Cardinality in mathematics
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Cardinality is just a way to say that two sets have the same number of things.
Now, it is tempting to say that both of the sets {a,b,c} and {x,y,x} have 3 elements and, therefore, they have the same cardinality but this requires “3” to be a middle-man in this process. So we skip the middle man and determine equal cardinality by just making associations. {a,b,c} and {x,y,z} have the same cardinality because I can make the associations:
* a <-> x
* b <-> y
* c <-> z
And with this association 1.) Every element of the set {a,b,c} is used exactly once and 2.) Every element of the set {x,y,z} is used exactly once. So I can say that {a,b,c} and {x,y,z} both have the same cardinality without “3” having to even be a thing! In fact, we *define* the cardinal “3” as “Things with the same cardinality as {a,b,c}”.
This “not having to have numbers to compare size” thing is important because in math you frequently run out of numbers with which to count! If I only have numbers to compare sizes with, then how do I compare the size of the set {0,1,2,3,4,…} with the set {…-2,-1,0,1,2,…}? There’s not enough numbers for this, but I can still say that they have the same cardinality through the association
* 0 <-> 0
* 1 <-> 1
* 2 <-> -1
* 3 <-> 2
* …
This gets even more important when we find things *bigger* than the simple infinity of {0,1,2,3,…}. If I am restricted to using numbers, then I can’t really say that the set {0,1,2,3,…} is *smaller* than the points on the interval (0,1) but, through a clever argument, I can show that any association between these two sets will always fall short of including everything in the (0,1) interval and, therefore, (0,1) is bigger than {0,1,2,…}. Cardinality as associations is very powerful and flexible.
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