# Cantor Paradise, Riemann Utopia,

and

Riemann-Dirac Set Theory

○ In the history of mathematics, it often happens that we start learning a theory in a complicated way and later find out that there is an easier way of doing the same. An example is the Laplace transforms, which converted the differential equations into algebraic equations. I feel it is time we did the same with set theory.

○ Paul Dirac has done many things beautiful in mathematics, I take his delta functions and make heavy use of it. Georg Cantor uses natural numbers for developing the ZF theory, I use the real numbers as the basic elements to start with, and develop a set theory I call RD theory (Riemann-Dirac set theory). Riemann comes into the picture from the fact that Laplace transform of delta function occuring at log n is n^(-s).

○ In a nutshell, RD theory is about the subsets {f(n)} of real numbers, called numerals. Numerosity is the term used to represent the size of numerals. Here are two significant results:

1. If f(n) and g(n) are two inverse functions, then g(aleph0) is the numerosity of {f(n)}. For example, numerosity (size) of {log(n)} is 2^aleph0. Note that according to ZF theory, the cardinalty (size) of {log(n)} is aleph0. This is a substantial difference between the two theories.

2. Statement of prime number theorem in RD theory: The numerosity of the set of prime numbes is log(aleph0).

○ In Riemann-Dirac set theory, there are no verbal arguments, and all derivations are algebraic.