History of Infinity
The concept of infinity was the one that had both philosophers and mathematicians perplexed. In modern times, infinity can be manipulated and used in calculations but Greeks approached it as a philosophical idea.
According to a The History of Infinity by Allen, Donald, Anaximander, a Greek philosopher might have been the first to record the idea of infinity. Apeiron he called it, which means unbounded or indefinite. This happened a long time ago. To put it into perspective, somewhere between 610 C to 546 BC.
It wasn't until the 17th century that Infinity found its way into Mathematics when then the symbol for Infinity and Infinitesimal Calculus was introduced. In the 19th century, the study of Infinite Sets and Infinite Numbers led to the discovery that these sets can be of various sizes.
The phenomenon of Infinity passed from Greeks to Arabs to European Mathematicians before finally emerging as the concept we know of today.
Infinity Is Not A Number
A really common misconception is that Infinity is a number. The largest number to ever exist. It's not. Rather it’s a name given to a phenomenon. In fact, according to Math Forum, there is no 'largest' number. Googolplex is a number where we stop at, the largest named number we know. It is ten raised to googol power, where googol is 10^100. Which makes the big googolplex equal to 10^(10^100).
Another cool thing to know is that there’s no number larger than Infinity. But since Infinity is not a number, we can’t call it the largest. If you still aren’t convinced and would like to think of Infinity as a number, then you might as well believe 1 to be equal to zero. Because describing infinity as a number leads the crazy things like proving 1=0.
Infinities of Various Sizes
In Fault In Our Stars, Augustus said, "...some infinities are bigger than other infinities,".
Turns out he was right. Mathematically at least. Some sets of infinities are actually bigger than the others. You might already know that since the natural numbers are unbounded, they must be infinite. But how large is that set? Certainly not larger than all the real numbers.
Real Numbers include negative integers, all the rational numbers, and irrational numbers as well. This set of infinity is greater than the set of natural numbers.
According to an article posted by Scientific American, Cantor, a German Mathematician showed that all the real numbers between 0 to 1 are greater than all the natural numbers that exist!
Countable VS Uncountable Infinity
Mathinsight.Org defines countable infinite sets as those whose elements can be put in one-to-one correspondence with the set of natural numbers. In simpler words, all the members of this set can be listed. These elements would follow some sequence.
For example, the natural numbers are countably infinite. If I start from, say, 1, after counting for a finite amount of time I should get to another natural number and I would. Counting from 1 and for 20 seconds might get me to 25.
This is what makes this infinity set countable.
Uncountable sets on the other hand aren't countably infinite. Real numbers, for example. These sets contain too many elements to be counted off an infinite amount of time, and the members of the set can't be listed.
Consider numbers from 0 to 1, where should you start to list them? 0.1? Not precise enough. 0.0000001? Could be made smaller. To be considered as countable, the elements must adhere to a sequence, must correspond to natural numbers but they don't. We aren't even sure what the first element is. This set would therefore be Uncountable.
And there you have it, the astonishing idea of infinity tackled!