But I'll leave you with this little tidbit for a possible future discussion...
∞≠∞
By the way, I was serious about this....
If anyone is interested in this, then I suggest some study in number theory, real analysis, and set theory.
Okay, so here's how it goes. Any set of numbers has what is called
cardinality. Cardinality is nothing more than the number of elements in a set. For example, take the set of even numbers between, and including, 0 and 10: {0, 2, 4, 6, 8, 10}. The cardinality of this set is 6 (it has 6 elements).
Now, with the definition of cardinality in hand, let's talk about infinity. For our set, let's take the positive integers (1, 2, 3, ..., ∞). This is denoted by a "fancy" capital Z+. (I say "fancy" because on the chalkboard it is written with two diagonal lines instead of one, and the + signifies the positive ones.) The cardinality of Z is... you guessed it... infinity. The cardinality of this set is denoted by the Hebrew letter
aleph. More precisely, aleph-null (the letter aleph with a subscript zero).
Now... let's take for our set,
all the integers (not just the positive ones, but the positive, negative, and the element zero). You may think that we just doubled our set and added another element (zero) so it must be larger. Can anyone guess what the cardinality of this set is?
It is still infinity (aleph-null)!
This kind of set, such as the set of positive integers, and the set of all integers, is further defined as being
countably infinite, - it is denumerable. For example, we can start with the number zero, then increment one positive and one negative, and keep repeating this forever as we count (0, 1, -1, 2, -2, ...).
We can count them. And the cardinality of the two sets I gave as examples can be proven to be the same if we can define a
one-to-one and onto mapping between the two sets (which we can).
Now, you may have guessed that there are sets that are infinite and
not countable. For example, there are an infinite number of numbers between 0 and 1. Try to count them. Here, I'll get you started: 0. But where do we go from there? See... we can't count them!
Okay, almost done...
Now let's take as our set the set of all
real numbers (R). (Z and Z+ are subsets of this set by the way, and R is a subset of C - the set of all complex numbers. Just an aside....) With the set of all real numbers, R, in our mind's grasp, can anyone guess what the cardinality of this set is?
Yup. Infinity. But...
Do you think it is the same as the infinity of Z? The answer is
NO!
The cardinality of R is
not equal to the cardinality of Z!
(The cardinality of R may be aleph-
one, but it is still unproven - see the
continuum hypothesis).)
So how would we prove this you might ask? If they are one-to-one and onto, then the cardinality is the same. But we are not able to define a one-to-one and onto mapping between the two sets, which means the cardinality is not the same. Mathematically, you start by assuming you
can make this mapping, and do a proof by contradiction.
The moral of all this is that sometimes infinity does not even equal infinity. There are different "degrees" of infinity!The person who first realized this was
Georg Ferdinand Ludwig Philipp Cantor. He suffered from mental illness. Coincidence...? I don't think so....