What is Superformidability in Mathematics?

Superformidable is a term that was popularized by the mathematician and polymath John Horton Conway. It is a playful way of referring to a certain type of mathematical object, which is a generalization of a formal system.

In mathematics, a formal system is a set of rules for constructing and manipulating mathematical expressions. For example, a formal system might include a set of axioms (propositions that are assumed to be true without proof), a set of inference rules (which allow us to derive new propositions from given ones), and a set of symbols (such as 0, 1, and +) that we can use to build expressions.

A superformidable is a formal system that has the property that every statement that can be made within the system can be proven either true or false using only the rules of the system. In other words, if a statement cannot be proved either true or false using the rules of the system, then it is not superformidable.

Superformidability is a strong condition that not all formal systems satisfy. For example, the standard system of arithmetic (which includes the natural numbers and the usual operations of addition and multiplication) is not superformidable, because there are statements about the natural numbers that cannot be proved either true or false using only the rules of the system.

John Horton Conway was interested in superformidability because he believed that it might provide a way to understand the nature of mathematics itself. He thought that if we could find a superformidable formal system, we might be able to use it to prove the consistency of all mathematical truths, and thereby gain a deeper understanding of the foundations of mathematics. However, despite much effort, no one has yet been able to find a superformidable formal system that is powerful enough to prove all mathematical truths.