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What is the point of things like the zero factorial, aleph null, etc.?

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"0 factorial"? You means 0! ? 0!=1 is useful so that you don't have to write an exception for 0 every single time. There are a lot of formula involve n! for n>=1, but for the exceptional case n=0 the correct value is 1, so you define 0!=1 so that you don't have to write the exception each time. The hidden reason why that pattern keep coming up is because n! come from you needing to multiply by a bunch of factors, starting from 1, so when you are not multiplying by anything it's the same as multiplying by 1.

Or maybe you mean 0^# ? That's read as zero sharp.

We need different "infinity" because infinite sets have different cardinality. You need to specify how big a set is, and even among infinite set they don't have the same size. Aleph_0 is just the cardinality of natural number, the smallest cardinality. The Aleph sequence is for naming these cardinality. In practice, only Aleph_0 and c=Beth_1 is used, and maybe Aleph_1 . Other cardinality is too abstract so you pretty much only encounter them in context of set theory. We have no ideas what Aleph_1 set look like, and it's actually impossible to decide with our current standard axioms; but it does not matter.
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Combinatorially, *n*! represents the number of rearrangements (permutations) of a set with *n* elements. When *n* = 0, we have the empty set on our hands and since there is nothing to rearrange there, the number of permutations on the empty set is 1. Therefore 0! = 1.

Furthermore, it turns out that having 0! defined to be 1 is very useful for many other things, like having a nice general expression for the Taylor series.

As for the infinities, it's not that we *need* them. The existence of multiple infinities is a logical consequence of allowing ourselves to talk about infinite objects. We have a choice: we either do not have any infinite objects, or if we do have infinite objects, then we have to have infinite objects of different sizes.

A large majority of people agrees that being able to talk about infinite objects (such as the set of integers or the set of real numbers) is a very useful hing. There are people who disagree, which is why we have fields like finitistic and ultrafinitistic mathematics.

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