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## Numbers

In this section of the website we're going to present the axioms for several sets of numbers, and we're going to present models for those axiom systems.  Construction of a model which satisfies a system of axioms provides a (limited) proof that the axioms are consistent.  The proof is rather limited, however, because construction of the model requires implicit use of another axiom system; that's typically the axiom system used in set theory.  Consequently, the "proof" actually assumes the consistency of the axioms of set theory.

The reason for putting this here is really just amusement -- I find the constructs given here entertaining, and working out the details is fun.  However, despite the fact that I've put this in the "basics" section of the website, it is really not "basic" material at all, and in fact assumes familiarity with concepts which are typically not introduced until rather late in a typical undergraduate math curriculum.

Finally I should mention that there may very well be errors in the axiom systems or models presented here.  Since moving, I haven't been able to locate the reference texts which have this general derivation in them (stuck in a box somewhere), so I haven't been able to double-check what I wrote here against "authoritative" sources.  I looked up the Peano axioms on line before putting them on the web page, but for the rest of the axioms I "winged it".   Caveat lector!

 Notation Brief discussion of the notation used on these pages.  For the most part, it's reasonably standard notation as used in mathematical logic. Not written yet. Natural numbers In which the game begins.  The Peano axioms are presented, and a model is constructed which is shown to satisfy the Peano axioms.  Addition, multiplication, and a comparison relation are built using the basic axioms. Integers The natural numbers are extended to the positive and negative integers.  An axiom system for the integers is presented and a model is constructed using the natural numbers as a basis. Rational numbers A set of axioms for the rational numbers is presented, with some explanatory notes as to why we need so many.  And then a model for the rationals is constructed; the model is based on the integer model we developed earlier. Real numbers A modification to the axioms for the rationals is presented, which turns them into axioms for the real numbers.  We then go on to construct a model for the real numbers. Cardinalityof theReal Numbers We present a brief discussion of the uncountability of the real numbers, and point out some of the difficulties which result.  We discuss the countable set of available names for individual numbers.  The question of what you might hit if you threw a very sharp dart at the number line is considered.

Page created on 11/18/2007