The Continuum Hypothesis was the number one mathematical challenge set by Hilbert at the start of the 1900’s. It was only ever solved in the negative. In this post we explore the background to the problem in more detail.
The continuum hypothesis is based on the mathematics of ‘infinite sets’. The prime example of this are the ‘whole’ numbers, that traditionally appear on the number line.
The problem arises when we ask the question, “how many numbers are on the number line?”. This leads to an apparent paradox as reciprocal space between zero and one contains all real numbers. Despite this George Cantor, grandfather of fractal geometry, was able to construct a mathematical proof through a process he called ‘bijection’. By taking the example of the sets of whole numbers, and comparing it to the real number set, he could determine that the whole number represent an infinite set that is ‘smaller’ than the real numbers. However, what he could not determine was how much smaller.
This problem led to the conjecture of the continuum hypothesis, which posed an interesting question. Is there an infinite set that is greater than that of the whole numbers, yet smaller that that of the real numbers. Whilst this challenge was the great set out to be solved in the 20th century, it has only been shown that under the current axioms of our mathematics a solution is not possible. therefore, logically, if a solution should be found, then it may well require a radical overhaul of our mathematical perspective. For the nature of infinity lies at the heart of all numeration, right down to the simplest of calculations.
This is because any number can be divided by 1. The fact that a fraction is an expression of a division of two numbers means that is the higher number
One of the keys to understanding the nature of infinite begins with a comprehension of the different types of numbers.
When most people think about numbers, we often relate the concept to objects in our physical realty. In our view it is a function that is intrinsically founded in the nature of consciousness itself. The ability to distinguish one thing from anther, is the first step to identifying similar objects and counting them.
Space is like that. We are surrounded by energy, atoms that are in constant fluctuation, yet the realty we are experiencing is formed of countable objects.
Historically the definition of cardinal numbers has altered. Sometimes it includes zero and the negative numbers but not always.
The next class of number are called ‘Ordinal’. These are used to categorise or arrange objects according to there quality.
For example, from large to small, or the top winners of a race. The terms first, second and third…and so on, are used for ordinal numbers, whereas cardinal numbers are denoted by the numbers one, two, three… etc.
The key difference here is quality (ordinal) Vs quantity (cardinal). It is interesting to note that the great geometrician, Euclid, suggested that cardinal numbers actually do not exist. Instead he defined the first unit as the number one. All other numbers are just a multitude of this first initial unit.
Natural numbers - ℵ0
These two types of Cardinal and Ordinal Numbers form the set of infinite whole numbers. The Latin word integer means whole. As the name suggests, theses number that only be can be expressed without any decimal or fraction.
It is this set that is first defined in the continuum hypothesis, as aleph 0, mathematically expressed as the Greek letter ℵ0. We can establish that all other number series such as square numbers, will increase more rapidly than the set of sequential whole numbers. As they are less ‘dense’ and leave more gaps, the infinite series is contained within the infinite set of whole numbers. But is there an infinite set that is bigger than ℵ0?
Real numbers - ℵ1
Aside from countable numbers, there are also rational numbers that can be expressed as a fraction. In fact all cardinal numbers can also be expressed in this way. However, there is another type of number that defies expression. Irrational numbers such as √2 and √3 cannot be expressed as a fraction, however they can be expressed in algebraic terms. The golden ratio for example, can be express as, (√5 ± 1) ÷ 2. Within the irrational numbers, is another subset term transcendental numbers. These numbers cannot be constructed mathematically using any formula. The most famous example is the number π (pi). This is a geometric constant that expressed the ratio between a circle diameter and circumference. All of these numbers, including the whole numbers are contained within the infinite set of real numbers.
All real numbers with the exception of √-1 (number i), which form the the complex number plain, can be found on the number line. This infinite set is referred to as ℵ1.
Now that we have a clearer overview of the two types of infinite sets ℵ0, all whole numbers, and ℵ1 all real numbers, we can begin to look at the background to the continuum hypothesis.
The first point to consider whne trying to answer how many numbers are on the number line, is how can we count them. George Cantor in his famous diagonal proof showed that it is actually impossible to count all numbers.
Using the most simple binary system, we can see if we can list all possible numbers. the list is construed so the number of variations grows. Lets call the set S. Next we employ a simple rule which takes the first number in the first row, second number is the second row, and so on, and f inverse the digit. A one becomes zero and a zero becomes one. This creates a new number, that as it is based on set S cannot be found in the set. If a number exists outside of S the S cannot be listed, and so is uncountable.
As numbers cannot be listed, we need to find another way of qualifying them. Cantors solution was to look at the density of numbers, or more importantly the numbers missing from an infinite set.
How big is infinity?
How big is infinity? You might think that infinity is infinity and that’s it. However, mathematics tell us a different story.
Let’s take the infinite set of whole numbers, that proceed in series, on into infinity. Each one is evenly spaced from the next. We can compare the sequence to another, such as that to the 2xs multiplication table. This sequence professes, but misses out certain numbers. Yet these missing numbers are present in the set of whole numbers. As both series can carry on to infinity, they are called an infinite set. Yet we can see that the whole number series contains more numbers than the 2x series. So the whole number series is ‘bigger’.
There are a infinite number of whole number set, just built from the nature of multiplication. Just as the 2x table produces a predictable number sequence, so the same goes for 3x, 4x, right the way up to ∞x. (More on the multiplication of ∞ can be found on our post, 4D squaring.)
However as we move further up the multiplication scale, we find that there are larger gaps between the numbers. The numbers are ‘less dense’ than the whole number series, (which in a sense is 1x multiplication table). Whilst we cannot measure the ‘length’ of infinity, we can understand its nature through observing the missing elements. In doing so we find that any infinite set based on multiplication will contain less numbers in its set than the series of sequential numbers.
Cantors ‘Binary tree’ was one of the first examples of missing density of a number set. A line is divided into 3 parts and the center section is removed. The process is repeated for the two new lines in the next step, and so on into infinity. As more space is removed the number line gets less dense.
The nature of missing numbers can be qualified by a process developed by George Cantor, call bijection. This refers to a one-to-one correspondence of the numbers. Previously we saw how all multiplications tables produce a number series bigger than 1x. Through bijection we mathematically establish a one-to-one relationship with each step, which provides the density ratio between the whole numbers and any other whole number series.
In this way Cantor could show that the whole number set is the ‘foundations of all other whole number series, and therefore there can be no other whole number set with greater density than the sequential numbers.
That is until we begin to include the other real numbers, of course.
Having established that that densest possible infinity set that can generated by whole numbers are the sequential series of numbers. Whilst infinite in nature, the process of bijection provides the logical basis for this assertion. However, when we begin to include the complete set of real numbers the situation becomes far more complicated.
Larger than whole numbers
The infinite set of whole numbers is given the value ℵ0. Is it within this infinite set that all other sets of whole number sequences occur. However, the same cannot be said if we introduce the REAL numbers.
For example, we can map the 2x table using bijection, and see that as the process carries on into infinity, is will be half as dense as the sequential numbers. We can say the density ratio is 1:2.
Now lets do the same experiment with the fraction ½ or 0.5. Starting from zero in the first step the whole numbers progress in sequence. However, the fractional series only progesses at half the rate. the result is exactly the opposite density ratio to the 2x series. the whole number series is only half as dense as the fractional series.
Fractions and Reciprocals
The infinite set of Real numbers is defined as ℵ1 and it can be mathematically proven that it is a denser type of infinity than the whole numbers. This is clearly demonstrated by the fact that any number above 1 will have a reciprocal value.
If we thanks about how fractions are constructed, we find one number is divided by another. Notice this key difference between the whole numbers that are compared by addition or multiplication. We write a fraction as one number over another. If the larger value is on top, the fraction will appear above the number 1. if the smaller number is on top, then the result will be a reciprocal number between one and zero. This is consistent with all fractions. In fact even a whole number can be expressed as x/1, which gives its reciprocal value 1/x.
Bigger than infinity
We have seen that when we begin to include fractions, we can create an infinite number series that has a greater density than the whole numbers.
Therefore, ℵ1 > ℵ0.
But how much greater is it?
We can perform the bijection process on the set of real numbers by creating a second number line at 90° to the first. Then we can draw a diagonal line from any two points and find correspondence. In fact infinite correspondence. Even the tiniest section between one and zero has more numbers within it than the entire infinity of whole numbers. We have already shown that these numbers cannot be counted in the traditional sense. We can only measure them through density. This is fine for whole number series but when is comes to the real numbers, we run into serious problems.
The fact that all real numbers above can be expressed as a reciprocal value between 0 and 1, leads to a paradox. The numerical space has a greater density than all of the other number line above one, including real numbers. Yet the space between 0 and 1, can hold the same number of infinite fractions, as the space between 1 and 2. So the same goes for each number the gets added sequentially. So which has the greatest density? The space between zero and one, or the space between zero and three?
You should think that the space between zero and three should have 3 times as many numbers. Yet each number that appears in the numerical space above 1 will have a reciprocal value, a one-to-one correspondence, to a point within the space between 0 and 1.
This paradox was determined by Bertrand Russel. In set theory, any definable subclass of a set is a set in its own right. As the subset of reciprocal space is denser than the entire infinity of numbers on the number line, we run into a mathematical contradiction when trying to identify the density of ℵ0.
Is was at the turn of the 1900’s that the continuum hypothesis was present by Hilbert as the number one mathematical problem of the century to needed to be solved. Despite prize money being offered, the closest mathematicians could get was to prove that under our current mathematical axiom, the Continuum Hypothesis is impossible to solve.
Whilst we can perceive the fact that there are more real numbers than integers. The difficult arises when we try to ascertain the density of real numbers compared to the integers. If we cannot ascertain the size of ℵ0 then how can we make any kind of analysis. From the perspective of density, the real numbers are infinity dense, wherever we look.
Attempt to overcome this paradox were presented in 1908, by Ernst Zermelo, who tried to alter the foundational axioms of set theory in order to overcome the Russel Paradox. However, Gödel’s incompleteness theorems, showed that even with Zermelo adaptation of the axiom that lie at the foundations of set theory, a solution to the continuum hypothesis can never be found.
This directly challenged Hilbert’s attempt to encourage mathematicians to find a consistent set of axioms for mathematics.
To this day the Continuum Hypothesis remains unsolved, and is consider unsolvable. That is unless some new kind of mathematical construct were to be developed that could overcome the paradoxes present by the illusive nature of infinity.
Now that you have been introduced to the concept of the continuum Hypothesis, let us pose the problem.
We know that the set of integers is a smaller infinity than the set of real numbers. In fact all whole number are derived from the real numbers, and all whole number series derived from the sequential whole numbers. Each set contained within the other.
But is there an infinite set that exists between the REAL numbers and the integers?
That is the question presented by the Continuum Hypothesis.
One of the fundamental problems to overcome is notion of trying to measure something that has an infinite density at any point of its construct. How can you tell if a set exists in between two types of infinite sets if you cannot determine the size of one of the sets?
Furthermore, it appear that we cannot even determine the density of even the reciprocal space between one and zero.
It is clear that in order to solve the incompatibility of mathematics and the infinite nature of numbers, requires more than just a solution based on our current axiom. What we need is a fresh new approach to the problem, which will inevitably question everything we thought true of our current mathematical system. Such a revolution will be as greater shock to the mathematical community, as the discovery of the atom was to scientists who in the early 1900’s believed they had already discovered everything there was to know about the universe.
Solving the Infinite
Our solution to the Continuum Hypothesis is exactly that. A new geometric axiological approach to the filed of mathematics, that explains the nature of infinite number sets, and can even put a boundary around the whole set of real numbers. In this way we can quantify the reals, and propose a logical solutions to the apparently illogical paradoxes presented by infinity.
More about the Maths of infinity
This article is part of our new theory, ‘Maths of Infinity‘
Read the main article or browse more interesting post from the list below
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