Infinite Sets and the Continuum

Once an inquiry into the nature of the natural numbers has been made, one can only notice that this is the beginning of a much longer journey than could have been expected. Once an adequate theory for the natural numbers has been adopted, it then falls on the mathematician to find a satisfactory construction of the order and algebraic operations of the continuum of real numbers.

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 We propose not only a canonical construction of the natural numbers, but also a canonical construction of real numbers and an enveloping set theory to provide optimal constructions of all mathematical objects. A real number is defined as an infinite set of natural numbers. The structure of the continuum of real numbers is defined in terms of set operations and the lowest upper bound function in a construction that is a logical extension of the construction of natural numbers.

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These constructions can be generalized to a canonical construction of all types of objects.