In none of my articles have I actually tried to define infinity. Loosely it is the concept of never ending; something that goes on forever. It occurs in many equations that describe the natural world yet it continues to be on the edge of our general understanding of the universe. Despite its frequency it is still a tough concept to deal with. It is hard to visualize infinity and it is often a pain to deal with mathematically and computationally. This is my third article to try and tackle the concept. Read on....
Ordinal vs Cardinal
In mind my there is only one infinity. At least that's what I thought was the most obvious conclusion to draw. Little did I know that there are different types of infinity, it wasn't obvious that this should be true and after much reading it turns out to be a highly debated topic (even among contemporary academic mathematicians). Cantor divided infinity into two different types, one is called 'ordinal', the other 'cardinal'. The first is similar to thinking of infinity as a number you can to. while the second concerns the size of a set of numbers. Start from one and count to infinity and you are tending towards to the infinity that Cantor called ordinal. The second type comes from asking how many numbers are in a particular set of numbers, specifically we might ask "what is the size of the set of all natural numbers?" Obviously there is an infinite number of elements in the set of natural numbers. So the countability of the set of natural numbers can be denoted by some symbol (most commonly the Hebrew symbol Aleph with a subscript zero).
The next question is to ask if these two quantities are the same and whether they can ever be different, and what are the implications of such seeming absurdities? It did not seem obvious to me that there could be two different types of infinity, at least not at first. Coming from a mostly naive and ignorant perspective I tried to create thought experiments that would ''surely disprove'' the existence of multiple infinities (see the section "thought experiments" below).
High School Logic
At school there is little distinction made between integers and reals (decimal numbers); sure they look a bit different but that's all there is to it, and surely 1 is always the same thing as 1.0? I couldn't see any reason why they should differ. Both inhabit the same number line, the decimal form is merely a different way of writing the same thing. In fact, the decimal form allows for many numbers that lie in between every integer. To cut a long story short: integers are a subset of the reals. The rationals (fractions) allow for integers divided by themselves to give numbers that don't exist as integers. That is to say that integers are a subset of the rationals. Nothing new here.
As we progress through our mathematics education we (might) uncover some strange facts about the set of real numbers, such as there are some numbers which can never be found algebraically. That is, they never form the solution of a polynomial and are thus very different from integers or rationals. Despite all the clever games we can play with fractions using addition, subtraction, multiplication and division there are some numbers that can never be found in this way. To restate it: these numbers are nonalgebraic and are known as transcendentals (e.g. pi and e). The rationals (well, the algebraics) plus the transcendentals make up the reals.
Perhaps this should be enough to tell us that integers and reals are far different beasts and that their countability ought to be different. However, a common confusion (at least true in my case) is that infinity is a limit that one tends towards in counting, but it wasn't something that I had considered as a measure of the countability of a set. The infinity at the ''end'' of the real number line was the same as the infinity at the ''end'' of the integer number line. It feels strange to say that the infinity associated with the reals is greater than that of the integers. Infinity is just the unending list of all numbers, or at least that's how it seems from simple high school logic. This number at the ''end'' of the number line is better understood if we call it ordinal infinity, and to that extent it is the same ''number'' in both cases, but as understood from the context of cardinality then there are more reals than integers. This is there the confusion is resolved by understanding that ordinal infinity is not the same as cardinal infinity.
Given that, it then seems fair to suggest that there are more rationals than integers.
Are there more rationals than integers?
So our conventional sense of wisdom would lead us to believe that there are more rational numbers than integers. Seems an easy one to deduce right? And that there are more reals than either of these two. Well that last part is correct, but there are just as many rationals as there are integers which is hard to believe as it is hard for us to see how it could be true. The infinity associated with the rationals is the same as it is for the integers. Expressed another way, the countability of both sets of numbers is the same. The technical phrase for this a ''bijection'', or isomorphism, between the rationals and the integers; that is to say that the integers can be put into a onetoone correspondence with the rationals. And as guessed from above there is a bijection does not exist between the reals and the integers.
The fact that such a bijection does not exist is enough evidence to show the reals have greater countability, at least that was true for Cantor. That said, there is another argument which states that this difference is not enough to prove that the (cardinal) infinity associated with the reals is greater than that of the integers. Which comes back to trying to understand what it means to have one infinity which is greater than another. Infinity is a concept, or a limit, rather than an actual number so treating it like a number is perhaps going to lead us into fallacy or contradiction. There is a lengthy discussion of this on Wikipedia, I point the reader there for further discussion.
Now I present two thought experiments that I came up with while trying to understand infinity. In retrospect I think my 2 thought experiments illustrate the difference between ordinal and cardinal infinity, this was clearly an accident but is amusing all the same. I will also admit that separating the two is still a contentious idea (see Wikipedia for more discussion).
Thought experiment 1
What follows is a visualization and attempted rationalization of infinity with regards to the different fields (sets of numbers).
Take, or imagine, a blank piece of paper. Now draw a small straight line somewhere in the middle of the paper. At one end write a zero then at the other end draw the symbol for infinity (8^T). We will use this drawing to visualize a distance between zero and infinity, I will also represent this diagram as the set: [0,inf). This set only contains 2 elements, zero (bounded) and infinity (unbounded, not really a number).
This was my first attempt to rationalize the concept of infinity and was something of a reaction to Cantorian set theory: "it doesn't make sense to have one infinity which is bigger than the other". I wanted to place infinity at some 'fixed' distance away from zero and that the 'fastest' way between the two was a straight line. In set notation above I am implying that one can jump from zero to infinity in one step. This may not be possible but let's accept it for this thought experiment. There are, at least, some hyperbolic geometries that allow infinity to be part of a bounded shape (however, from memory, the metric tends to zero step size as you approach the boundary at infinity).
Now let's imagine we can make a half way jump, such that there is some point in between, ''half infinity''. Let's write that as a set like this: [0, A, inf). There are now two steps to infinity, each is step is ''half of an infinity''. We could draw a notch or dash on our straight line to represent this halfway point. This is a crude idea but let's keep going. Let's repeat this process by splitting step sizes again, and adding further dashes to our straight line; now we would have the set [0, A/2, A, 3A/4, inf). We could do this sort of process infinitely so and end up with a set that looks eerily familiar: [0,1,2,3... , inf), the natural numbers. Our straight line is full of dashes and is simply the number line that we see at school.
The step sizes here would tend to zero if infinity is a ''fixed'' distance away. If I start from zero and count to infinity it shouldn't matter if I use integers or reals as the final destination is the same. Little did I know that this was Cantor's ordinal infinity. I reckoned that if I divided my line up into small enough sections then I'd be able to reproduce the set of real numbers, and on top of that I'd be able to preserve infinity as an immutable concept that is the same for both integers and reals.
Thought experiment 2
Another attempt to force infinity into my way of thinking was to start back with the straight line analogy. Instead of this being the real number line it is merely a line that is the 'quickest' way to infinity (a nebulous concept I came up with to try and rationalize/ visualize infinity). This line is the original one I suggested above except the length of the line now represents the size of the underlying field of numbers. So this straight line represents the set [0,inf) as before and all progressively 'larger' sets, ie sets that have more numbers, are now represented as arcs of ever increasing size that connect zero and infinity.
That is to say that the sets presented above where I increased the number of elements in the each set, until we retrieved the natural numbers, will now be represented by arcs. It would stand to reason then the set of rational numbers provides an arc of greater size than that of the naturals, it naively seems that there are more of them. That means there should be another arc of greater length that represents the reals. Well, as said above, the rationals and integers are sets with the same countability moreover there is no set which countability between the integers and the reals. We can visualize an arc that sits between the two, the arc of integers and the arc of reals, but apparently it does not exist. That but doesn't stop us imagining one and trying to play some number games as if though it existed. It is also possible to imagine an arc that is larger than the reals too.
Well the point of this was to consider what happens if there is just one, absolute, infinity and how would the different fields of numbers represent with regards to this concept. Each set is represented by ever increasing arcs as the number of elements becomes larger. Essentially, the destination is the same but larger paths take longer to get there. There are two key considerations, to state it naively (perhaps closer to physics terminology than maths terminology), there is the length of the path and the size of the steps. One can imagine a path of infinite length, and / or, one can imagine a path of fixed length which is divided up infinitely so.
One view would be to say that the integers can be represented by a path of infinite length but finite stepsize, while reals are represented by an infinite path length where the stepsize is infinitesimal (tending to zero). The infinity (the destination) that both sets are tending towards is the ordinal infinity as presented above in the first thought experiment while the length of the arcs are an illustration of cardinal infinity and clearly shows that the set of reals is larger than that of the integers.
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