Ebook The foundations of general arithmetic
We assume as known the whole machinery of formal logic, such as the notion of set class, relation, propositional function, correspondence, formal equivalence, counting, and the like. We shall use this machinery to investigate a kind of class called a "collection" whose elements consist of "entities" which in a given collection are either all "objects" or all "marks," and certain special types of propositional functions associated with the collection.
The reason for introducing these terms is to avoid confusing the general use of the words "class" "element" in our reasoning about collections and entities with the particular collections and entities themselves.
Loosely speaking, by "marks" we mean bare symbols which are distinguishable from one another, but which have no direct connotation. By an "object" we mean something which can be denoted by a "mark." The distinction between "mark" and "object" is somewhat vague; to state it intelligibly would be to solve one of the major problems of epistemology.
Our ultimate aim is to lay the foundations for a precise definition of an "arithmetic" analogous to the postulational definition of an abstract group. For the present, by an "arithmetic" we mean any system wherein 1.) All operations possible can be carried out in finite number of steps. 2.) Division is not always a possible operation. 3.) Unique factorization into primes is always a possible operation.
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