system
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human
[Mathematics] What number refers to the number of possible orderings of natural numbers?

The number of unique orderings of a set of length n is n! and as such it seems logical to me that the number of unique orderings of the natural numbers is Aleph null factorial, but what does it mean to take the factorial of an infinitely large number as infinity minus one is still infinity(i think). I was also thinking about the Continuum hypothesis which says that the cardinality of the set of real numbers is 2 raised to the Aleph null. x! grows faster than 2^x so would that make the number of unique orderings of the natural numbers to be larger than the set of real numbers(assuming that we are able to take the factorial of Aleph null)? Apologies if my question makes assumptions which are incorrect, but thanks for reading!
gpt
A rather simple way to encode numbers between 0 and 1 to all possible orderings of natural numbers would be to create decimal number between 0 and 1, and type each integer in order, in base-2, then use 5 as separator. So if your ordering starts like 1, 5, 16, 4, 2, ... Then we'd have our encoding of

    0.1510151000051005105...

Or to replace 5 with clearer separator

    0. 1 | 101 | 10000 | 100 | 10 | ...

It's quite easy to see that if n'th digit of two orderings differs, then the encoding differs, so it's an injection from all orderings to the interval ]0, 1[, and as such, there are at most as many orderings as there are real numbers.