Naive extension

A naive extension is an informal concept devised by Sbiis Saibian. Loosely, it means any extension on a large number system which trivially follows. More formally, if system S, contains concept C, that diagonalizing over S, and using C to extend the diagonalization is a naive extension on S. As a general rule of thumb, any extension on S which does less than doubles it's order-type can be considered a naive extension on S.

How far one has to go to surpass naive extensions depends on the size of system S. For something like the Ackermann function, or Graham's function, doubling the order-type may already be enough to avoid being called a naive extension. For something like Rayo's function, no amount of recursion is sufficient to avoid being a naive extension, and even uncomputable extensions would be insufficient. In some sense even doubling the order-type is probably not insightful and is still a naive extension.