Infinities Galore

What do you see when you picture Infinity? Perhaps you imagine a very, very large number. If so, please don't. Infinity is not a number at all, but a concept. Infinity is actually a seeming oxymoron, an unreachable limit of a never-ending series of numbers. A simple example is a list of all the positive integers: 1,2,3....., where the dots indicate that this list never stops. And, to further complicate matters, there is not just one infinity, but instead multiple ones, each larger than the one before.

A warning is in order here: The concept of Infinity is initially quite mysterious and unintuitive, because in the mathematics of our daily life we deal with actual numbers such as the cost, weight, and speed of a new cell phone. Thus thinking about the enigma of Infinity might cause headaches, hallucinations, or other adverse reactions. So proceed at your own risk!

An often-used example, The Infinite Hotel, illustrates the properties of the smallest infinity. Here is my version: 

Picture a hypothetical hotel with an infinite number of rooms, named imaginately Room 1, Room 2, ... This is the hotel's high season, and so the hotel is completely full with, of course, an infinite number of guests. Then, very late one night, a young couple shows up. These two are clearly in love (or at least lust), and just as clearly in desperate need of a private place with a bed, and so ask the hotel manager for a room. The aged night manager patiently explains to the young lovers that the hotel is full. But the couple entreats him to use all of his powers to try to find them a room. Well, who among us could not open his/her heart (and hotel) to such a plea? The old manager, remembering well the urgency of youthful desires, is deeply moved by the duo's plight. And he also knows that he could actually provide the lovers a room in the hotel. But doing that would require a lot (actually an infinite amount) of distasteful interactions with an infinite number of sleepy, irate guests. So with a heavy sigh (but a light heart!), the manager moves the guests in Room 1 to Room 2, those in Room 2 to Room 3, and so on right up the line. The lovers of course then get to move into Room 1. And since the hotel is infinite, none of the hotel's current guests will have to sleep out in the cold.

Clearly there's a problem here, since no real-world hotel could work like this. Though the room list of this hypothetical hotel will eventually reach any whole number we can think of, the list will never reach the non-number Infinity. And because Infinity is not a number, it is meaningless to use this concept in standard numerical equations; e.g., Infinity - Infinity = ?  

But surprisingly, there are many different kinds of infinity. The brilliant mathematician Georg Cantor showed in 1878 that infinities come in different sizes. Cantor gave the name Aleph-Null to the smallest Infinity: the set of all "rational numbers", which include the whole numbers in the hotel example above, plus all fractions like 3/4, 1/137, 20/21 that can be written as a ratio of two integers. Cantor showed that it is possible to count all of the Aleph-Null numbers, including the fractions, in a diagonal manner that would eventually arrive at any specified number. So he termed Aleph-Null the countable ("denumerable") Infinity. It turns out that even if you multiply Aleph-Null by itself, the result s still Aleph-Null. So what manner of infinity could possibly be larger than Aleph-Null?

Cantor showed that there is indeed an infinity larger than Aleph-Null, termed Aleph-One, that contains all of the "real" numbers. The real numbers include all the Aleph-Null rational numbers. But these real numbers 
additionally contain all the "irrational" numbers (like the square root of 2 and pi), that can't be expressed as a simple ratio of integers. These irrational numbers  can instead be represented only by a never-ending series of digits. Cantor demonstrated that Aleph-One is a non-countable infinity, since between any two rational numbers lie infinitely many irrational numbers. 
So Aleph-One is a larger infinity than Aleph-Null. Cantor's proofs clearly showed that Infinity could no longer be viewed as a single concept. In fact, Cantor showed that there are infinitely many infinities!

Could there be any infinities lurking between Aleph-Null and Aleph-One? Cantor believed the answer is no (and
long attempted to prove this), a conjecture now termed the "continuum hypothesis". It is not known whether Cantor was correct. Even worse, this question is an undecidable problem, of the type shown in 1931 by Kurt Godel's incompleteness theorems to plague the basis of arithmetic. So Cantor's conjecture can't be either proven or disproven within any standard (e.g., ZFC) arithmetic system. Sadly, we will thus never know for sure whether there is a long-lost middle sibling lying between the two smallest known infinities. 

So not only is Infinity a concept or limit rather than a number, there is actually an infinite number of infinities, each larger than the one before. When it comes to infinities, size does indeed matter.

(sciencequandaries.blogspot.com)