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
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).
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