A recursive universal function for the primitive recursive functions is constructed. It is based on the characterization of the primitive recursive functions given by Meyer and Ritchie, namely that they are the functions computable by programs in a certain language called LOOP. A by-product is a recursive function that grows faster than any primitive recursive, function. This is a desirable alternative to the so called Ackermann function because the construction is easier to motivate.
It is shown that for a large class of structures, monic
endomorphisms are induced by automorphisms of a larger structure
of the same kind. Many garden variety structures are
encompassed by this result e.g. groups, rings, vector spaces,
fields, metric spaces.
Ackermann's function is the classical example of a total computable
function which is not primitive recursive. By permuting
one clause in the definition of Ackermann's function we arrive
at a different function which also has this property, but which
in a sense made precise in the paper is the ``right'' way to
execute Ackermann's basic strategy.
A proof of the existence of a recursive universal function for the class of primitive recursive functions is given. This was first proved by Rosa Peter in 1934. In our proof we view the the problem as one of writing an interpreter for a language whose programs compute primitive recursive functions within a language whose programs compute partial recursive functions. Following an idea in the textbook of Davis and Weyuker, we write this interpreter program ``in macro". We devised a Godel numbering to make things come out simple. The upshot is that our interpreter (in macro) has only nine lines .
A discussion of the origin of Tallahassee's community information system, its mission, and progress.
This paper deals mainly with generalizations of results in finite cominatory mathematics to transfinite ordinals. Among other things a formula is derived for summing all the ordinals less than a given ordinal.