The Fundamental Theorem of Arithmetic without the Axiom of Infinity

Robert S. Wilson

Sonoma State University

steve.wilson@sonoma.edu

 

In 1931 Kurt Gödel wrote an important paper in which he showed that an axiomatic system which contained the axiom of infinity could not be proved to be consistent or complete. The trouble does not exist, however, for finite systems. As a result, one might be tempted to confine their research interests to solely finite phenomena, but there are difficulties even there.

For instance, if one is going to study, say, finite abelian groups, the fundamental theorem of arithmetic plays an important part in the development of that theory, and the fundamental theorem of arithmetic is a theorem about the natural numbers, which, as currently formulated, depend upon the axiom of infinity for their existence. Is it possible to prove the fundamental theorem of arithmetic without the axiom of infinity?

Of course it is. The fundamental theorem of arithmetic goes back thousands of years. The axiom of infinity, which states that the collection of all natural numbers forms a set, goes back only to the development of set theory which is much more recent.

The most important thing that is lost when you throw out the axiom of infinity is the closure property of natural numbers under addition and multiplication. Any set of non zero natural numbers that is closed under addition must be infinite. However, any computation that can be practically performed will require only a finite number of operations and a finite number of numbers.

All that is required is a slight shift in philosophy. Instead of developing a set that will contain the results of any possible computation, we can deal with any practical application by starting with the application and merely developing a set of numbers that will suffice for that particular application.

In particular, the fundamental theorem of arithmetic states that given a natural number greater than one, it is possible to break it up into a product of primes in a unique manner. So we start with a natural number greater than one, and at that point everything we need could take place in the set of all natural numbers less than our given number, which is a finite set.

What is required is to start with a basic axiomatization and prove everything we need to get to the fundamental theorem of arithmetic to make sure that it can be done without the axiom of infinity.

We use just four axioms. The first three are that subsets, unions, and power sets of known sets are sets, and the fourth is that the empty set is a set. Intersections and Cartesian products can be obtained using subsets and power sets.

Without the axiom of infinity, we can depend only on sets that can be developed from the empty set in a finite number of steps using these operations. Any individual natural number can be obtained in the standard manner in a finite number of steps, and so any finite collection of natural numbers will form a set.

The more one thinks about it, less reasonable the axiom of infinity becomes. The conveniences that it provides turn out to be nice but unnecessary luxuries.

The material is organized into definitions, axioms, and theorems. They are ordered so that each one is stated before it is used.. The material is organized into eight sections. Each section has a title which usually accurately reflects the material covered in the section.

Web pages offer unique opportunities for presenting mathematical material. The definitions, axioms, and the statements of the theorems in each section are stated in a single web page. The theorem numbers are links to the proofs of the theorems. In the proofs of the theorems, if a definition, axiom, or previous theorem are invoked, there is a link to it. The links to previous theorems go to the proof of the theorem, but since the definitions and axioms do not have proofs, the links form them go to their statements in the main web page for the section in which they are found. At the bottom of the main web pages for each section is a link to the next section. The title, "Finite Mathematics", in each section is a link to the table of contents, and any place you see my name in this web site is a link to my home page. Enjoy.

Table of Contents

1. Definitions and Axioms