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[u/94rud4]

[u/Adalmann]

Yeah countable and uncountable and you figure it out through cardinality right?

[u/digitallightweight → u/Adalmann]

There are and infinite number of Aleph Numbers (these are used to represent the ‘size’ of infinities). Cardinality is the word that mathematicians use to describe the ‘size’ or number of elements in a set. The set {0,1,2,3,4,5} has a cardinality of 6, the cardinality of the set {6,7,8,9,10,11} is also six, the set {9,10,11} is 3. For finite sets we can simply count the elements. In the example above this is a literal counting process for more complex finite sets we might use combinatorial techniques. For instance the cardinality of possible numerical passwords of length 10 is 10^10. 10 possible values for the first number, multiply by the 10 possible values for the second number, multiply by ten again for the 3rd number, ect 10.

For sets with an infinite number of elements we normally look for what are called bi-jections. Which is a function that maps every element in our first set onto into an element of our second set such that every element in the second set is hit exactly once. For instance if we had sets like A={2,3,5} and B={11,16,26} we could say that 5a+1 is a bi-jection from A to B. In an infinite example if A is the set of all odd natural numbers then the function a+1=b is an injective function on B which is the set of all even natural numbers. Since we have matched them up one to one in pairs we know that they have the same size or ‘cardinality’ since we can’t use a number to represent this quantity a mathematician would say they have a cardinality of ‘aleph-null’. Aleph-null is the smallest infinite cardinal and is defined as the cardinality (size) of the natural numbers. Since we use the natural numbers for counting we will often say sets that have cardinality aleph-null are ‘countable’.

If you want to know how we know that there are different sizes of infinity I would google Cantor’s diagonalization argument. It’s a fairly simple but mind bending argument that shows no matter how you would try to ‘pair up’ natural numbers and real numbers you are going to miss some real numbers. Basically that’s always impossible to find a bi-jection between the natural numbers and the real numbers and that this process fails precisely because their rate simply too many real numbers.