StuffThatMatters: Mathematics
Georg Cantor, the founding father of the wondrous discipline called Set Theory, was one of the greatest mathematicians of the 20th century, and as Prof. Marcus du Sautoy notes, maybe his mathematics was too ahead of his time, which contributed to its underestimated devaluation by his contemporaries.
In here, let us discuss one of the greatest mathematical discoveries by Cantor, which is a rather mind-blowing conclusion for us all, and yet seemingly irrefutable (to date, at least).
Taming the infinity
Before Cantor, not many mathematicians worked with infinity, and it was widely thought that infinity was beyond the scope of mathematics. However, it was Cantor's genius that finally included the infinity, in fact, the infinities, into the paradigm of modern math.
We are all pretty familiar with the symbol ∞, which is the symbolic representation of infinity. However, to think that there's only a single infinity, would be a premature conclusion.
Hilbert's Hotel, a hotel with infinitely many rooms
In his 1947 book, Cantor devised this beautiful and easily understandable veridical paradox (a statement with a very intuitive set of premises, but a bizzare and counter-intuitive conclusion), which he attributed to one of his mathematical heroes, David Hilbert. And this is what changed how we see infinity, forever.
Premise: Let us assume that there's a hotel, run by Hilbert, which has infinitely many rooms! So, the room numbers can be written as the series below:
1 , 2 , 3 ... ∞
To write it as a set, it'd be:
{ x : x ∈ [ 1 , ∞ ) }
Now, let us say that on a typical evening, Mr. David Hilbert sees that every room in the hotel is full, and there's no room left for anyone else. But then, 3 groups of guests arrive, and the fun starts.
Group 1: Finite number of guests
Ok, so at 7:00 PM, 5 new guests arrive at the hotel, and request Mr. Hilbert to reserve a room for each of them. Now, the hotel is already full, as we've mentioned. But Mr. Hilbert makes a clever solution. He shifts the occupants of each room n to the room n + 5.
That is, the guy occupying room 1 shifts to room 6, the gal occupying room 2 shifts to room 7 and so on.
Naturally, rooms 1 to 5 are now empty, and Mr. Hilbert easily books them for the new guests.
Countably infinite guests
Now, let us say that at 8:00 PM, another bunch of guests arrive, only they are infinite in number this time. So, the guests too, like the rooms, are numbers from 1 to ∞. They request rooms to Mr. Hilbert.
Mr. Hilbert employs a yet cleverer solution, he makes each occupant at room n to the room 2n. So, the man at room 1 moves to room 2, the lady from room 2 shifts to room 4 and so on. Since n is a natural number, 2n excludes odd numbers from the list.
So, all the odd-numbered rooms are now empty, and the guests now comfortably enter the rooms.
Conclusion: Finite infinities can be equivalent.
Uncountably infinite guests
Ok, so at 10:00 PM, another group of infinitely many guests arrive, with each guest denoted by a real number (R) between 0 and 1. This time, there's trouble.
Say, Mr. Hilbert empties the entire hotel for the newly arriving guests, and now all the rooms are empty. Then, he makes the following arrangement:
Guest 0.1 moves to Room 1
Guest 0.2 moves to Room 2
...
Guest 0.n moves to Room n
Now, all the rooms are now full.
But there are guests left! Between 0.1 and 0.2, there are infinitely many numbers, 0.11, 0.12, 0.1113646247 and 0.164924848246284638 and so on! So where are they going to stay now?
Conclusion: This demonstrates a very important property of the infinities, real number infinities are larger than integer infinities.
Infinitely many infinities: power sets to the rescue
So, let us say that Mr. Hilbert now grows really wary, and decides to build an ever grander hotel, with uncountably infinite rooms.
Now, let us say that the third group of guests are grouped together to form a set S. So, n(S) = ∞ (i.e the cardinality of the set, or the number of elements in the set, is infinite).
But, what about P(S), i.e the power set of S? (I explain power sets at the end of this post, refer to it if you're not familiar with it). We know that the cardinality of a power-set is given as:
n( P(S) ) = 2 ^ ( n(S) ).
Now, the same group of guests arrive again, and demand rooms. Mr. Hilbert can now easily accommodate them all, with the following process:
Guest 0.1 shifted to Room { 0.1 }
Guest 0.172646 shifted to Room { 0.172646 }
...
Guest x ∈ ( 0 , 1 ), x ∈ R shifted to Room { y : y ∈ ( 0 , 1), y ∈ R }
So, since 2 ^ n(S) > n(S), there must be rooms still left, even after accommodating these guests. That is, when each guest is shifter to the room whose name is the singleton set containing the number associated with the guest, all those elements of the power-set containing non-singleton sets, such as { 0.1 , 0.2 }, { 0.1 , 0.4 , 0.6 } etc are now empty!
Naturally, infinitely many rooms are still empty.
Now, there's a power-set of the power-set itself, so if a group of people now arrive at the hotel, with each person associated with an element of the power-set of the power-set set of the real-number infinity between 0 and 1, the hotel will be blocked once again, with infinitely many people left without rooms.
Conclusion: So, there are infinitely many infinities. At first, we see that the power-set of the real-number infinity set is larger than the latter, but then again, every power-set can have its own power-set. And since there can be indefinitely many, infinitely many power-sets, there are infinitely many infinities (proved).
A bit of explanation
Although set theory is the very basic thing behind this, non-mathematics students may not be familiar with it. So, here's a brief discussion.
Set: A set is roughly any group of things, known as its elements. These can be numbers, letters, names and anything else. For example, S = { 1 , 2 , 3 } is a set of 3 elements. If an element e is in a set, it is written as e ∈ S (i.e e belongs to S).
Cardinality: The cardinality of a set is the number of elements it contains. It may be finite or infinite. It's written as n(S).
Subsets and supersets: Subsets are sets such that, every element of the subset P are present in the original set S. But not vice versa, usually. They are written as P ⊂ S.
Likewise, supersets are sets such that, every element of the set S are present in the superset Q. They are written as S ⊂ Q.
Power-sets: A power-set, P(S), is a collection of all possible subsets of a set S. It has a number of elements given as 2^n, where n is the number of elements on the original set. For example, let S = { 1 , 2 , 3 }. So, all possible subsets of S are given below:
{ } (empty set, with no elements)
{ 1 }
{ 2 }
{ 3 }
{ 1 , 2 }
{ 1 , 3 }
{ 2 , 3 }
{ 1 , 2 , 3 }
Naturally, the power-set is a set containing all these. So,
P(S) = { { } , { 1 } , { 2 } , { 3 } , { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 1 , 2 , 3 } }
And clearly, n = 3, so number of elements in P(S) = 8 = 2^3.
Set theory is a very broad theory, and so there's a lot more to explain, but I guess this can do for our purpose here.
References
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