A formalisation of the arithmetic of transfinite ordinals in a multisuccessor Equation calculus.
thesisposted on 19.11.2015, 08:55 by H. P. Williams
This thesis presents a syntactic development of the arithmetic of ordinal numbers less than This is done by means of an Equation calculus v/here.all statements are given in the form of equations. There are rules of inference for deriving; one equation from another. Certain functions, including a countably infinite number of successor functions are taken as primitive. New functions are defined by substitution and primitive recursion starting with the primitive functions. Such definitions constitute some of the axioms of the system. The only other axioms are two rules concerning the combination of successor functions, Fundamental for this development is the axiom. In this system a multisuccessor arithmetic is developed in which it is possible to prove many of the familiar results concerning trans-finite ordinal numbers. In particular the associativity of addition and multiplication as well as multiplication being left distributive with respect to addition are proved. It is shown that each ordinal in the system can be represented in Cantor's Normal Form. An ordinal subtraction is defined and a number of results involving this are proved. It is shown that this subtraction is, in a number of respects, an inverse to addition. In particular the key-equation is proved. As in Professor Goodstein's formalisation of the primitive recursive arithmetic of the natural numbers this equation is important as it allows a difference function to be defined for which a zero value is equivalent to equality of the arguments. Inequality relations are defined and some results concerning them proved. In Chapter II it is shown, using a suitable coding, that this arithmetic can be reduced to the primitive recursive arithmetic of the natural numbers. Chapter III gives a meta-proof of the consistency of the system. Also submitted with this thesis is a paper The Synthesis of Logical Nets consisting of NOR units which is the result of work on a logical problem which was done at the same time as work for the thesis.