Zeno's Paradox and the Infinity Proof: Why Infinity Doesn't Exist | All about God's Love
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Zeno’s Paradox and the Infinity Proof: Why Infinity Doesn’t Exist

Quite a few years ago I took a group of A level philosophy students to a small university in Wales which was keen to promote its philosophy department.

Presentations on a number of philosophical topics were given to A level students from various schools and sixth form colleges possibly interested in pursuing a degree in philosophy. Among these was an account of the philosophical problem of what is called Zeno’s paradox, which ‘proves’ that movement is impossible. Since clearly movement is possible, the ‘proof’ is paradoxical.

You are probably well aware of the story, but I’ll present it briefly in case it would be helpful. A tortoise and Achilles have a race. To make it more fair, the tortoise is given a head start. But, actually, the tortoise is not given much of a lead and, in practice, it will be easy for Achilles to catch up and overtake it. But he can’t ever do this, according to the paradox. The problem is that, in order to overtake the tortoise, Achilles will first have to reach the point where the tortoise started. But when Achilles reaches this spot the tortoise will have gone further forward. Achilles then has to get to the tortoise’s new position. But when he gets there the tortoise will again have moved forward. This continues indefinitely – or, rather, infinitely. Each time Achilles reaches where the tortoise had got to, the tortoise has moved on. Achilles can therefore – the argument goes – never catch up with the tortoise.

Another way of putting the problem is that, to reach the tortoise, Achilles has to get half-way to where the tortoise is at any given point. And once Achilles reaches the half-way point, he then has to reach half-way to where the tortoise now is. And when he reaches this point, he then has to… I think you get the point. This happens to infinity, so Achilles gets closer and closer to but never catches up with the tortoise.

In fact, according to this logic, no movement is possible at all because, in order to move to the tortoise’s starting point, Achilles has to get half-way there. And in order to get half-way there, he has to get half-way to this point. But in order to get to this point he has to get half-way there too. And so on. In other words, Achilles has to travel through an infinity of points even to get started. Since he can’t do this, he can’t move at all – and nor can anybody or anything else.

I was placed among a group of students from various schools given the task of discussing this problem. It happened that the lecturer who presented Zeno’s paradox was in my group. I suggested that, if infinity doesn’t exist, the paradox doesn’t exist and Zeno’s problem vanishes as a mirage. My suggestion was swept aside with the words, ‘You can’t just dismiss infinity!’

Logically, with respect to the paradox there are three possibilities: (1) that movement is not possible; (2) that infinity does not exist; (3) that Zeno’s paradox is faulty philosophy. The first possibility can confidently be rejected. The third can also provisionally be rejected, since many philosophers still regard Zeno’s argument as pertinent. This leaves us with the second possibility: that infinity does not exist. This possibility has not to my knowledge been carefully considered.

In this article, I’d like to show that infinity doesn’t exist. Without infinity, there is no paradox for Zeno: if there are not an infinite number of points to traverse, but just a specific distance, then there is no problem in making the first step, in getting half-way to anywhere, or in catching up with and overtaking a slower moving object (given sufficient time).

I have a number of arguments to prove that infinity doesn’t exist – which are perhaps all the same argument posed in different ways.

1 Infinitely small

An infinitely small unit is an impossibility. For a unit to be infinitely small it would equate with zero. How big is an infinitely small space? It’s no space at all, because if it took up any space whatsoever it would not be infinitely small. An infinitely small space therefore does not exist. Being able to talk about an infinitely small unit, as if it has existence, does not give it existence – any more than talking about Martians or superheroes, or parallel universes, make them a reality.

Here’s another way of demonstrating the same point. You might suppose at first glance that 9.99 recurring is an infinitely small distance away from 10 and represents an infinitely small fraction, but 9.99 recurring is actually the same number as 10. This is easy to demonstrate. What is 10 divided by 3? The answer is 3.33 recurring. And what is 3.33 recurring multiplied by 3? It’s 9.99 recurring – or 10. The two numbers are the same.

Since Zeno’s paradox is dependent on the notion of infinitely small spaces, this argument should be enough to neutralize it. But I’d like to take the opportunity to show that infinity has no existence with respect to largeness as well as smallness.

2 Infinitely large

There is no such thing as an infinitely small unit. But could an infinitely large space exist? Yes, if the universe is infinite, it might be argued. If it’s not, an infinitely large space does not and cannot exist. So could the universe be infinite? I shall argue that it can’t be.

For the universe to be infinitely large there would have to be two points, let’s call them star A and star B, that are an infinite distance apart. You can’t have an infinite universe unless there are points that are infinitely far apart, because if the furthest two points are only a finite distance apart then the universe is not infinite.

But as soon as two points are situated in space they have a spatial relationship with each other, and the space between them cannot be infinite.

Two points situated in space must have a point which is approximately half-way between them, which we’ll call star C. If that place is measurable as half-way between them, however enormous the measurement, the distance is finite.

You might want to argue that there is an infinite distance between star A and this half-way point, star C. Then find the point, star D, half-way between A and C. Is this also an infinite distance from star A? Then do the same again. And again. Keep going an infinite number of times (supposing such a thing exists). Are the two points still an infinite distance away? They must still and always be an infinite distance away. This means of course that there cannot be a half-way point between stars A and B in an infinite universe.

From the location of star A, in which direction is star B? Star B is in no direction in relation to star A if it is an infinite distance away. If you could edge any closer to it at all, you wouldn’t be an infinite distance away. To be an infinite distance apart the two points can have no spatial relationship with each other. Remember, it’s not that any direction will be in the direction of star B (which could be true if space were finite and curved in four dimensions as the earth is in three dimensions), but that no direction leads in the direction of star B. If they have no relationship with each other – if in relation to star A star B is neither up, down, left, right, forward or backward (or any other conceivable direction), it has no existence relative to star A.

Travelling in all directions from star A, you’ll never reach anywhere that has a spatial relationship with (i.e. is theoretically measurable in relation to) star B. Similarly, travelling in all directions from star B, you’ll never reach anywhere that has a spatial relationship with star A. Imagine light travelling in all directions from stars A and B. However far the light travels, it can never reach infinity. A finite distance can never become an infinite distance (see argument # 3 below). The idea of an infinite universe erroneously assumes that finity can become infinity since stars (or points in space) are deemed to exist which are in the ‘same’ universe an infinite distance away.

Star A * [finite distance]… [becomes infinite]… [finite distance] Star B *

There is no reason to suppose (in this imaginary infinite universe) that there are only two ‘areas’ (star A and star B and all points stretching indefinitely from both in all directions) which are an infinite distance apart. How could the number of such ‘areas’ in an infinite universe be limited to any finite number? There must be an infinite number of locations an infinite distance apart, all being like a separate universe in themselves, having no spatial connection with the other infinity of ‘universes’ which are an infinite distance away.

Star A * [finite distance]… [becomes infinite]

Star B * [finite distance]… [becomes infinite]

But where are the boundaries of these apparently separate (because spatially unconnected) universes? There are no boundaries, since they go on forever. If they go on forever and have no boundaries they must be infinite. This means that there must be points an infinite distance away. But we’ve been through all this before…

Star A * [finite distance]… [becomes infinite]… Star AZ * [an infinite distance from star A]

Star B * [finite distance]… [becomes infinite]… Star BZ * [an infinite distance from star B]

As soon as infinity is postulated, infinity proliferates into an infinity of infinities.

But none of this is possible. A finite distance can never become an infinite distance. Travel from one point as far as you like, you never get an infinite distance away. There cannot therefore be two points an infinite distance from each other.

Two points plotted in space may be a very long distance apart, but they can’t be an infinite distance. As soon as points are plotted in space, i.e. given a location, the distance between them is theoretically measurable. There is no such thing as an infinitely large space since finity can never become infinity. Space cannot be infinite. Infinitely large is as impossible as infinitely small.

If infinitely small units are deemed to exist, there is no possibility of movement. If infinitely large units are deemed to exist, locations in space must exist (in infinite number) that have no relationship with other objects separated infinitely from them.

3 Going on forever

But, you might object, we can keep halving – or doubling – a number forever. There is no limit to how small or big we can go. This is true, and it gives the illusion that infinity is possible. But although you can keep halving or doubling forever, forever never arrives: you never get to infinity. You never reach a number that is an infinite distance from the number you started with. However big the number is, it is still a number and has a relationship with the numbers that come before and after it.

This is just like the finite universe argument. However far away the points are, they never reach infinite. Keep counting and never stop; count in huge leaps; quadruple the number a trillion times with every count; count until the numbers are as long as a galaxy, as long as space itself – you’ll never get to infinity. Finity never becomes infinity. Very, very big numbers are possible, as are very, very small fractions – but not infinitely big numbers nor infinitely small fractions.

Infinity is a concept. We have no trouble knowing what it means (even though we often use it loosely in an erroneous way, saying for example that something is ‘almost infinite’). Potentials can be infinite: possibilities are infinite; but a possibility is not an actuality. Infinity has no actual existence. Nothing in the universe can be infinite. No infinity, no Zeno’s paradox. Sorry, Zeno, and philosophy lecturer in Wales.

A paradox remains however, regarding not space but time. Time, from the present, can go on forever and infinity can never be reached – an infinite time will never have passed. That is true of the future. But what of the past? If time had no beginning but always existed, this would mean an infinity has already passed – a notion that is untenable with regard to space. But if time does not go backwards for ever, it must have started at some point. If the universe started at some point it would mean that something was created out of nothing: that there was no space, time, or matter, and suddenly these things came into being. Either time goes back for ever, or started at a certain point. Either is equally impossible.

Even if an infinite God exists (not bound by time and space, which are finite) who, at a certain moment, created space, time and matter from nothing, what was God doing before this? God, being infinite, must have been around long before the creation of the universe, in the gardens of infinity – just hanging around.

Zeno’s paradox is traditionally thought to postulate the impossible: that, given the existence of infinity, movement is impossible. But the real importance of the paradox is the infinity proof; or, if you prefer, the finity proof: for any movement in space to be possible, for any activity at all to be possible (since all activity involves movement), and for all objects to exist in relationship with each other – infinity can have no existence.

Source by Jonathan H Livingstone

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