Infinity

Infinity x Infinity

Infinity x Infinity

infinity x infinity

What is infinity x infinity? The problem of defining the infinite does not stop people asking questions about it. One such question which crops up regularly is “What is infinity x infinity?”

Some answer infinity squared which is to beg the question, some say it is 1 and others 0 which begs the question – what are they doing?

As maths deals with clearly defined objects and the infinite is not clearly defined then we have to be prepared from the outset for a range of potential answers and try to sift them down to one which seems to work best in the most circumstances. The first task we have is to find a suitable definition for the type of infinity we are going to use. It would have to be the infinity N as understood by the natural numbers or infinity R as understood by the Real numbers. As these are two different infinities we may find that we have two different results for the problem, infinity x infinity.

We know infinity R is a higher order infinity than infinity N. That is, there are more members in infinity R than in infinity N. Just to recap from earlier posts, the number of even numbers is the same as the number of even + odd numbers (both infinite sets), whilst the number of Real points on a line between 1 and 0 is the same as the number of points on a line between 0 and 100, or between 0 and a million for that matter. These apparently different infinite sets are actually equal in size, yet the set N and the set R are different in size and can be shown to be so.

Cantors proof that Set R is bigger than set N

First of all we need to put every Real decimal between 0 and 1 into a form that it has infinite length. So 0.5 becomes 0.500000… for example. We can also assume decimal parts such as pi – 3 which would give us 0.1415…, the decimal part of root 2 which is 0.4142… and recurring rationals such as 1/3 which is 0.3333…..

Now given an infinite list of such decimals between 0 and 1 we can begin to pair them off with the Natural numbers.

So,

0 – 0.3333…

1 – 0.1415…

2 – 0.5000…

3 – 0.4142…

. .

. .

. .

Take a diagonal line through the decimal parts, then extract this line to create another decimal. Add 1 to each digit of this decimal and we have generated a new decimal which we have not yet listed. Though we have paired an infinite number of decimals with the infinite Natural numbers, there are still decimal numbers out there for which we haven’t accounted. This tells us that the real numbers are greater than the naturals.

So, in considering the question, infinity x infinity we must first ask, which infinity are we looking at. Infinity N or infinity R?

If infinity x infinity is referring to N then we can see that the answer is infinity N.

Proof.

Consider the powers of primes. We know there are an infinite number of prime numbers and we also know that each power of primes numbers will not replicate a number in any other prime power list.

2,4,8,16….

3,9,27,81…

5,25,125,625….

7,49,343,2401..

We can now generate a table of p and powers of p which shows and infinity x infinity.

P = prime number , n = natural number.

So infinity x infinity = infinity.

The acceptance of such a conclusion is no more problematic than accepting there are the same number of Natural numbers as there are Even numbers.

If we now consider the same problem, infinity x infinity, but with reference to the infinity of the Real numbers we find the result to be the same.

It is far less intuitive however, and a lot more interesting.

Assume a line containing an infinite number of points which is perpendicular to a similar line such as an x,y axes. Now we know that there are the same number of points in a line no matter its length. We also know that there are the same number of points in an area no matter its size, the question we are now faced with is, are there the same number of points on a line as there are in an area? Answering this question will give us an answer to the problem of infinity x infinity where the infinity in question is that of R.

The answer it appears is yes from a proof provided by Cantor after work by Peano. It is a highly technical proof and those further interested can find it at wikipedia.

The conclusion is that infinity x infinity always equals infinity.

Infinity: The One and the Many

Infinity: The One and the Many

The third type of Infinity we will look at is the infinity of the one and the many. Rather then commence a process whereby we aim to prove the existence of the infinite, with this Infinity we begin by making the assumption that it exists. Basically we make the assertion and then seek to justify our assumptions.

It is very difficult to capture the essence of this infinite. Commentators often conjure up different words and ways of expressing the idea of this type of infinite, using terms such as denumerable, super denumerable or simply, extraordinary. It is one thing making an assumption that the infinite exists but it is something else when we look at the consequences of that assumption.

So how do we begin to define the infinite of the one and the many? First of all consider a line. Now ask the question, how many points are there on that line considering the definition of a point to be it has no length or breadth; that is, a point has zero dimensions. We find that we have similar problems to that of the divided stick which we encountered in the previous article on the infinitely small. For if we divide the line into an infinite number of points of zero breadth then how is it possible that an infinite number of zeros has any length at all. Also if the point does have some length then an infinite number of them joined together would be infinite in length. Despite the fact it seems quite reasonable to make the assumption that a line is made up of an infinite number of points we see already that such an assumption leads us into logical difficulties.

If we investigate further and try to resolve the difficulties our initial premises have created, we discover that the problems just become more embedded and increasingly difficult. For example, we can now show that the number of points on the line always equal the number of points on a different line no matter how long the line is.

Infinity-The-one-and-the-many

If we take any point in area Z and draw a line to the point O then we cross through the line CD and the line AB. In doing this we can see that each point on the line CD maps uniquely onto a single point on the line AB. Also if we start at the point O and draw a straight line out to area Z we get a similar result, in that each point on the line AB maps uniquely onto a single point on the line CD.

Now if we introduce the point Z and create a smaller cone using the points A and B we create a new line EF which is a subset of the line CD. We can now repeat the logic from the previous argument. By drawing a line from area O to the point Z we can uniquely map all points on the line AB with the points on the line EF and, conversely all points on the line EF map uniquely onto the points of line AB. So we have proven that the number of points on the line CD is equal to the number of points on the line AB, and that the number of points on the line AB is equal to the number of points on the line EF. Therefore the number of points on the line CD is equal to the number of points on the line EF. It only takes a small consideration from this position to realise that there are the same number of points on every line no matter how long the line is.

This conclusion is somewhat problematic and can also be applied to real-world situations. For example, consider time compared with moments in time. Analogously time would be the line and each moment is a point in time. We can conclude quite quickly that there are the same number of moments in my lunch hour as there are moments since the beginning of the creation of the solar system.

The Infinite of the one and the many is best represented mathematically by the real numbers, R. If we look at the infinitely small we can see that the process of perpetual division will never help us reach the points such as root 2 or pi. The assertion that a line contains an infinite number of points has some validity because contained within the assumption is that the real numbers all exist.

The Infinitely Small

The Infinitely Small

The first article on infinity looked at infinity N the natural numbers and the type of infinity most often associated with the infinitely large. This second article is going to view the infinitely small and the problems and paradoxes associated with it.

The infinitely small has some parallels with the infinitely large and also some parallels with the third type of infinity which will be discussed in the final article. First of all the mathematical expression: The infinitely small is represented by the class of numbers known as the Rationals, which I shall call Q.

Simply, a rational number is one which can be expressed as a fraction, so that it is one natural number divided by another. So the Rationals include for example ½, 1/3 as well as the Natural numbers themselves which can be expressed as 2/1, 4/1 etc.

infinity

The Rationals are associated with infinite smallness and it is the constant search for the smallest size which goes on and on which makes this type of infinity similar to the infinitely large. We can take some object – assume a line – divide it in half, divide in half again and again until we complete the infinite series and then enquire what it is we are left with. Just how small can things be?

We could also divide the line into thirds or fifths or sevenths or any fraction (and we will do infinity itself in article three!) and the process, though not the numbers would be the same.

The infinitely large gave us great difficulty both in definition and conception in its mathematical and metaphysical form, is the infinity of the small any different? Can we make a coherent mathematical definition of the infinitely small and if so can we find some correlation of the mathematically infinitely small with some feature of reality?

The difficulty of creating or arriving at the infinitely small has parallels in the process of arriving at or creating the infinitely big. Each step, or downsize makes the object smaller, but no matter how small a thing is we can always envisage the object being a half its size and so the process goes on. No matter how many cuts we make there is always smaller still to come. If we make infinite cuts and end up with infinitely small pieces in infinite quantity then we have once again, as with the infinitely large, used the term infinity to create the infinitely small. We have defined the infinite by using the infinite.

Conceptually then we run into the same type of problem we have with the infinitely big. A logical definition is impossible and conceptually we can only imagine a never ending process which is potentially infinite rather than ever becoming actually an infinite.

When we try to put the infinitely small into some kind of real world situation the conceptual problems become even more acute. For example, imagine the problem known as the divided stick. Divide it in half again and again an infinite number of times and what is it we would be left with? It seems we would have an infinite number of pieces of stick infinitesimally thin. But what can that mean?

Do the pieces of stick have width or not. If they have any width at all then an infinite number of them would reassemble to have an infinite length, but if they have zero width then no number of them, including an infinite quantity could reassemble the stick into something with any length. Once again we find our intuitions stretched. Just as we have problems truly comprehending the mathematics of the infinitely large so we find similar conceptualizations problematic with the infinitely small. It seems we can know what it is we are discussing yet fail each time we attempt to define it or clarify what it is we know.

Many have claimed the infinite is unknowable, yet it seems clear that we know what it is we do not know. This is another paradox of the infinite, the paradox of thought about the infinite. We think we know it, but we don’t. We can see that we don’t really know it, but we can’t really give it up. We are in a position where we can acknowledge the infinite but have to accept that we cannot do anything with it. Any attempt to define the infinite is doomed to fail because given our own limitations we can only grasp those concepts which are suitably limited themselves.

But this just gives an even deeper paradox. If we cannot come to know the infinite, then more seriously we cannot come to know that we cannot come to know anything about the infinite. Yet this is what we have just done. We have defined the infinite as that which is beyond definition. This is a contradiction in itself and a limiting factor on what is essentially the unlimited.

Aristotle claimed that the infinite was central to the scheme of things. It would seem as though our own finitude is as much a part of the study as a study of the infinite.