We leave you a challenge: to convince yourself that the **"complementary to one set over another"** It does not need new axioms to exist. In fact, consider a set A and a set B. Let's think of the complement of B with respect to A. We write: A - B = {x: x belongs to A but not to B}. This set is nothing more than the set of sets that belong to A and satisfy a property P (x).

Now the property "x is a set of A that does not belong to B" defines a set by Axiom 2. Therefore, the complement of B with respect to A exists because of the axioms already accepted. Similarly, you may be convinced that the other sets of the challenge also exist because of the accepted axioms.

With the meeting axiom we can form the "tender" set, generalizing the concept of the "even" set. Given sets A, B, and C, we define, with the help of the meeting axiom, set {A, B, C} as the set of sets {A}, {B}, and {C}. Notice that the set {A} exists because of the pair axiom that says {A, A} is set. That is, {A, A} = {A} is a new set. Similarly, the sets {B} and {C} also exist and, therefore, by the meeting axiom we can form the meeting set {A} È {B} È {C} = {A, B, C}.

Now we are interested in the orderly tender. Given sets A, B and C, we define the ordered pair (A, B) using the Axiom of the Pair and now we define a new "ordered pair" which is the "ordered tender": ((A, B), C). note that

((A, B), C) = **{{{{A}, {A, B}}}**, {**{{{A}, {A, B}}}**,Ç}**} **

Can you check if the keys are correct? Note that it is much easier to think of the ordered tender as ((A, B), C), although this "ordered pair" means the "complicated" set above right.

Mathematics is full of situations like this, that is, definitions by **"recursion"**. We define a recursion-ordered term to be "a certain ordered pair plus a third set". In the same way we can define an ordered quadruple, an ordered fivefold,…, an ordered "n-double", etc. If a physicist imagines a quantity as a set, then he can easily imagine a point in space as an "ordered n-double" of quantities. For example, (x, y, z, t), the quadruple of the three quantities that give the spatial location of a particle and the instant it occurs. We now know that (x, y, z, t) = ((x, y, z), t) = (((x, y), z), t). A physicist then has at his disposal **"so many coordinates"** as many as you want. Superstring theory considers 11 to be a very plausible number for the correct number of coordinates in our universe.

The geometers also have at their disposal the numbered point space of Analytical Geometry invented by Renée Descartes and Pierre de Fermat.

We could now take the avenue of geometry or the avenue of physics. But we still don't have a good theory of the universe of numbers. We'd rather have a little patience and calmly discover the fundamentals that will help us lift the Math building.

We now need the "power set". That is, we need a new axiom that guarantees us the existence of "parts of a set." It is an intuitive notion of the parts of a set. But why can we "think of them"? It is precisely Axiom 6 that allows us to suppose that the parts of a set are legitimate sets for our thinking. In the next column we will go into detail about the existence of the parts of a set.

Back to columns

<