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
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.
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