Degrees of Infinity
The Inequality of Endless Sets
An infinity can be defined as something having either no upper limit or no lower limit (one might make the mistake of defining infinity as that which has neither an upper or lower limit, but to say this is to overlook a set, such as "postive integers" which has a definite beginning at zero, and extends from there to infinity).
This definition is simple enough. Infinity plays an obvious and important role in mathematics (the subject of this particular article) as well as in many other fields, such as physics or logic (though admittedly, the applications of infinites in other fields generally tend to boil down to applied mathematics).
This being said, it shouldn't be too much of a stretch to assume that all infinites are equal. Equally infinite.
The idea that this might not be so dates back to the very beginning of set theory and the ideas of Georg Cantor in the 1870's.
The Alephs
The idea that infinite sets might not be equal, but instead may be ordered in terms of degrees is one that seems overtly counterintuitive. After all, how can one infinite number possibly be greater than another infinite number?
The counter-logic to this, though, is incredibly intuitive, and deceptively simple.
First, some terminology should be cleared up. With the idea of "degrees" of infinites, a set of terms arose in order to define them - the aleph numbers.
The first degree of infinites is known as aleph-null (from the first letter in the Hebrew alphabet). The second is aleph-one and so on.
The fundamental ideas here are those of "countability" and "one-to-one relationships". The standard, aleph-null infinities are those which are considered "countably infinite," which may very well seem like a logically contradictory term, but is not so.
Countable Infinites
Countable infinity simply means that the sequence of infinite numbers can be counted sequentially (though without end). The integers, then, are aleph-null numbers, as they can be counted: 1,2,3,4,5,6,n+1...
The even numbers: 2,4,6,8,10,n+1...
The primes: 1,2,3,5,7,n+1...
These numbers can all be counted without end, and they all succomb to a one to one ratio with each other - that is, the elements of each set can correspond to elements in the others, and progress in an orderly fashion forever. For instance, the one to one ratio of integers and even numbers begins: 1 to 2, 2 to 4, 3 to 6, 4 to 8. The first integer is matched with the first even number, the second with the second, and so on.
While the numbers themselves may not be equal, this system can continue forever, the ratio remaining the same.
Uncountable Aleph-One's
What, though, of those infinity of numbers in between two integers, such as 0 and 1? After all, with just a bit of rational thought, it becomes clear that in between these two numbers possesses an infinite set of other numbers (0.01, 0.114, 0.00000456, 0.32432423423344, etc...).
While these numbers are also infinite, they are considered by mathematicians to possess a higher degree of infinity (aleph-one), as they are uncountable (because in between any two of them, there is always an infinite number of others), and cannot be counted one to one with any infinite set.
Can one possibly say, then, that the number of aleph-one numbers are greater than aleph-null numbers? Not exactly. They are both infinity. One can be considered greater than the other, however, and that is what makes this idea of cardinality of infinite sets so mathematically interesting.
The cardinality of infinities does noe end here, of course. This is just one example of a conclusion that might be drawn from this idea. It should be able, at the very least, to allow one to realize that perhaps not all infinities are equal, and that mathematicians have their work cut out for them in making sense of these things.
References:
"Aleph-0" Wolfram Mathworld.
"The Colossal Book of Mathematics." Martin Gardner. W.W. Norton & Co. 2001.
Infinity Symbol in the Lawn
|
More in Science & Nature
Latest Articles