Georg Cantor and Different Levels of Infinity – It’s True!
Sometimes when people think too deeply about things they can suffer depression or discover something new that was never discovered before. Georg Ferdinand Lugwig Philipp Cantor (say that fast 3 times in a row) did both.
Countable Infinity
Cantor, a German Mathematician born in 1845, created set theory now well known in mathematics. One main area he investigated is a one-to-one correspondence between sets.
As an example, let’s take the set of natural numbers (N) (1, 2, 3, 4, 5, … and so on – all positive numbers) and the set of integers (Z) (…-3, -2, -1, 0, 1, 2, 3…). If we pair up each natural number (0 and 1, 1 and 2, -1 and 3, 2 and 4, -2 and 5…) with each integer you can match each natural number with each negative number for both entire sets. You may wonder how this can be since the natural numbers don’t contain any negative numbers. On the surface you might say there are more integers than there are natural numbers however, these are sets that go on forever – infinite sets (infinity is not a number). Cantor argued (showed) both sets are countable infinite sets. Right away, this goes against our common sense. Each of theses sets are infinite, yet we can count their members which are the same for each set. It is said these sets have the same cardinality.
Also, we can also pair up the infinite set of rational numbers (Q) or quotients of integers without using 0 in the denominator (fractional numbers like -1/9, -3/5, 1/2, 2/3/ 3/4 etc.). It turns our we can make a one-to-one correspondence with fractions and the set of integers (Z) as well. Strange.
Uncountable Infinity
But, if we try to match up the infinite set of real numbers (decimal numbers like 3.14159, 0.45345, 12.34543322, -320.4333, etc), the real numbers are much larger than say the infinite set of integers. We cannot make a one-to-one correspondence between them. So, we can only conclude that the infinities for each of those sets (real numbers verses integers) are different. We cannot count the real numbers. These infinities are different levels (sizes if you must, even though infinity is not a number) where the infinity for real numbers is much larger than the infinity for integers. We call the real numbers an uncountable infinite set. In fact, there are an infinite different levels (sizes) of infinity. Don’t think about this too much or your brain will rot. Yet, it’s still interesting to think about.
Cantor was in and out of sanatoriums at the end of his life for depression. He died from a heart attack in a sanatorium on January 6, 1918. He was 72 years old.
Read more details about infinite sets here.