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Numbers |
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. |
Cardinality of the Real 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. |