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

It’s the smallest infinite cardinal, by definition.

Cardinal numbers measure “how many things” are in a set (broadly speaking), aleph null describes the size of any set whose elements can be assigned a positive whole number in a 1-to-1 manner. Such sets are also called “countably infinite”.

Take the set of even numbers for example

* 1 -> 2

* 2 -> 4

* 3 -> 6

* and so on and so forth.

Every even number can be assigned to exactly one positive whole number, so we say the cardinality of the set of even numbers is aleph-null.

It’s the cardinality of the natural numbers, so if a set has cardinality “aleph naught”, then it is “countably infinite”. Some infinite sets are “bigger” than this, and that’s when a cardinality like “aleph one” might come into play.

This stuff can be confusing, so don’t give yourself a hard time if you don’t understand it!

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).

It is the cardinality of the set of natural numbers. Also known as the garden variety “infinite”. There are higher aleph numbers and “more infinite” infinities – for example, the set of real numbers have an aleph number higher than 0. In addition to being infinite the “normal way” by always being able to name a larger number than the previous, there are also infinite real numbers *between* any two real numbers, not matter how close they are.

One of the more mind-bending ideas in mathematics is this: infinity comes in more than one flavor, and some infinities are larger than others.

For an example of the simplest, most common type of infinity, think about the natural numbers. These are the numbers 1, 2, 3, and so on. If you start at the beginning and count each number in order, you will never reach the end. This collection of numbers is infinitely long.

For a more complex example, think about the real numbers. These are the kind of numbers you use in a calculator – things like 1.0, -2.3, 450,000.15, and π. Like the natural numbers, this collection is infinitely long – you can count from 1.0 to 2.0 to 3.0 and so on, and never reach the end. However, when I counted from 1.0 to 2.0, I skipped some numbers. I skipped 1.1. If I want to count _all_ of the real numbers, _in order_, I have to include 1.1. So I could count like this: 1.0, 1.1, 1.2, and so on, forever … but doing that I’m still skipping numbers. I skipped 1.01. So I could count 1.01, 1.02, 1.03, and so on … but doing that, I’m _still_ skipping numbers, such as 1.001.

So, these two examples clearly have some really extreme differences. With the natural numbers, I can count forever and never reach the end, but with the real numbers, I can’t even make any _progress_ – I can count forever and never even get from 1.0 to 2.0. In fact, it’s not even possible to decide what number comes next after 1.0. The list of natural numbers is infinitely long, but the list of real numbers is both infinitely long _and_ infinitely _deep_. So these two groups are both infinite in size, but clearly one is larger than the other.

The term “aleph-null” refers to the smallest kind of infinity – anything that is the same “size” as the natural numbers. This is also called a “countable infinity”, because you can count one at a time in order and make measurable progress.

The term “aleph-one” refers to the second kind of infinity – anything that is the same “size” as the real numbers. This is also called an “uncountable infinity”, because you can’t put it into order and count through it one at a time.

And as these terms suggest, there are kinds of infinity even larger than these, called aleph-two, aleph-three, and so on.

aleph null is the ‘amount’ of integers.

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If you look at the whole positive numbers, like 1,2,3,4,5… (continuing with no end), you’d look at it and you could say:

“That’s infinity numbers.”

and that is correct, but it is perhaps a bit vague. Might there be different kinds of infinity? If so, maybe we can have more details.

The extra bit of detail we can say is “That is a aleph_0 (aleph null) numbers.” or “Those numbers are countably infinite.”

By “countably infinite”, we mean that you can come up with a system for listing/counting them, such that your system won’t miss any numbers. For the whole positive numbers, that’s easy, just go up the list.

Some other infinite lists are also ‘countable’ in this way, like all the negative numbers, we can just count dowards. Or all the prime numbers, we cna say “the first prime number, then the 2nd prime number, then the 3rd…” and we won’t miss any.

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Now at this point, you might wonder if it is possible for an infinite set to *not* be “countably infinite”.

Well, consider all the numbers between 0-1. This includes all the ‘proper’ fractions (like half, a third, 4 ninths, etc etc), but also other numbers like the square root of all those fractions, some small enough fractions of special numbers like like pi/4, and so on.

And we can always find another number in there, by taking 2 of them, and averaging them: if both numbers were above 0 and below 1, then their average will easily be in that range too, so ‘the average of 1/3 and sqrt(pi/4)’, and then the average of that number and e/9, etc etc, endlessly as far as we like, finding more and more numbers. And there are even more numbers than the infinity numbers we might generate with that method (I just mentioned it to show one way in which they are infinite).

Notably, the infinity of numbers between 0-1 cannot be ‘counted’ or ‘listed’. No matter what system you try to cook up to list them, you’ll always miss some. (You’ll acually always miss an infinite number of them.)

Many people think of the size of this kind of infinity as aleph_1 (although that can’t be proven).

First, ask yourself, what is means “a number of something”. In the most naive way — you point your finger to every element of some set and say a natural number. So if you do that to every element exactly once and you counted to, say, five, you know there are five elements.

There are sets, that when you try to count, you will never stop counting. These are infinite sets. Like a set of all numbers.

But let us think more deeply about this concept of number of elements in a set — so if you count something, what you do is you create a function, a relationship between two sets. If two sets can be connected in a way that every element from one set is assigned one and only one element from the other and vice versa (this is called a bijection), then we say, that these sets have the same number of elements. A set that can be joined with this kind of relationship with a set of natural numbers from 1 to 5 is said to have five elements.

Along comes George Cantor, that tries to do this more formal approach with infinite sets. Cantor noted, that while we can create bijection between some infinite sets (e.g. you can show that there is the same number of natural numbers as there is of even natural numbers just by using the function that multiplies by two), you cannot create a bijection between natural numbers and real numbers. So he proved, that the number of elements in these two sets is different. He called the number of elements in the set of natural number aleph_0.

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