On (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.
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.
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 have
written about that before (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.
In 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, etc.
Both situations
create the same effect on our eyes and in our heads but they are situations
that should be very different if the perspective of the World of Mathematics is
considered.
How do we deal with the
straight line world?
On (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
Let m m ((a,b))=b−a (0,1)∪(3,5) .
It seems quite reasonable to define
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 measure of the union to be the sum of the Lebesgue measures
of each individual interval.
If we segment the naturals, then we get that the set of the natural numbers is
bigger than the set of even/odd numbers and this idea is not only intuitive but is obviously also
true.
Take
the naturals from 1 to 5.
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 belongs to N/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.
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 the slices of the sets under analysis that correspond to an interval of Lebesgue measure five and
would be suitable to determine how big an ordered numerical set is when
compared to another, provided that the elements of each one of the sets under analysis be equally distributed along the real line.
________________________________________________________
References:
Rochford, A. (2013). There
are Almost no Rational Numbers. Retrieved January 1st 2014 from http://www.austinrochford.com/posts/2013-12-31-almost-no-rationals.html?utm_content=bufferc9d48&utm_source=buffer&utm_medium=twitter&utm_campaign=Buffer#!
Good Math, Bad Math. (2014).
The Banach-Tarski non-Paradox. Retrieved January 1st 2014 from http://scientopia.org/blogs/goodmath/2012/01/06/the-banach-tarski-non-paradox/
Pinheiro. (2009). Infinis. Retrieved
January 1st 2014 from http://www.protosociology.de/Download/Pinheiro-Infinis.pdf
