This statement is unprovable.

Godel is best known for the Incompleteness Theorem which reshaped our view of mathematics. This is part of the answer as to whether mathematics is a science of abstraction or an art. Godel showed that much of mathematics is beyond our understanding and that we are discovering mathematics.

Axioms are the key to deduction. When you get a set of axioms, you can deduce the solution to any problem within the system. It is assumed that we can create axioms on our own as we do in art. What if these axioms are discovered but are incomplete?

Set theory opened a door to evaluate mathematics in terms of logic. Sets are a group of numbers or objects with properties. We can compare or manipulate them in relation to other sets. This is a set of apples {A}; this is oranges {O}. {A}does not equal {O}. There is also the empty set {}, which is, well, empty. It’s related to the null set – which can be empty or irrelevant.

Bertrand Russell introduced a Paradox showingÂ that naive set theory is internally inconsistent or incomplete.

The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox.

There is a set of apples. This set is normal because it is not a member of itself (the set is not an apple, it’s like a basket that holds apples). There is a set of all non-apples. This is an abnormal set because it is a member of itself (the set of all non-apples is not an apple, so it is a set of itself).

There is a null set {}. Russell defined the null set and set {A} which contains *everything* not in {}. A is a universal set.

Russell defined two subsets R and R’. {A} is an element of R if and only if {A} is not a member of itself. R’ (abnormal) includes sets that are members of themselves, which is {}. The empty set belongs to R’. The everything set belongs to R. Everything should be accounted for.

So is R normal or abnormal? Is R’ normal or abnormal? If R’ is not a member of R, then it must be a member of R. If R is R, then it is R’. If R is not a member of itself, then it is a member of itself. If R is a member of itself, then it is not a member of itself.

Russell defined this so that between R and R’ everything would be included. There is a contradiction or something is missing in everything.

Set theory and logic are incomplete. It cannot define everything without leaving out something.

There were early attempts to solve this paradox. David Hilbert and other great mathematicians like John von Neumann tried to reform the axioms of mathematics to solve Russell’s Paradox. If there was a problem with the original axioms, they could invent better axioms. If Set Theory remained incomplete, it meant that mathematics was incomplete.

It turns out a good deal of Mathematics *is* incomplete. Godel discovered that no formal system that formulates arithmetic (or is isomorphic to arithmetic) can be both consistent and complete. He initially proved that number theory is consistent but incomplete. If it is inconsistent, then no theorem is consistent so you can completely prove anything you want. Inconsistent systems are useless however. If the system is consistent then it is incomplete. the theorems are valid but cannot prove everything: This statement is false. It is true and unprovable.

The Incompleteness Theorem forces us to reconsider the philosophy and origins of mathematics. We probably discovered mathematics empirically rather than inventing axioms. Instead we find out that the rich mathematical system using natural numbers is complete in such a way. Mathematicians then discover axioms to explain that incompleteness and make the mathematically system more complete – but then they discover more incompleteness. We discover axioms, as in science. Godel essentially argued that there is an independent mathematical reality that we discover and learn about – just like in physics.

I believe very simple systems are complete and consistent but are powerless compared to richer mathematical systems that Godel was talking about.

It has pragmatic uses too. We now know there are limits to our mathematical systems and there are no ways to invent better axioms. Alan Turing discovered a limit in computer science called the halting problem. This is a decision problem that encounters Godel’s Incompleteness Theorem. Given a program system and finite imputes, a computer cannot solve certain problems in finite time. It will continue calculating infinitely. He invented the “Turing Machine”, an idealistic computer that can test for the halting problem.

## Leave a Reply