One of the most important mathematical objects is the set, with its many useful properties.
We can look at the size of the set by wondering how many members it contains. The answer obviously can be found by counting them.
Then we can take another set and compare its size with the first one, again by counting the members. We call this size the cardinality of the set.
One can wonder, do I have as many fingers on my left hand as on my right hand?
Well, by counting first left and then right, we find that our hands have an equal number of fingers. But there are or have been primitive cultures that couldn’t count beyond three. Now what?
They could solve the problem by putting the fingers of the left hand against those of the right hand. Where after, they would find that there were no fingers left without a counterpart. So even these primitives would have found the right answer: the number of fingers left and right is equal.
This conclusion has been achieved by the simple researchers without counting. They did this by applying what mathematicians call a one-to-one correspondence. Or even more trivial: by mapping the members of the left-hand set to the members of the right-hand set.
George Cantor started around 1870 his research on infinite sets. He, of course, was aware of the fact that an infinite set could not be counted. Nevertheless, he would compare the size of infinite sets with each other and realized that this should be done by mapping the members of the sets in question.
Maybe we should pause here shortly to ponder about infinity, since it is not an easy subject to grasp. It is, in fact, rather counterintuitive. Most of this is caused by our inclination to consider infinity as a quantity or a number.
It is not. You can’t calculate with it, cannot multiply or divide by it, at least not in the usual sense. Infinity is a concept that requires its own arithmetic.
But let’s look at the set of natural numbers. And from there to the set of even numbers. It seems obvious that the latter set is smaller than the first. Half of it, one should say.
For every quantity up to the number N, no matter how big N is, this is true: the even numbers in such a set are half the quantity of the natural numbers. But when we go to infinity all the way, it is no longer true. The sets are of equal size. We say that the sets have the same cardinality.
The set of primes looks even more pronounced. Primes are ‘thinning out’ further on the number line. The further we go, the rarer the primes get. But the mathematicians state that the cardinality of the set of primes is the same as that of the set of the natural numbers.
How can this be?
Well, the answer lies in the fact that all these sets are countable. What can we say, for instance, about the set of primes? This: there is no biggest prime number. There is a simple proof that there is no limit to primes. There is always a bigger one. We never run out.
And because of this, the primes can be mapped one by one to the natural numbers. Another prime? Here is another natural number to map it with. And so on, up to infinity.
This is what countable means when we are comparing sets. Hilbert’s hotel is an entertaining lesson about grasping infinity.
Cantor knew all this of course. But still he pondered about the cardinality of infinite sets.
The ‘smaller’ sets of even numbers and primes appeared to be of equal cardinality as the set of natural numbers. But what about the rationals?
The rationals seemed to be abundant. We can make any number of fractions by combining any number of natural numbers in the numerators and denominators. Shouldn’t that scale out the cardinality of counting numbers?
Cantor showed us it didn’t. By writing the rationals systematically down in a square of infinite horizontal and vertical length, he could draw a line through them without missing a single one, thus mapping all the rationals to the counting numbers.
What more numbers are out there? The irrationals, of course.
The numbers that we can’t trap into a simple form, like the naturals or the rationals.
Now we are in the realm of the real numbers. A lot of these numbers are to be represented in a decimal form, since there is no simple way to write them down.
Cantor took up the challenge of ‘counting’ the real numbers. He observed the interval (0,1) and found infinitely many real numbers in there. A closer look dazzled him, and he used his now well-known method of writing all those numbers down in his infinite square (horizontal infinite and vertical infinite). After writing this infinite number of reals down -no small feat-, he tried to ‘count’ them, which is the same as mapping them to the set of natural numbers. But then he found that he could create a new real number by drawing the diagonal of this enormous square. Starting at the left, top, and proceeding to the right, bottom, he changed the number of each crossing. Doing so, he had created a new real number that was not previously on the list. The number could not be the same as the first number, as it differs at the first position. It could not be the same as the second number, since it differs at the second position. It could not be the same as the third number since… And on and on.
Of course, this new number could be added to the list, but then a new diagonal would repeat the trick. And so infinitely many times.
So Cantor discovered and proved that the infinity of real numbers is bigger than the infinity of natural numbers and therefore cannot be counted.
Zooming in on his interval (0,1) Cantor found that there is no gap between the numbers in this interval. It isn’t like the natural numbers, where you have to step over to the next number. In the realm of the real numbers, there are no gaps. Between any two numbers, there is always another to be found. Numbers are just passing by fluently.
That’s why he called this a continuum. Like dimensionless points on a geometric line.
Cantor wanted to have a ranking order to the infinities and he stamped the cardinality of the first set (the counting set) as ℵ₀ pronounced as aleph-0, from the Hebrew letter ℵ.
Somehow, he could also prove that the cardinality of the real numbers (the continuum) is equal to that of the power set of the set of natural numbers.
Each set has a power set. That is the set of all subsets that can be made out of the original set. E.g. {1,2,3} has {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}, {Ø} as subsets.
The cardinality of the power set appears to be 2n where n is the number of members in a set.
So, {1,2,3,4,5} has 25 = 32 members in its power set.
The power set of the set of natural numbers – P(N) – is 2aleph-0, a breathtaking ‘number’.
Now, Cantor defined his hypothesis.
He knew
- The set of natural numbers has cardinality aleph-0 (this by definition)
- The continuum is of size 2aleph-0
And from this, he formulated his CH: 2aleph-0 = aleph-1
Or simply rephrased: the size of the continuum is aleph-1.
More informally: Cantor was wondering: does a set exist with higher cardinality than aleph-0 but with a smaller size than 2aleph-0 that would be entitled to the aleph-1 position, so that the ranking order should change into aleph-2 (or higher) for the real numbers?
He thought there wasn’t such a set, and exactly that is his Continuum Hypothesis.
Of course, Cantor was puzzling over which numbers made his continuum so huge. He gave it a further look. The rationals had been covered already and found countable. What about the irrationals? These numbers can merely be written in decimal form, as no recurring pattern in the decimals ever occurs.
We know that, for instance, √2 can’t be written as a fraction. Pythagoras discovered this to his dismay, for he thought that numbers should behave properly. Anyway, the rationals and irrationals together form the set of real numbers, and since the rationals are countable, it’s obvious that the irrationals are responsible for the extreme abundance of the real numbers.
Cantor realized that all those numbers, natural numbers, rationals, and a lot of irrationals are in fact algebraic numbers. Such a number is the root of a polynomial of finite degree in x, with rational coefficients. This sounds more intimidating than it is.
Here is one: ax5+ bx4+ cx3 + dx2 + ex + C.
And here is another: ax2 + C.
The terms higher in x disappeared because their coefficient is 0; when a coefficient is 0, the term drops out.
We pick up the expression x2 – 2 = 0.
The root of this polynomial is √2, which makes √2 an algebraic number.
Likewise we find that 6x-2 gives the rational 1/3 as an algebraic number, and even the imaginary number i as the root of x2 + 1 = 0 is algebraic.
Lots and lots of numbers have been covered now, and they all belong to the set of real numbers. But then Cantor was able to prove (this proof goes beyond the intent of this essay) that the set of algebraic numbers is countable and thus has cardinality aleph-0.
And cardinality aleph-0 means, equal to the cardinality of the set of natural numbers, the counting set. In other words, it is not the set of the algebraic numbers that gives the continuum its higher cardinality.
So what did he have left? There were, of course, still some transcendental numbers, but that seemed like a joke. Joseph Liouville proved in 1844 the existence of transcendental numbers. These numbers are not roots of any polynomial, meaning they are not algebraic numbers. But although the proof of their existence was there, apart from some specially constructed examples, there weren’t any known. The numbers π and e were under suspicion, but it would take until 1882 before π was proved to be transcendental, years after Cantor’s struggle with the matter.
This is what the mathematicians report these days about transcendental numbers:
< Although there are infinitely many transcendental numbers, only a few of them are known. Strangely, it is not so hard to prove they exist, but it is ferociously hard to prove that a given number is transcendental. />
Cantor’s ultimate conclusion was (and had to be) that the abundance of real numbers should not be sought in the natural numbers, not in the rational numbers, and not in the algebraic numbers, but that the uncountable richness of the continuum lies among the elusive transcendental numbers. Highly unexpected, but true nevertheless.
It is important to underline once more that his Continuum Hypothesis does not express the ‘size’ of the set of real numbers but foremost the order of infinities. The set of natural numbers has cardinality aleph-0, and Cantor thinks that there is no set with cardinality > aleph-0 and < the continuum’s size. So the continuum’s cardinality (which is 2aleph-0) must be aleph-1. And that’s his hypothesis.
In fact, this is a special case of the later proposed generalized continuum hypothesis(GCH), which asserts that for all α, 2aleph-α = aleph-α+1.
But it is precisely this case that Hilbert put on his famous list of open problems.
It may seem that Cantor’s interval (0,1), however big is dwarfed by the total size of the set of real numbers. But that is not true. We can make a one-to-one correspondence from the numbers in the interval to each number in the set of real numbers. Odd things are going on in infinity, and applying our intuition to these matters often leads to wrong conclusions.
How can the CH be proven?
Kurt Gödel had a way of stepping forward with disappointing messages. Earlier, he brought bad news to Bertrand Russell with his incompleteness theorems, which undermined Russell’s long effort to place mathematics on a complete logical foundation.
In 1938, Gödel showed that the Continuum Hypothesis cannot be disproved within the formal axiomatic framework on which the hypothesis was based. He believed the CH to be false and that we should wait for an extension of the framework to prove this.
In 1963, Paul Cohen showed that the CH also cannot be proved within the aforementioned formal framework, and that is where we are today.
Not until the formal axiomatic base is properly extended can the CH be proved or disproved.
Eef
2022-10
France
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Editor’s comment:
This article is a completion of the same article published in october of 2022. Although it has ‘V2’ in its title, it replaces the previous version of the article.