All of us are exposed to Arithmetic from an early age. That is, the basic business of addition, subtraction, multiplication, and division. Most of us got some exposure to Algebra somewhere along the line in school. For many it represented a fork in the road. My reaction to Algebra was "cool". That put me irrevocably on the path toward nerdiness. Others found it a slog, or worse. They became part of the "I'm bad at math" crowd.
Beyond Algebra is Trigonometry. And beyond that is Calculus. The big four: Arithmetic. Algebra, Trigonometry, and Calculus, provide a "degree of difficulty scale ranging from Arithmetic (easy) to Calculus (hard) that we are all familiar with. But there are other kinds of math.
I wasn't exposed to set theory until I got to High School. But it has now migrated down to elementary school, or so I'm told. Geometry seems non-mathematical but generally gets lumped in with math. Logic is also some kind of weird step-child of math. So these oddballs, set theory, geometry, and logic, are kinds of mathematics that many people have at least some exposure to but which they also view as something other than "pure" math.
It turns out that once you get past the basic four there are lots of these oddball branches and offshoots of mathematics beyond the three I listed above. And these other oddballs and offshoots are generally lumped together as "higher mathematics". The general perception is that all of them are really hard. After all, they are called "higher" mathematics because, with few exceptions, no one studies them until after they have mastered Calculus and Calculus is really hard.
And, for many kinds of higher mathematics, this is a totally accurate characterization. Matrix Mechanics makes Calculus look like a walk in the park. Yet it is the mathematical foundation of everything Particle Physicists do. And that's why 99.99% of us are not Particle Physicists. But there are also parts of higher mathematics that are not all that complicated. They are just ideas that regular people never get exposed to. One of them is infinity. And that's what I want to expose you to.
We all have some idea what infinity is. It's the number beyond all numbers, it's so big. Or, to put it into more concrete terms, you start counting. You go one, two, three, and so on. You keep going and going and going. And at some distance beyond the last number you count up to is the last number, infinity. It's bigger than any number you can actually count to. That's people's idea of infinity.
People have an intuitive idea that "infinity" is a weird number. But they tend to underestimate its weirdness. For one thing, it's not a specific number. Say it was a specific number. Call it "I". Now add one to "I". This "I+1" number is also a specific number and it's bigger than infinity. But infinity is the biggest number there is. So that's impossible. So forget about the idea of infinity as "a number like any other number". It just isn't. It's these sorts of problems with infinity that make it hard to wrap your arms around it.
But mathematicians wrestled with all these kinds of issues and found a way to work with infinity anyhow. They resorted to set theory. A "set" is just a group of objects. One of the fundamental attributes of sets, the mathematical construct, is that you need to have a way to determine if a particular object, called an "element", is in a particular set or not. There are several ways to do this but I am only going to very briefly cover the two most common.
First, you can just list all of the elements of your set. You can say "my set consists of 'A', 'B', 'C', . . ., and 'Z'". The second way is to provide a rule: "My set consists of all of the letters in the alphabet". For many small sets the "list all the elements" method works just fine. But we are going to be dealing with large sets so we will use the "rule" method. And the elements of the sets we are going to be talking about will consist solely of numbers. So no "the set of all Presidents" or "the set of all cars I have owned".
Another attribute of a set is the number of elements in the set. It may be zero. The set may be empty. That is perfectly legal. But all of our sets will have elements in them, lots of elements. In fact, they will have an infinite number of elements in them. And the technical term for the number of elements in a set is its "cardinality".
And we can compare sets. Two sets may be equal (each consists of exactly the same elements) or unequal (there is an element that is in one set but not the other). We can also have subsets (all the elements in one set are found in the other set) or supersets (some of the elements in one set are not in the other set but all of the elements in the other set are in the one set).
There is lots more that could be said about sets but that's all we need. Oh! There is a bit more we need to know about cardinality. If two sets are equal (contain exactly the same elements) then the cardinality of the two sets is identical. But what about sets that don't contain exactly the same elements?
Let's say that we have two unequal sets but we want to know if the cardinality of both sets is identical. If we can make up a rule that associates exactly one element of the first set with exactly one element of the second set then we can say that those elements have been put into a "one to one correspondence".
And let's say that we make up a rule that creates a "one to one correspondence" for every single element of the first set with an element in the second set. And let's say the same rule can be used to do the reverse, to create a "one to one correspondence" for every element of the second set with an element of the first set. And to be clear, there is only one "one to one correspondence" for each element of each set. We don't ever use the same element twice.
Anyhow, if we succeed in doing this then the cardinality of the two sets will be identical. And you are probably saying to yourself at this point "this all seems needlessly baroque". But it is necessary to deal with the fact that infinity is not a single specific number. With that out of the way, let's go to work.
The "counting" numbers: one, two, three, etc., are called "natural" numbers by mathematicians. So if we create a set containing all of the natural numbers its cardinality will be infinity. That seems like a lot of work to get somewhere that seemed obvious from the get go, but trust me, it's all necessary. Because now the fun starts.
The next step up in the hierarchy of "numbers" from the natural numbers is to "integers". Take each of the natural numbers, and add "plus" to its name and add it to a new list. Then add the new number "zero" to the front of the list. Finally, take each natural number in turn again. But this time add "negative" to it's name before adding it to the list. So, add "negative one", "negative two", "negative three", etc. And let the negative numbers increase as we move left and also put them on the front of the list.
Now, instead of having a list of "natural" numbers starting at "one" and marching off to the right as they get bigger and bigger, we now have a list with a central number, "zero", and two lists of numbers, one marching off to the left and the other marching off to the right. It's just that the list of numbers marching to the right now have a "plus" added to each name and the list of numbers marching to the left now have a "minus" added to each name. That all seems straight forward. But now let me throw a monkey wrench into the works.
Let's make a second set consisting of all of the integers. The question is: Is the cardinality of both sets (natural numbers and integers) identical or not? The obvious answer would be "no". The natural numbers are a subset of the integers. We can form a "one to one correspondence" between each of the numbers in the set of all natural numbers and an element in the set of all integers. Just match "one" from the set of all natural numbers up with "plus one" in the set of all integers.
Proceed step by step with "two" and "plus two", "three" and "plus three", and so on. (This would take an infinitely long amount of time. But that doesn't matter. Obviously the process works. And that's all we need.) We end up with everything lining up nicely with respect to the "natural numbers". But at this point we also have "zero" and all the negative numbers left over in the "integers" set, and they are not matched up with anything. So obviously the cardinality of the set of all integers is greater than the cardinality of the set of all natural numbers.
There is just this one tiny little problem. Take "zero" from the "integers" set and match it up with "one" in the "natural numbers" set. Take "plus one" in the "integers" set and match it up with "two" in the "natural numbers" set. Now take "minus one" in the "integers" set and match it up with "three" in the "natural numbers" set. Keep going matching "four" in "natural numbers" to "plus two" in "integers", "five" in "natural numbers" to "minus two" in "integers", and so on.
It turns out that we can create a "one to one correspondence" between the "integers" set and the "natural numbers" set. So what should we do? If we do the "one to one correspondence" one way we decide that the cardinality of one set is greater than the cardinality of the other set. If we do it another way we decide the cardinalities are identical. And, if we are clever, we can even come up with a rule for creating a "one to one correspondence" that makes it look like cardinality of the "natural numbers" set is greater than the cardinality of the "integers".
What's really going on all goes back to the fact that infinity is not a single specific number. And what mathematicians decided to do about that was to decree that as long as at least one rule existed that put every element of one set into exactly one "one to one correspondence" with every element of the other set and that as long as the "one to one correspondence" went both ways then the cardinality of the two sets was identical. To say that this is not intuitive is a vast understatement. But there it is. But wait. There's more.
"To infinity and beyond" was the signature line of Buzz Lightyear, a character in the movie "Toy Story". Anyone who has seen the movie (and everyone should) knows that Buzz was not the sharpest tool in the box. Still, Buzz brings up an interesting question. Is it possible for anything to be "beyond" infinity? And, spoiler alert, the answer turns out to be "yes". What's beyond infinity turns out to be infinity. But this second kind of infinity is distinct from the first kind.
Mathematicians call this first kind of infinity, the one based on integers and natural numbers, "Aleph Naught". "Aleph" is the first letter of the Hebrew alphabet. "Naught" is just British for "Zero". So, Aleph Naught can be roughly translated as "A Zero". And you will also sometimes see "Aleph Null" or "Aleph Zero" used. They are just variants for the same thing and "Aleph Naught" is the form most commonly used by mathematicians, so it's the one I am going to use.
And mathematicians asked themselves: "is there a bigger infinity than Aleph Naught infinity?" Their early tries at finding one were a failure. Like us, they started with natural numbers and moved on to integers. Their next step was to try "rational" numbers.
Any number that can be represented using the combination of an integer and a fraction consisting of a natural number in the numerator and a natural number in the denominator is a rational number. Minus twenty-seven and five thirty-sevenths is an example. So we get all the integers plus all the numbers between two adjacent integers that can be created by making use of an additional fraction.
And there would seem to be a whole lot more rational numbers than there are integers. There are literally an infinite number of rational numbers between each and every pair of adjacent integers. And remember, there are an infinite number of integers. So it would seem that the cardinality of the set of all rational numbers would be infinity multiplied by infinity. So the cardinality of the "set of all rational numbers" surely must be greater than the cardinality of the "set of all integers".
Alas, it was not to be. I am going to skip over the details but some clever mathematician found a way to create a rule that put the "rational numbers" set and the "integers" set into a "one to one correspondence". So, nope. The cardinality of the rational numbers is Aleph Naught. But wait. It gets worse.
What we have been talking about so far can be thought of as the "number line". All of the rational numbers represent dots on a single straight line. How about if we go multi-dimensional? How about a "number plane", or a "number space", or even a "number hyperspace" with many, perhaps an infinite number of, dimensions? Still, no joy. It turns out that there is a rule that will put all those points in all those dimensions into a "one to one correspondence" with our original set of integers. And that means the cardinality of the whole mess is still Aleph Naught.
So we're screwed, right? There is only Aleph Naught. Wrong! We just haven't gotten creative enough yet. Beyond the rational numbers are the "real" numbers. It turns out that if you take a continuous line and plot all the points on it that can be described by rational numbers (integer plus a fraction consisting of a natural number in the numerator and a natural number in the denominator) not all the points on the line are marked.
The rest of the points on the line represent the location of various "irrational" numbers. (The "real" numbers are what you get when you combine all the rational numbers with all of the irrational numbers.) And there are lots of irrational numbers. The most well known irrational number is pi.
And what we generally associate with pi tells the tale. Pi is equal to 3.14159 and on and on forever. In fact, it doesn't matter how many digits we list, the value will still not be exactly correct. The exact value of pi, expressed as a decimal number, is a number with an infinite number of decimal places.
But when it comes to rational numbers there is a trick. With this trick we can represent any rational numbers with complete accuracy. And to do this we only need to list a finite number of digits. We do this by breaking the decimal representation of a rational number into two parts. Each of these parts contains a finite number of digits so the whole contains a finite number of digits.
To represent a rational number with complete accuracy in the usual way requires an infinite number of digits. But the decimal representation of a rational number contains a quirk that is always present. With an irrational number like pi the digits never repeat no matter how far you go. With a rational number the digits start repeating if only you wait long enough as you work your way along the number. And once a particular number has gotten into the repeating part that's all it does forever after.
We can use this quirk of rational numbers to our advantage. The process is pretty obvious. We break the number down into a non-repeating part on the front and a repeating part on the back. We list the non-repeating part as is. It will usually contain an integer sub-part to represent the "whole" part of the number and the part of fractional part that does not repeat. (Either or both of these sub-parts may require listing no digits.)
This non-repeating part is supplemented by a repeating part. The repeating part consists of a finite number of digits that are repeated an infinite number of times. Mathematicians indicate this by drawing a bar across the top of the first set of repeating digits. Then they just drop the rest of the number. Note that we are getting rid of an infinite number of digits while retaining complete accuracy. Nice trick, isn't it?
So, in example above (minus 27 and five thirty-seventh's) we would have "-27.". That's the non-repeating part. This would be followed by "135" with a bar over it (or so my calculator tells me). I don't know how to get the blog formatting software to put the bar over the three digits so you'll just have to use your imagination.
As indicated above, it turns out that any rational number can always be rendered exactly in decimal form by splitting it into the non-repeat and repeat parts. (BTW, sometimes the repeating part is just "0" repeated an infinite number of times.) I'm not going to go into the process necessary to turn a rational number into a decimal number in this "non-repeating part plus and infinite number of repeating part" form. Nor am I going into how to do the process in reverse. Just trust me that it can be done.
Now we are in a position to note that an irrational number is one that has no part that repeats indefinitely when rendered as a decimal number. Given two specific rational numbers rendered in decimal form it is easy to see how to construct lots of irrational numbers that lie between the two. (Hint: The rational numbers have repeat parts.)
Annoyingly, it is also easy to construct as many rational numbers as we want to, such that they lie between any two given irrational numbers. You just note the early parts that coincide. Then, starting with the decimal place where they diverge, you just toss in repeating sections that result in numbers that lies between the two. This is another example of where this whole "not a single specific number" business makes life complicated.
Anyhow, a mathematician named Gregor Cantor came up with a way to crack the case by making use of the fact that real numbers can be represented accurately as decimal numbers if you are willing to include an infinite number of decimal places. In 1874 he came up with a proof that a number bigger than Aleph Naught existed.
The easiest way to explain what he did is now referred to as the "diagonal argument". Write down a list of all real numbers in decimal format. Don't forget to include all of the infinite number of digits necessary to represent them with complete accuracy. It doesn't matter what order the list is in. It is just important that the list includes ever single real number. (This is impossible to do in real life but this is what is called a "thought experiment".)
Now consider the first number in the list. Change the first digit in the number to anything other than the correct digit and save it as the first digit of a new number we are constructing. Move on to the second number. This time change the second digit to anything other than the correct digit. Again save this digit as the second digit of the new number we are constructing.
Move on through the list. For each number we change the next digit along to be anything other than the correct digit and save it in the correct place in the new number we are constructing. Again, this is impossible to do in the real world but the process is easy to understand.
When we are done (we never will be but, again, thought experiment) we will have constructed a number that does not exist anywhere in our original list. (Remember, we constructed the new number so that it differed from every single number in the list of real numbers by at least one digit.)
But this is impossible. Our original list was carefully constructed so that it included each and every real number. This contradiction means that the cardinality of the set of all real numbers is greater than the cardinality of the set of all integers. So there is an Aleph One and it is bigger than Aleph Naught.
So there are at least two infinite numbers. There is Aleph Naught and Aleph One. And it turns out that there is a way to construct Aleph Two and so on. (I'm not going to go into the process because, frankly, I don't understand it.)
So there you have it. There is a "beyond" beyond infinity. "Infinity" presumably refers to Aleph Naught and there is an Aleph One that is "beyond" Aleph Naught. You have now mastered some way cool mathematics that is so far beyond Calculus that you could say "it's infinitely far beyond Calculus". So go forth and win bar bets and wow your friends and family at parties.
Pretty cool, hunh?
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