MATHEMATICAL CIRCLE: On the Banach-Tarski paradox and the size of the ordered numerical sets

Today we got to know that the Banach-Tarski is still a concern for human kind.

Thomas LaRock, who is a member of one of our social networks, sent us a call like this: WARNING: DEEP THOUGHTS AHEAD --> There are Almost No Rational Numbers.

We then visited the website (Rochford, 2013) to read the following:

A counterintuitive construction shows that not every subset of the reals is measurable. The counterparts of such odd sets in R2 are intimately connected to the famous Banach-Tarski paradox.

Curious enough, we then directed ourselves to (Good Math, 2014).

There we stumbled on the following (Good Math, 2014):

How about this? Take the set of all natural numbers. Divide it into two sets: the set of even naturals, and the set of odd naturals. Now you have two infinite sets,
the set {0, 2, 4, 6, 8, ...}, and the set {1, 3, 5, 7, 9, ...}. The size of both of those sets is the ω - which is also the size of the original set you started with.

We then found the roots of the problem: Size for mathematicians, in terms of numerical sets, is the cardinality of those sets, which is measured from a human perspective: If we cannot count by hand or by any instrument that we have invented, in its totality, then it is all infinite, therefore of the same size… .

We ourselves have written about that (Pinheiro, 2009): They are mixing up things.

Basically, when we enter the World of Mathematics, we have to use a special language developed to deal with it, so that our brain adapts.

This special language cannot be formed from terms that point to more than one world object, since we need maximum objectivity in Mathematics: All is supposed to be abstract and perfect.

This way, we cannot simply say that the cardinality is infinity and we then have the same size of set when we have even natural numbers or the whole of the natural numbers.

The cardinality is infinity in what regards our capability of counting, of seeing, and etc.

This way, both situations create the same effect on our eyes and in our heads, but they are situations that are very different if the perspective of the World of Mathematics is considered.

Notice that the World of Mathematics is very different from the World of the Computers.

Computers are so close to us and our logical reasoning, when that is purest, that some people think of replacing us with them.

The World of Mathematics however is completely inaccessible to us: It lies there, on paper only, as it is, if so.

Notice that we have created sets inside of it that we cannot really describe with our vain human tools (we can mention them and the collective unconscious gives us the right pick, but we cannot list all the elements of the naturals, for instance, and this is even the topic of our discussion).

How do we deal with the straight line world?

We go again to (Good Math, 2013):

Take a line segment. How many points are in it? It's infinite. So, from that infinite set, remove an infinite set of points. How many points are left? It's still infinite.

Apparently, in the same way that we deal with the sets of numbers!

There is obviously only one solution to this problem: Segment all.

Lebesgue, with his infinitely perspicacious mind, had already noticed that somehow. See (Rochford, 2013):

The idea behind the Lebesgue measure is that the size of the interval (a,b) ought to be equal to its length, b−a. The construction of the Lebesgue measure generalizes this notion to a much larger class of subsets of the real numbers.

Let m denote the Lebesgue measure. From the previous discussion, we define m((a,b))=b−a. In order to extend the definition of the Lebesgue measure to a slightly larger class of sets, let us first consider the size of the set (0,1)∪(3,5). It seems quite reasonable to define

m((0,1)∪(3,5))=(1−0)+(5−3)=1+2=3.

We see that this idea readily generalizes to finite unions of pairwise disjoint intervals (recall that two sets are disjoint if their intersection is empty, that is, if they do not overlap). In that case, we define the Lebesgue mesure of the union to be the sum of the Lebesgue measures of each individual interval. More concretely, if a1<b1<a2<b2<⋯<an<bn, then

m(∪ni=1(ai,bi))=∑i=1nm((ai,bi))=∑i=1n(bi−ai).

We need however more than what Lebesgue has proposed to understand why there is probably no paradox once more and we are just confused for not being coherent/consistent or skilled enough on observation and logical reasoning.

It is obviously the case that if we segment the naturals, then we get that one is bigger than the other, and this idea is not only intuitive, but is obviously also true.

Take the naturals from 1 to 5, for instance.

When we consider them as a whole, we have {1, 2, 3, 4, 5} in this interval.

When we consider just the even ones, we have {2, 4} instead.

When we compare numerical sets, we must compare them by parts and comparing one part is equivalent to comparing the whole if they are regular things.

This way, 2N is smaller than N because when we consider the accumulation criterion {x ͼ SET/1 ≤ x ≤ 5} we get fewer elements in 2N and that is a reasonable sample (of size that is acceptable).

There seems to realistically be no mystery in any of it and we are simply uttering wrong things when we say that the cardinality of the reals is the same as the cardinality of the rational numbers, for instance.

The cardinality of the reals is greater than the cardinality of the rational numbers, quite trivially, since we add numbers to each small interval.

Marcia’s Measure would then be an interval of Lebesgue measure five, for instance, and would be suitable to determine how big an ordered numerical set is when compared to another.

____________________________________________________________

DO YOU WANT TO PLAY LOVE GAMES? GET THE BOOK HERE!