On local arithmetical functions and their application for constructing types of peano's arithmetic

This chapter discusses how the properties of local functions can be used to get results concerning models of Peano's arithmetic. The concept of a local relation (or function) is defined. The basic property of these functions is that the recursive definition, which defines their iteration, can be formalized in Peano's arithmetic, a property that does not hold for arithmetical functions whose arguments and values are sets of natural numbers. This property is proved in the chapter. It is shown how the results of the property can be employed to get certain types of elements with respect to the theory of Peano's arithmetic. The concept was used to get results concerning measurable cardinals, the proof of which involved iterating a function whose arguments and values were proper classes. The chapter outlines some of the results concerning models and types of Peano's arithmetic, which are made possible through the use of local functions.