Gödel's incompleteness theorems

Blog » For every mathematical statement S, exactly one of the following two claims is true:


1. S can be proven
2. S cannot be proven


Every statement S has a negation which is also a statement:


~S


So, furthermore, for every statement S exactly one of the following two claims is also true:


A. ~S can be proven
B. ~S cannot be proven


(To put it another way, we might also say:


A. S can be disproven, or
B. S cannot be disproven)


Putting these together, for every statement S exactly one of the following four claims is true:


1A. S can be proven and ~S can be proven
1B. S can be proven and ~S cannot be proven
2A. S cannot be proven and ~S can be proven
2B. S cannot be proven and ~S cannot be proven


Case 1B uncontroversially indicates that S is "true" and case 2A uncontroversially indicates that S is "false". These cases are relatively straightforward and can be put aside.

If case 1A holds, both S and ~S can be proven. Thanks to something called the principle of explosion...