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

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 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.

Infinity – The Infinitely Big

The infinite is a subject of great interest. Studied since man has been able to think, argued over for equally as long, yet ask anybody if they understand the infinite then most people would respond with a yes. It is full of contradiction, full of surprises and lies at the heart of most philosophical discussion. Mathematicians disagree as to what status the infinite should be afforded and scientists tend to sweep it under the carpet whenever it arises. Full of paradox and unable to be defined, the infinite is both understood and misunderstood in equal measure, often by the same people.

infinity

So what is the problem? This short series is not the best place to go in to great detail but is designed as an introduction to one of the most perplexing objects ever studied. Aristotle said that the infinite was central to the scheme of things. What he meant was that if we truly want to understand the world then an understanding of the infinite is crucial.

There are seven different types of infinity which are easily identifiable. Three of them are mathematical infinities, three metaphysical infinities and the final one is probably in the metaphysical camp. That is not to say that it is indisputable that they are mathematical or metaphysical objects but they are candidates for being so depending on who you are talking to. In this article I shall deal with just the first and probably most commonly known infinity and its metaphysical correlate.

This first infinity is can be expressed by the sequence:

1+2+3+4+5…..

where it is assumed the last term in the sequence is the object we seek to define; infinity. This is the infinite as expressed by the natural numbers so is called infinity N.

Note that it is impossible for me to define the infinite in clear unambiguous terms. The dots which appear after the five in the sequence mean ‘and so on and so on an infinite number of times.’ So in my definition of infinity I have had to use the thing I am defining to define it. (It is a good job you know what the infinite is or I could never explain it!)

The string of numbers represents a mathematical infinity. Now it is controversial as to whether this is a true mathematical object; we can call it N and let that mean ‘all numbers’ but it is by no means clear that we could ever gather together all numbers into one group. What we can say is that for whichever number we have, no matter how big, we can always find a number which is bigger. Because of this inability to collect together all objects of N into one group this type of infinity is often referred to as the potential infinity. It is only ever an infinity in becoming and never actually gets to the point of really being infinite.

The metaphysical equivalent of N is in objects whose size or distance or temporal duration have the same internal structure. So for example we may encounter this type of infinity in the real world if we were to have an infinite number of some object, or if we were to travel in a straight line away from earth – to infinity – or if we were to look back or forward in time to an infinite past or forwards to an infinite future. If any of these situations do occur in the real world then that would be a metaphysical example of infinity N.

The kind of concepts we will most often see being associated with this type of infinity are for example, unlimitedness – (As in unlimited hosting accounts), endlessness, immeasurable, eternal and boundlessness.

Despite this account of infinity probably being the most well known it is also probably the least likely to be represent anything which is truly ‘infinite.’ As a mathematical object it fails to exist in any coherent way and as a metaphysical or real world object it is probably impossible that it should exist to describe any aspect of reality.

Think what it would mean to have an infinite number of a thing; grains of sand for example, or any object in infinite quantity. Infinite distance seems unlikely and an infinite period of time either forward or backwards doesn’t quite stand up to logical scrutiny. There are also a number of paradoxes we run into if we treat this kind of infinity seriously.

For example: How many even numbers are there in comparison to natural numbers? The answer would seem to be less or we might even think a half but that is just not true. If we pair off the numbers against the even numbers we discover that we have the same amount of both.

0..1..2..3..4..5….6….n
|    |   |   |    |    |     |       |
0..2..4..6..8.10.12….2n

It seems as though there are as many even numbers as there are ordinary numbers and we know it is true because there are an infinite number of each yet…. it just doesn’t seem quite right..

We can devise more examples of infinite trickery. For example, think of a hotel with an infinite number of bedrooms. The hotel is full and new guests arrive, no problem. Just move the occupants in room 1 into room 2, the occupants in room 2 into room 3 and so on and put the late arrivals in the first room. In fact you would never have a problem finding a room in this hotel, even if you arrived with an infinite number of friends on a bus with infinite number of seats. In this case you just need to point out to the hotel manager all he has to do is move the guests in room 1 into room 2, the guests in room 2 into room 4, the guests in room 3 into room 6; in effect the guest in room n to room 2n and there’s enough space created for you and your infinite number of friends to all stay the night at this magical hotel. (Pity the chambermaid, but not too much. She would work hard, but only for one shift, just think of all the tips she would receive!)

There is also the example of the two men who are destined to spend all of eternity, one in Heaven and the other in Hell. The keepers of the eternal realm however have a policy of allowing the resident of hell to spend Christmas day in heaven whilst the resident of heaven has to spend that day in hell just as a reminder of how fortunate he is. On the surface it would seem that one man spends a lot more time in heaven whilst the other spends most of his time in hell. But on closer inspection, because this is an eternal arrangement they both spend an equal amount of time in each.

In temporal terms there are a few ideas to think about which makes eternal past and futures seem incoherent. Imagine someone walked up to you and said; ‘2,…4,…1,…3 finished!’ and then explained that they had just recited the decimal part of pi backwards! This draws attention to the difficulty we have in contemplating an infinite regress of time. The forward part we can take on board a little easier. For someone to begin reciting the decimal expansion of pi and continuing ‘for all time’ is easier to conceptualize. We can just assume that no matter how far he has got, he will always have more to do. But for someone to have already gone through an infinite sequence to have arrived in the ‘here and now,’ strains our imagination more than we can cope with. Just when did our eternal reciter begin his task and at which point in the sequence!

This is the infinite of the very large and the associated paradoxes are the paradoxes of the infinitely big. The infinite of this kind does not stand up to scrutiny. It makes no real sense from a mathematical perspective and if it was taken seriously logically, it has nothing to offer us in our investigations into reality. It seems likely that this kind of infinity could not exist in the universe as an object of math or as a description of things, events or time.

That is the infinite in its most common conception and it should be clear that it is probably something of a falsehood both mathematically and metaphysically. I shall deal with the other types of infinity on this blog later in the next few days / week.

Infinity

Questions of infinity are some of the most difficult and incomprehensible known to man. For a subject which has its roots in the subject of mathematics which itself is based on the principles of logic we have in the infinite a topic which is in a class of its own. Few agree on what infinity is; it divides mathematicians, philosophers and scientists.

Is it a number, a concept, an existing thing and of all the types of infinity is there one which is truly infinite? There are many paradoxes surrounding infinity, not least the one of understanding infinity. If something is truly infinite is it possible to define it? And if we cannot define it then how can we claim to understand it? Yet we seem to have some grasp of it, yet the more we get to know the infinite the clearer it becomes that we probably do not know it at all. The infinite is a subject of minefields for the unwary, but in the attempt at understanding everything the infinite is central to everything.

Is time eternal, does space go on forever? Why is so easy to accommodate an infinitely full busload of people in a hotel with infinite bedrooms even though the hotel is full? And is God and infinity the same thing?