Infinity
Mathematics is solving problems through abstraction. One of the abstractions I had come across is infinity. Have you ever thought when and where infinity might end? Infinity is not a number, it is an idea or a concept. An idea of being endless and going on forever.
Infinite was a major stumbling block for many. It required a leap of faith that many mathematicians were unwilling to take.
Georg Cantor, one of the pioneers of modern set theory, started out on a problem which continued to vex him for the rest of his life: the nature of the continuum. The continuum is the ideal infinitely divisible space conceptually required for a theory of continuous change, or, in simple terms, it is a continuous series where all parts are very similar to their nearest neighbour, but the ends or extremes of it are different from each other. This describes something that changes gradually (little by little) from one condition, to a different condition, but without any sudden changes or discontinuities.
You can count numbers 1,2,3,4,5,……….. What is the biggest number you can think of ? 1000 ? Or a million ? Or maybe a zillion! But the biggest number lies somewhere in infinity which is still unknown.
Infinite series can be of natural numbers such as 1,2,3,4,5… etc.
Or, 2,4,6,8,10,….. which are even natural numbers.
Or, 3,5,7,9,11,….which is the series of odd natural numbers.
There is also an infinite series of prime numbers, 2,3,5,7,11,…..
The series of integers that consists of 0,1,-1,2,-2,3,-3,…..etc.
All these series go till infinity.
It intuitively seems that the infinite series of odd or even is half of that of natural numbers and that of prime numbers is even smaller and that of integers is double the size of it.
But are the infinities that is reached in every series the same? We’ll look upon that a little later. These type of series where you can easily list the numbers present in it are called as countable infinite series.
So, what are uncountable infinite series? These consists of real numbers or decimal numbers, irrational, numbers, fractions etc.
Are there more real numbers than integers? Whichever real number appears on the right-hand side, can’t we just find an integer to set opposite it, since there are infinitely many integers?
So, let’s take a list of real numbers, say,
0 . 1 3 4 8 2 9 5
0 . 5 7 3 1 4 6 0
0 . 7 4 0 0 9 3 7
0 . 1 2 9 4 4 0 9
0 . 4 0 7 5 8 8 0
0 . 8 1 6 3 1 2 1
0 . 6 0 5 9 4 3 6 ……..
Well, look at the numbers on the diagonal. It is 0.1704826 … Now change this by subtracting one from each of the digits. It becomes 0.0693715 … This number will never appear on the list because it differs in at least one of the decimal places from anything which appears there. So, there are more reals than integers.
If you take a number line, you have ……-3,-2,-1,0,1,2,3……. But, there is 0.5 between 0 and 1. But, hang on, there is 0.25 between 0 and 0.5. In between 0 and 0.25 there is a 0.237….. So, this list will never end.
You always, ALWAYS, find a real number between two real numbers no matter how deep you go between the numbers infinite amount of times. Thus, this kind of infinity that is reached is called uncountable infinity.
Cantor’s argument implied the existence of different types of infinity. He went on to develop a whole new theory of transfinite arithmetic, convinced that he had uncovered a powerful new principle of reality with profound physical and spiritual implications.
But, in 2016 Maryanthe Malliaris and Saharon Shelah published a 60-page paper in the Journal of American Mathematical Society that changed the thought of all infinities being equal.
In July 2017, Malliaris and Shelah were awarded the Hausdorff medal, one of the top prizes in set theory. The honor reflects the surprising, and surprisingly powerful, nature of their proof. Most mathematicians had expected that a proof of that inequality would be impossible within the framework of set theory. Malliaris and Shelah proved that the two infinities are equal. Their work also revealed that the relationship between the types of infinities has much more depth to it than mathematicians had realised.
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*photo: Numberphile (YouTube)
