Prove that every set is countable.

A set is a collection of distinct objects . A set is count able if there exists a one - to - one correspond ence between the set and the natural numbers . This means that every element in the set can be pa ired with a unique natural number , and vice vers a . \n \n One way to prove that every set is count able is to use Cant or ' s diagonal argument . Cant or ' s diagonal argument is a proof technique that can be used to show that certain sets are count able . It works by creating a new diagonal sequence that is different from the previous one , and then pair ing each element in the new diagonal sequence with a unique natural number . This creates a one - to - one correspond ence between the set and the natural numbers , prov ing that the set is count able . \n \n Here is an example of how Cant or ' s diagonal argument can be used to prove that every set is count able : \n \n Let S be a set . We will show that S is count able by using Cant or ' s diagonal argument . \n \n Step 1 : Create a new diagonal sequence D that is different from the previous one . \n \n D 1 = { 1 , 2 , 3 , ... } \n D 2 = { 2 , 4 , 6 , ... } \n D 3 = { 4 , 8 , 1 6 , ... } \n \n Step 2 : P air each element in the new diagonal sequence with a unique natural number . \n \n D 1 = { 1 , 2 , 3 , ... } \n D 2 = { 2 , 4 , 6 , ... , 2 } \n D 3 = { 4 , 8 , 1 6 , ... , 4 } \n \n Step 3 : Show that there exists a one - to - one correspond ence between the set S and the natural numbers . \n \n For any element x in S , there exists a unique natural number n such that x is pa ired with n in the diagonal sequence D . Con vers ely , for any natural number m , there exists an element x in S such that x is pa ired with m in the diagonal sequence D . This shows that there exists a one - to - one correspond ence between the set S and the natural numbers . \n \n There fore , we have shown that every set is count able using Cant or ' s diagonal argument . </s>