The Great Mysticism in Logic and Mathematics

Several great minds in history have amused themselves with some glorious, seemingly entirely different problems of mysticism.

Interestingly, these multiple thoughts, issues and paradoxes, boil down to a single phenomenon. Examples are several - starting with Greg Cantor’s diagonal argument, Alan Turing’s halting machine, Kurt Godel’s incompleteness theorems and a plethora of logical paradoxes. The amusement continues to some extent in the works of Einstein and in the recent excitement around quantum computing as well.

Such tendencies seem less frequent in the earlier times with Euler, Riemann, Newton, Euclid etc focusing on more practical problems.

The core phenomenon

These thoughts from Godel, Turing, Russel and Cantor are the multiple reflections of the same phenomenon. It is not a problem or issue. The core phenomenon is about partitioning a set into two. What does it actually mean?

Partitioning a set into two

The simple rules for ensuring consistency of the process:

1. Initial state is one set containing one or more different elements in it.
2. End state is two sets whose union is the first set and whose intersection is an empty set.
3. There is no intermediate state where the un-partitioned initial set exists along with the resultant sets. The states before and after partition are mutually exclusive even if one of the resultant sets is empty.
4. The partition creates a boundary between the two sets. The boundary is not a set.
5. All the sets (source and result) and the boundaries exist in the same logical space

Now let’s use these rules to throw some light on our mystique problems.

Problem 1: This statement is false.

This violates the consistency rule #2, because it has a non-empty intersection.

Problem 2: A program can’t determine if a program can halt.

This is isomorphic to the problem #1 and so violates the rule #2.

Problem 3: A theorem can’t be proved using an axiom, if the theorem negates the axiom.

This violates the rule #3, because it considers the states of divided and undivided to exist at the same time.

Problem 4: A set of sets which do not contain themselves as a subset, can’t exist.

This is isomorphic to problem #3 and violates the rule #3

Problem 5: The diagonal argument shows that there are multiple infinities.

This is inconsistent because it violates the rule #4 due to considering the boundary between the sets as a set itself.

Problem 6: P = NP?

This issue is due to partitioning a set into two sets which belong to different spaces that observe different laws. This violates the rule #5.