Oct 132014
 
Infinite sets

Infinite sets

Is infinity a valid mathematical concept?

It is inconceivable that infinity might not be a valid mathematical concept. It is used extensively throughout almost all braches of mathematics. Top mathematicians have accepted and used the concept for hundreds of years. Calculus would not exist without it. The very foundation for modern mathematics is the axiomatic system known as ZFC, which guarantees the existence of at least one infinite set. On top of all this, there are obviously an infinite number of numbers and so any argument against the concept of infinity must be invalid.

I used to accept these arguments, but concerned by the many resulting paradoxes, particularly those in Hilbert’s hotel (where infinity + anything = infinity), I began to question these views. Perhaps we should have ‘endless’ or ‘open-ended’ sets instead of infinite sets where an actual infinity must exist?

Particularly concerning to me is that when the fundamental laws of mathematics break down, as in the example of Hilbert’s Hotel, these are dismissed as paradoxes. Why are they not identified as contradictions or inconsistencies that highlight a flaw in the premise that ‘infinitely many’ of something can occur?

These problem areas are not difficult to find. For example, we can show that any set of unique (non-repeating) natural numbers, excluding zero, must contain at least one number that is equal-to or greater-than the size of the set. If this rule holds, then the set of ‘all natural numbers’ must contain one or more natural numbers of infinite size. But natural numbers cannot be ‘infinite’ by definition and so we have a contradiction.

The solution is to have another rule that says there are different rules for finite and infinite sets. But no explanation is ever provided for why does (and how can) such a rule suddenly not hold when the set contains ‘infinitely many’ elements? It seems rules have been created to provide arguments for ignoring any apparent contradictions and to support the idea that infinity is a valid concept.

 

The root of the problem

The concept of infinity is used extensively throughout mathematics. Behind all of this is an appreciation of infinity derived from the argument:

As natural numbers are endless, there must be an infinite number of them.

Initially this sounds convincing, but on closer examination the problematic nature of this statement soon becomes apparent.

One definition of infinite is ‘limitless/endless’ or ‘without bounds’ and the sequence of natural numbers is certainly limitless (within the physical constraints of any system that uses them; note that even the human mind is a physical system). If we try to construct the set of all natural numbers by extending a finite set, we fail. It can be extended indefinitely, but at every stage of the process the sequence remains finite. All we can do is assume the infinite set already exists, and not worry about its construction.

Another definition of infinite is ‘non-finite’. This implies that an endless (infinite) sequence can form a completed (non-finite) collection. Only if the collection can be completed will it be valid to use phrases like ‘an infinite number of’ and ‘infinitely many’.

The first definition implies that no end point can be reached, as the sequence is endless. But the second definition appears to contradict this as it allows a theoretical end point called infinity. Perhaps the sequence can still be ‘endless’ and be confined within a completed object? But it is difficult to see how this could occur and it cannot be shown how this can be constructed.

 

Proving infinite sets exist and proving they don’t

We can use a mathematical representation of endless recursion or iteration to prove that an infinite object does exist. By showing that these processes do not end we can claim they are not finite and thus they must be infinite. But such proofs are only valid if it is possible for a process with no defined end point to somehow occur ‘infinitely many’ times.

Proving an infinite set cannot exist is difficult or easy depending on your point of view. If the set of natural numbers was a normal (finite) set, we could simply add 1 to the highest number to generate a number not already there. This would prove by contradiction that the set does not contain all natural numbers. But as we have no insight into the structure of an infinite set, any counter proof of its existence can be rejected on the basis that normal rules do not apply to this type of object.

For example, Cantor uses a diagonal argument to prove there is no bijection (one-to-one correspondence) between the set of real numbers in the interval (0,1) to the set of natural numbers. This proof is achieved by showing that for any given set of real numbers between 0 and 1, another real number can be constructed that is in the interval but is not in the given set. However, if we extend our given set to be the whole interval (0,1), then the diagonal argument says we can construct another real that is in the interval and yet not in the interval! Therefore it could be argued that what this really proves is that any interval of real numbers cannot exist.

 

Proving infinite sets do not exist with diagonal arguments

Consider a set containing the three numbers:

66454517
       3
     507

It is always possible to construct a number not already in the set by going down a diagonal and changing each digit.

For example, first we will start by forming a number from a diagonal. For our ‘units’ column we will take the ‘units’ (least significant) digit from the first number (this gives us 7). Then we get our ‘tens’ digit from the next number (we can assume a leading zero in front of the 3), and so on. In this case we get 507, which just so happens to already be in our set.

Next we change each digit in our found number. Each digit can be changed to any other digit. And so 5 could change to 9, zero could change to 1 and 7 could change to 1 giving us 911.

This method of changing each digit in a diagonal will always generate a number not already in the set for any sized set. This means literally ANY sized set of natural numbers. Therefore we can never have a completed set of ALL natural numbers as this method proves there will always be numbers not in the set.

Does the counter argument offer any insight or glimmer of an explanation as to how a set of unique natural numbers can contain more elements than the value of the largest number in the set, or how there can not be a largest number in the set? No, it simply makes the priori assumption that ‘infinitely many’ is a valid concept and it uses this to supposedly disprove our argument. It says it is true that you will create a number not in the set, but it will have ‘infinitely many’ digits. Thus it is not a natural number, ergo it should not be in the set.

Even if we accept the starting position that ‘infinitely many’ is valid, the counter argument is still self-defeating. Some digits created by our diagonal algorithm will be a finite number of digit-places away from the units-column digit, and other digits must be an infinite number of places away from the units-column digit (if we are to have an infinite result). This forms an inconsistency. This type of inconsistency should not occur because our algorithm is uniform. How and why this inconsistency occurs cannot be explained.

Is there a counter argument to this? Not really… whenever an inconsistency (or contradiction) like this is encountered, it is shrugged off and ignored by the simple act of calling it a paradox. This means it cannot be explained right now, but it will be shown to not be an inconsistency or contradiction when we have a better understanding of infinity. One very weak argument is that infinity is an axiom making it is true by definition, end of story. Another equally weak argument is that infinity is well-defined resulting in entirely consistent mathematics. In other words, based on blind faith that infinity is a valid concept, and even though the concept is not understood/defined well enough to explain the many paradoxes, all counter arguments to its validity are deemed to be false.

 

Consider a simple problem involving an infinite set

Why do we need completed sets such as the ‘set of all integers’ or the ‘set of all natural numbers’? After all, instead of saying a variable belongs to the set of all integers we could say its ‘type’ or ‘class’ is integer, meaning it matches the description of what an integer is.

Also, infinity can create problems that otherwise would not exist. For example, if we accept the infinite set of all natural numbers exists {1, 2, 3, 4…} , it is valid to ask “what is the percentage of odd numbers in this set?”.

The obvious answer of 50% is problematic because it implies an even number of elements whereas infinity is supposedly neither odd nor even. Another option is to say the cardinality of the set of all odd natural numbers is exactly the same as the cardinality of all natural numbers, which is far from a satisfactory answer.

Without infinity we can simply say the percentage of odd numbers in the first n natural numbers is

100*floor[(n+1)/2)]/n      where n>=1           {NOTE: ‘floor’ means ‘round down’}

Without infinity this provides a complete solution for all natural numbers and we no longer have any problems.

And so for problems like this, we could re-phrase questions to refer to ‘the sequence of natural numbers up to n, where n is unlimited’ rather than referring to ‘the set of all natural numbers’.

 

What about other uses of infinity?

The use of the word infinity can be described as misleading at best. More accurate terminology should be used in its place. For example, instead of saying

as x tends to infinity, f(x) tends to whatever

it would be more correct to say

as x increases, f(x) has a limit given by whatever

The article ‘investigation of infinity in mathematics’ explains why even the simplest and most common uses of infinity are problematic. It argues that all uses of the concept should be completely removed from mathematics.

 

Summary

  • If an endless process (of iteration/recursion) is offered as proof that an infinite object can be constructed, then a contradiction is created (as a completed infinity implies we can achieve an end)
  • Despite this contradiction, showing that recursion or iteration can be endless is claimed to prove that something can be non-finite
  • An infinite set cannot be constructed
  • As we do not understand how it is built, we can only make assumptions about the properties of an infinite object.
  • This lack of understanding makes it difficult to prove or disprove it as being a valid concept.
  • Diagonal arguments are accepted when used to prove a set is not countable, but rejected when used to prove an infinite set cannot exist
  • The concept of infinity adds nothing but mysticism. It leads to problems and paradoxes that do not exist without it.
  • The intangible nature of infinity lacks the precision and clarity traditionally expected from mathematics

 

Selected extracts from “Controversy over Cantor’s theory

Gauss’s views on the subject can be paraphrased as: ‘Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn’t belong in mathematics’.

Many mathematicians agreed with Kronecker that the completed infinite may be part of philosophy or theology, but that it has no proper place in mathematics.

Mayberry has noted that “The set-theoretical axioms that sustain modern mathematics are self-evident in differing degrees. One of them – indeed, the most important of them, namely Cantor’s axiom, the so-called axiom of infinity – has scarcely any claim to self-evidence at all”.

  One Response to “Infinity and infinite sets: the root of the problem”

  1. Excellent article. I am experiencing some of these issues as well..

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