feat of mathematically bringing into existence a set from other axioms that no longer contain any hypothesis of existence. Let us not therefore prolong this agony of nonexistence of sets. We bow to the reality of our powerlessness to produce sets out of nothing. Let us convince you that there are sets from now on. This undisputed truth will be our Zero Axiom.

Note that, on the other hand, let's not overstate our desire for some set in mathematics. The assumption that there are sets does not authorize us to state that there are more than one set. The most we can say about the existence of anything is that of the axioms 0, 1, 2, and 3, only Axiom 0 assures us that there are sets, but it does not tell us how many sets there are.

We must then ascertain what consequences follow now from these four axioms, and especially if we can conclude that there are many sets in mathematics. Let us remember Axiom 2, which stated that if The were a pre-existing set then there would also be the set B = {x belonging to The: A (x)}, which can be understood as “B is the subset of The formed by the sets x that belong to The and that satisfy the property THE" Now we can immediately think of a simple property A (x): x x.

That is, the property we are thinking about is that x meets the condition of be different from yourself. So we define the set B = {x belonging to The: A (x)}= {x belonging to The: x ≠ x }. You already realize that there are no sets that belong to this set, because there is no set that is different from itself. So when we think of the subset of the sets of the set The which are different from themselves, we are thinking of a set “empty" We then discover that the first set to present to us is precisely the empty set, that is, a set that has the property that whatever set X of the mathematical universe, X does not belong to the empty set. Let's baptize this first set that came into existence Æ. It is interesting that this fact is precisely the first set of which we become aware of is a “set with nothing inside”.

Mathematics is like that, it is always amazing. Want more surprises? Well, let's now deduce that there are then, from the empty set Æ, infinite sets in the mathematical universe. Think of Axiom of the Pair and Axiom 3: so how Æ exist we can form the set { Æ, Æ} = { Æ}. Again, using the Axiom of the Pair, and the Axiom 3, we deduce that there is the set {Æ, { Æ}}. Now we stop no more: successively applying Axioms 2 and 3 gives us the infinite sequence of sets: Æ, {Æ}, {Æ, { Æ}}, {Æ, {Æ}, {Æ, {Æ}}}, … .We are ready for the following definition and recognition: 0 = Æ, 1 = {Æ}, 2 = {Æ, { Æ}}, 3 = {Æ, {Æ}, {Æ, {Æ}}},… You have just been introduced to the famous “natural numbers”Which, in turn, have just been born and become the first inhabitants of the mathematical universe. Also note the fact that 1 = {0}, 2 = {0, 1}, 3 = {0, 1, 2},… CHALLENGE: read this column very carefully, and as many times as necessary, until you are absolutely convinced that you understand what "natural numbers" are. The last challenge is to ask him to demonstrate that the natural numbers are all two by two. So there are infinite mathematical objects!

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