I’m trying to get to grips with what this term means. I’m interested in mysticism, philosophy etc and it seems like a fascinating idea especially considering the symbolism of the Aleph.
I have absolutely no maths knowledge… Just a GCSE in maths and some mental arithmetic.
Edit; thanks everyone for your useful responses!!
In: Mathematics
Sooo, you have sets of elements like {1,2,3} or {cat, dog, mouse}. Some sets are finite (like the previous examples) but sets can also be infinite, like the set of all natural numbers {1,2,3,4,…}.
So one interesting question you can ask about a set is “how big is it?”
If it’s a finite set, it’s pretty easy to answer – just count the number of items. But what if the set is infinite?
So we come up with a new idea – let’s see if we can compare two sets to see if they are the same size. Supposed you have a stadium and a large group of people. You want to know what’s bigger – the number of people in the stadium, or the numbers of seats. One way is to count them, but a simpler way is to just make each person sit somewhere. If there are vacant seats – the stadium is larger. If there are standing people – the group is larger. If each person has exactly one seat – both numbers are the same.
We can apply this to infinite sets. Supposed you have to infinite sets A and B. If you can create a mapping from of elements from A to B so that each element of A is mapped to exactly one element of B and vice versa, then it means the sets have the same size (we call it cardinality, by the way).
It’s easy to apply this to finite sets, for example {1,2,3} has the same size as {cat, dog, mouse} using the mapping 1-dog, 2-cat, 3-mouse.
However, you can also find some interesting things, for example you can map all the natural numbers to all the natural even numbers, using the mapping x <-> 2x (e.g. 1<->2, 2<->4, 3<->6, etc.)
This means that a set that is “half” of the original set is the same size as the original! Mind blowing, huh?
So maybe all sets are the same size? Well, turns out that’s not the case. Without going into details, you can show that you can’t find *any* such mapping between the natural numbers and the real numbers. This means that the set of real numbers is necessarily *larger* than the set of natural numbers.
So we call all the sets that have the same size as the natural numbers “countable” (because we can match their elements with the natural numbers, meaning we can “count” them ad infinitum). The size of these sets is called “aleph null”. It is called that because it is the smallest infinite cardinality (aleph is the first letter of the Hebrew alphabet, and null means zero).
Latest Answers