Investigation of infinity in mathematics

Introduction

Should infinity be used in mathematics?

Mathematics claims to be a precise and logical discipline and yet it makes extensive use of the intangible notion known as infinity. The following investigation examines this apparent anomaly.

Has mathematics succeeded to master this concept or do the many paradoxes indicate failure?

Understanding infinity: the definition

There is no single clear definition for infinity, but there are a multitude of unclear definitions.
At the heart of these definitions lie three core concepts

1. Infinity is a magnitude beyond any finite value
2. Infinity is the idea that something is endless
3. Infinity is not a number

Concept 1 requires a magnitude greater than any finite value. For this to be possible there would need to be a limit on finite values. As there is no such limit, this concept makes no sense.

Concept 2 is the idea that if something is endless then we can say it is ‘infinite’. But we have already used the word ‘endless’ so what is the point of using this other word? It would appear that by using the word ‘infinite’ we conjure up the idea that an endless thing can somehow end at a place called ‘infinity’, or that an endless sequence can somehow exist in ‘its entirety’ whereupon it will have ‘infinitely many’ parts. We cannot conjure up these notions if we just use the word ‘endless’.

But these ideas are problematic because if something never ends it cannot reach an end point, called infinity or anything else. For example, the sequence of natural numbers (1,2,3,4…) has no end point, regardless of how far we extend our view of this sequence. The sequence has no last term, and so it cannot exist in ‘entirety’. As there is no last term, we cannot have a completed set/collection of natural numbers. Furthermore, there is no mathematical way of ‘reaching infinity’ or of constructing ‘an infinite number of numbers’.

Even if we assume it is possible to collect all natural numbers together into a single object (the size of which we call ‘infinity’), we can easily prove this object cannot possibly exist. For example, we can use variations on Cantor’s diagonal argument to construct numbers not in the object (see ‘Infinity and infinite sets: the root of the problem’ for an example).

Another example is:

(1) If natural numbers start from 1, then any non-empty set containing any amount of natural numbers without duplicates must contain at least one number equal to or greater than the number of elements in the set.

(2) Hence for a set of natural numbers to contain ‘infinitely many’ elements, it must contain one or more numbers of infinite value. But natural numbers cannot have an infinite value by definition and so this forms a contradiction with (1).

The usual (and very weak) counter-argument is that ‘infinite’ objects do not follow the same rules of logic that apply to finite objects, they follow rules that we define and devise for them. It is akin to saying fairies exist and any attempt you devise to disprove this using simple logic will be wrong because they do not conform to your logic or your real-world rules.

It is often argued that infinite means the same as endless by definition, and that it does not matter what words we use as long as the meaning is clearly defined. But ‘infinite’ is not clearly defined and it carries with it the inference that a completed infinite number of elements can be realised. This leads to statements such as since numbers never end, there must be an infinite number of numbers.

The word endless conveys the notion of never being able to reach an end point. It infers that a completed infinity can never be achieved. Perhaps we would not be using expressions like ‘infinitely many’ if we had always used ‘endless’ instead of ‘infinite’. I am not saying the concept of ‘endless’ is in any way valid; indeed I would argue it is not. I am pointing out that it does not carry the same inferences as the word ‘infinite’.

Concept 3 claims infinity is not a number. If so then what is it, and why is it always used in relation to numbers?

In summary, the core concepts at the heart of all definitions lack clarity and rigour.

Understanding infinity: the history

The ancient Greeks studied the paradoxes of infinity and Aristotle concluded that no actual infinites could exist. However, he did allow what he called ‘potential infinites’ to exist to cater for things like the endlessness of natural numbers and so that a geometric line could extend from a point without end.

Calculus was developed in the 17th century. It was so closely associated with infinity that many believe it could not have been developed without it.

The same century saw the birth of probability theory, which also involved infinity. The resulting paradoxes were viewed as intellectual challenges rather than flawed logic. For example,

In an infinite sequence of coin-tosses, any specific sequence will occur infinite times. Does it follow that the probability that an infinite number of coin-tosses all land heads up is zero?

In the 1870s, Georg Cantor invented ‘set theory’ which introduced the concept of infinite sets, such as the set of all natural numbers. An infinite set uses infinity as a number in respect of its size (or ‘cardinality’). The paradoxes in David Hilbert’s Infinite Hotel highlight the absurd nature of infinite sets.

Arguments against infinite sets include

• The notion of infinity does not obey basic mathematical rules (as per Hilbert’s Hotel)
• It cannot be shown how such a set can be constructed
• It can easily be proved that such a set cannot exist (e.g. by adding 1 to the biggest non-infinite number in the set)

In the early 20th century Zermelo-Franekel set theory (ZFC) was established. Within ZFC it stipulates several so-called axioms. Some mathematicians question whether these should be called axioms because their descriptions are complicated and difficult to comprehend. One of these axioms asserts that an infinite set exists, implying this is an obvious truth and no further justification is necessary. It simply ignores all arguments against infinite sets.

Today ZFC is widely accepted as the foundation of modern mathematics.

Understanding infinity: the conclusion

The notion of infinity has no usable definition. Despite this, infinity is still used like a number.

When infinity is used as a number, then
anything + infinity = anything_else + infinity
This simplifies to
anything = anything_else

And so depending on how you use infinity, you can effectively prove that anything equals anything else!

Concepts in mathematics do not need to exist in the real world. This means we can accept all sorts of abstract ideas as long as they obey logical rules, such as the square root of minus 1 for example. But it does not follow that we should accept ideas based on illogical rules that amount to ‘anything equals anything else’.

Even if we ignore the difficulty in constructing an infinite object, we cannot ignore proofs that show infinite objects cannot exist. For example, we can use variations of Cantor’s diagonal argument to show that for any set of numbers, it is always possible to construct a number not in that set. By contradiction, this proves an infinite set cannot exist (see ‘Infinity and infinite sets: the root of the problem’ for an example).

If the concept of infinity were truly as ambiguous, contradictory and nonsensical as it appears, then many paradoxes would occur. And they do.

On the weight of evidence, the only sensible conclusion is to reject infinity as a valid concept. Ideally, every occurrence of infinity in mathematics should be removed. The same applies to the concept of ‘infinitesimal’.

The next sections explain how infinity is misused in mathematics and suggest how infinity can be replaced.

Words to replace infinity

Since infinity has no understandable meaning, it should be replaced by words that do. The same goes for infinitesimal. Interestingly, since infinity cannot exist, there is little point in using the word ‘finite’ in expressions such as “prove XYZ is finite”.

In many cases the word finite can be replaced by limited, and the word infinite can be replaced by unlimited. Another acceptable term is “as x increases”.

We should not use words like “forever” and “eternally” as these are just other ways of saying “infinitely”.

Some phrases can appear to make some sense until you really think about them. For example, consider the expressions “as x tends towards infinity” and “as x approaches infinity”. Assuming we can use infinity in this context, then it does not matter how much we increase x because we will still be an infinite distance away from infinity. The same applies if we decrease x or leave x unchanged.

Since increasing or decreasing or not changing x all result in being the same distance from infinity, it follows that we cannot ‘approach’ it. This argument shows how ridiculous the concept of infinity is.

Alternatives for these expressions are discussed in the section “removing infinity from calculus”.

Other meaningless phrases include “infinitely many” and “an infinite number of”. For example, the statement “there are infinitely many primes” is incomprehensible because the meaning of “infinitely many” cannot be defined. The statement “the sequence of prime numbers never ends” is understandable as it means we will never reach an end point.

The expression “integrate from 0 to infinity with respect to x” is meaningless whereas “find the limit of the integral as x increases indefinitely from 0” can be understood. Similarly, “the sum of the series is infinite” is meaningless whereas “the series is divergent” is clear and precise.

Removing infinity from geometry

It is often claimed that geometric lines are infinitely long and that the Euclidean plane extends infinitely far in every direction.

But Euclid himself was very careful to not use terms like ‘infinitely far’. Aristotle had earlier concluded that an actual infinity could not exist, and so Euclid would talk about lines going on ‘indefinitely’ in a plane.

It is also often claimed that all lines contain an infinite number of points.

But a point is simply the identity of a position in space. A line is simply the representation of a path through space. A point has no size property, and so the question of how many points can fit on a line is nonsensical.

In any real or abstract system, there must be a smallest part of ‘space’ that cannot be sub-divided. The size of this smallest part does not need to be stated explicitly, it can remain as a variable parameter. Then a point could be said to refer to the position of one of these smallest parts of space. A line could extend indefinitely, and a line segment could be defined as a path comprising of a number of these ‘smallest part of space’ points.

A line does not consist of ‘infinitely many’ points.

Removing infinity from calculus

Calculus involves finding the limits of expressions as values diminish or increase. It does not involve substituting zero or infinity into expressions.

Suppose we try to find the limit of the expression 1/x as x increases. We can see that as x increases, the expression 1/x will get closer to, but never be equal to, zero. Therefore we can say that the limiting value is zero. What we cannot say is that 1 divided by infinity is zero, as this would be using infinity as a number, which is nonsensical.

The notation for “as x tends towards zero” (written x→0) is not ideal. Some mathematicians say, “as x goes to zero” which implies that it might actually get there. Others talk about x becoming “infinitesimally small” which is another nonsensical notion.

The notation should indicate that we are looking for the limit of an expression as the value of x diminishes. It should not suggest that we find the limit by substituting zero into the expression. There are many expressions for which substituting zero does not work, for example sin(x)/x.

The nonsensical phrase “as x approaches infinity” could be replaced by just “as x increases”.
Therefore the notation beneath ‘lim’ (for limit) could simply be

x↓                         meaning “as x diminishes”
x↑                         meaning “as x increases”

Similarly, where infinity is used as the upper term in summations and integrals, the symbol for infinity (∞) could be replaced by the up-arrow (↑), which could be read as “an indeterminable increase”.

The traditional approach to the introduction of Integral Calculus is to consider the area under a curve as consisting of the sum of the areas of a series of rectangles. You get a more accurate approximation of the area by increasing the number of rectangles, which involves decreasing the width of the rectangles.

The area under the curve is the limit that sum of these rectangular areas cannot reach. It is not correct to say that the area is reached when we have an ‘infinite number of’ rectangles. Indeed, if this were possible then each rectangle would have an ‘infinitesimally’ small width giving it a zero area, and our area under the curve would then be the sum of ‘infinitely many’ zeros which would be zero.

Removing infinity from set theory

Instead of saying a number or variable belongs to the set of all integers we could say its ‘type’ or ‘class’ is integer, meaning it matches the description of an integer. Another approach could be a simple change of terminology. Instead of infinite sets we could have open-ended sets. This makes it easier to appreciate it makes no sense to compare the sizes of open-ended objects.

Where a problem refers to all numbers of a particular number type, we could change the wording to avoid infinity and to add clarity. For example, instead of asking

What is the percentage of odd numbers in the set of all natural numbers?

What is the percentage of odd numbers in the natural numbers? {Assume ‘the natural numbers’ is sequence of natural numbers from 1 to n where n is unlimited}

Such re-phrasing enables precise mathematical solutions to be produced for problems where infinite sets are of no use (see ‘Infinity and infinite sets: the root of the problem’ for more information).

All concepts that use infinite sets should be dispensed with. Notably, this includes ‘countably infinite’ and ‘transfinite numbers’.

Removing infinity from probability theory

We can say an event can repeat an indeterminable amount of times, we should not say it occurs an infinite number of times.

Someone might ask: “but what if we have a true random number generator that outputs a real number between 0 and 1, then surely the set of all possible outcomes has to be infinite?”

The interesting point here is that a set of all possible outcomes has to take into account all results that are possible, not just results that have already been ‘encountered’. Therefore how can we describe such a set? One answer would be that its size could extend indefinitely. This means that the size is not defined; it is not the same as saying the size is infinite.

It is a bit like saying if we divide 1 by three then how many threes will we have after the decimal point? You can keep increasing them as much as you like, i.e. indefinitely. This is not the same as saying there are an infinite number of them.

Removing infinity from irrational numbers

Irrational numbers are real numbers that cannot be represented as terminating or repeating decimals. Examples include pi and the square root of 2.

The decimal expansion of an irrational number lacks precision, as the number can never be represented in its entirety. This shows why the decimal system is inadequate for representing irrational numbers.

And so how can we fully express irrational numbers in a framework that we can work with? The solution is to use symbols, such as π and √2 rather than their decimal expansions. We can then work with irrational numbers in an abstract framework. If we want to see a numeric result rather than a set of expressions containing symbols then we have to expand the expressions using the constraints of the real or abstract world to which we are applying it.

This problem has already been solved to some extent for the encoding of vector graphics in computer software. Vector images are made up from multiple objects. Each object consists of mathematical instructions that define shapes. And each shape is defined in terms of points and paths. This makes the image fully scalable without loss of quality. It only gets converted to discrete pixels when the image is rendered onto a real world object like a small screen or the large side of a building.

Removing infinity from mathematical models

This leads to an intriguing question about rendering mathematical objects in the real world. Given that the smallest discrete value that can be used on a screen is a pixel, what is the smallest discrete value that can ever be used in the real world?

As the real world is finite, there must be a smallest distance that can exist. For example, if we accept that a small movement cannot consist of ‘an infinite number of’ smaller movements, it is reasonable to assume it consists of a finite number of discrete movements, similar to quantum leaps.

If the smallest possible distance is a ‘Planck length’ say, then in a mathematical model of the real world we would only need to be able to represent distances to the exact number of Planck lengths. Indeed, if we were to represent distances smaller than a Planck length then our model would no longer accurately reflect the real world and any results found from this model would be untrustworthy.

But in such a finite world, what happens if an object of length 7 Planck lengths needs to be divided by 3? One solution could be to modify all operations (like ‘divide’) to work in a way that caters for a ‘smallest part’.  So this division might result in two objects with a length of 3 Plank lengths and a remainder object of length 1.

Does infinity exist in the real world?

If something were to exist in the real world that could not possibly be finite, there would be some justification for the argument that infinity might exist in some form.

But if the concept of infinity is nonsensical, then it should be possible to provide finite solutions to questions such as “if the universe is finite then what lies beyond its extremities?” Any proposed solution to this question will, of course, just be wild speculation, but here is an attempt.

Outside of the universe could be absolutely nothing. That is to say, the complete absence of all properties, including space, time, and everything else. Then to ask a question like how far does this nothingness go on for is meaningless since this nothingness has no property of depth or volume.

This may be a difficult concept to accept, but it does at least demonstrate that finite solutions can be proposed for even the most challenging real-world arguments for the existence of infinity.

Removing infinity from elsewhere in mathematics

In terms of specifying what needs to be changed, this article has only scratched the surface. Infinity appears explicitly or implicitly in most branches of mathematics, and it needs to be completely removed from all of them if we want precision and rigour to return to this discipline.

Author:
Date: 19th September 2014
Updated: 26th July 2016