Exercises on the Axiomatic Method [Hvidsten Chapter 1]

Last revised 29may13 \begin{document} \maketitle \section{Introduction} We here collect exercises appropriate for the introduction to axiomatic geometry based on Hvidsten's first chapter, but expanded by lesssons in this section of the course. All exercises on this page should be attempted by the student, and the completed exercises entered into the Journal. A selection of these problems will be collected for correction and advice. Some will be tested on exams or the final. The union, intersection, and complements of these subsets are non-empty. This document is updated frequently and should be consulted periodically for additions. It also contains comments and advice on doing the problems. Very Important Advice: You are directed to justify your solution to the problems on finite geometries (aka toy geometries) using models rather than attempting to construct proofs directly from the Axioms. Few beginners can do this successfully, and failed attempts are too difficult for your grader to correct. Intead, recall from the lessons that a categorical axiom system (one which has a unique model, up to an isomorphism) has the additional convenient property that theorems in the axiom system can be "read-off" the model. Thus, in solving the problems below you should first construct a model (don't forget to justify your discovery) and then you may assume that the axiom system modelled is categorical. The reason for this deviation from the Hvidsten Text is this. To deduce all of the theorem of Geometry (especially Non-Euclidean Geometry) from a set of axioms is beyhond the scope of this course. Instead, we follow the analytic method of using models, after accepting the fact that the Geometries we study are categorical. \subsection{Three Point Geometry} \textbf{ Exercises 1.4.3 through 1.4.5 } pertain to the finite incidence geometry consisting of exactly 3 points. This subject is treated in the lesson on \textit{ Toy Geometries }. You should translate the axioms by replacing "children", "likes", "flavors" with "points", "lines" and "incidence" and prove the problems using a model for the axioms instead of attempting a discursive argument. \subsection{Peano's Axioms for Arithmetic} Ever since Euclid presented the first set of axioms for geometry, mathematicians wondered about an axiomatic treatment of arithmetic. For a long time it was thought that the rules of arithmetic are so obvious and uncontroversial that they are simply part of the human mind. When a similar philosophical opion regarding geometry was proven false in the 19th century, axioms for arithmetic became serious business. Guiseppe Peano gave these \begin{itemize} \item \textit{Primitives:} Numbers $\mathbb{N}$, Successor \item \texttt{Peano 1:} $1 \in \mathbb{N}$ \item \texttt{Peano 2:} $(\forall n \in \mathbb{N})(\exists n' \in \mathbb{N})( n' \mbox { is the unique successor to } n ) $ \item \texttt{Peano 3:} $(\forall n, m \in \mathbb{N})(n \ne m \implies n' \ne m')$ \item \texttt{Peano 4:} $(\forall n \in \mathbb{N})(1 \ne n')$ \item \texttt{Peano 5:} If $ S \subset \mathbb{N} $ for which \begin{enumerate} \item $1 \in S$ \item $(\forall n \in \mathbb{N})(n \in S \implies n' \in S )$ \end{enumerate} then $ S = \mathbb{N} $. \end{itemize} As with Euclid, the interesting axiom is the Fifth Peano Axiom, better known as the \textit{ Principle of Finite Induction.} Hvidsten's \textbf{ exercises 1.4.3 through 1.4.12} treat Peano's axioms. \subsection{Exercises 1.5.1 - 1.5.3} Hvidsten's exercise sections contain questions best answered by a thoughtful essay. These are appropriate for students taking the course for the fourth credit hour. \subsection{Four Point Geometry} \textbf{Exercises 1.5.4 through 1.5.7} pertain to the finite incidence geometry consisting of exactly 4 points. This subject is treated in the lesson on \textit{ Toy Geometries }. You should do these problems using a model instead of attempting to compose as discursive argument. \subsection{Geometric Arithmetic} \textbf{Exercise 1.6.3} treats geometric arithmetic. This and similar examples easily located on the WWW are particularly important for current or future teachers of school math. \end{document}