1 Axiomatic Systems 1.1 Features of Axiomatic Systems One motivation for developing axiomatic systems is to determine precisely which prop erties of certain ob jects can b e deduced from which other prop erties. The goal is to cho ose a certain fundamental set of prop erties the axioms from which the other prop erties of the ob jects can b e deduced e.g., as theorems . Apart from the prop erties given in the axioms, the ob jects are regarded as unde ned. As a p owerful consequence, once you haveshown that any particular collection of ob jects satis es the axioms however unintuitive or at variance with your preconceived notions these objects may be, without any additional e ort you may immediately conclude that all the theorems must also b e true for these ob jects. Wewanttocho ose our axioms wisely. Wedonotwant them to lead to contradictions; i.e., wewant the axioms to b e consistent. We also strive for economy and wanttoavoid redundancy|not assuming any axiom that can b e proved from the others; i.e., wewantthe axiomatic system to b e independent.Finally,wemaywishtoinsistthatwebeabletoprove or disproveany statement ab out our ob jects from the axioms alone. If this is the case, we say that the axiomatic system is complete. We can verify that an axiomatic system is consistentby nding a model for the axioms|a choice of ob jects that satisfy the axioms. We can verify that a sp eci ed axiom is indep endent of the others by ndingtwo mo dels|one for which all of the axioms hold, and another for which the sp eci ed axiom is false but the other axioms are true. We can verify that an axiomatic system is complete by showing that there is essentially only one mo del for it all mo dels are isomorphic ; i.e., that the system is categorical. Reference: Kay, Section 2.2. 1 1.2 Examples Let's lo ok at three examples of axiomatic systems for a collection of committees selected from a set of p eople. In each case, determine whether the axiomatic system is consistentor inconsistent. If it is consistent, determine whether the system is indep endent or redundant, complete or incomplete. 1. a There is a nite numb er of p eople. b Each committee consists of exactly two p eople. c Exactly one p erson is on an o dd numb er of committees. 2 2. a There is a nite numb er of p eople. b Each committee consists of exactly two p eople. c No p erson serves on more than two committees. d The numb er of p eople who serve on exactly one committeeiseven. 3 3. a Each committee consists exactly two p eople. b There are exactly six committees. c Each p erson serves on exactly three committees. 4 1.3 Kirkman's Scho olgirl Problem Consider the following axiomatic system for p oints and lines, where lines are certain subsets of p oints, but otherwise p oints and lines are unde ned. Given anytwo distinct p oints, there is exactly one line containing b oth of them. Given anytwo distinct lines, they intersect in a single p oint. There exist four p oints, no three of which are contained in a common line. The total numb er of p oints is nite. 1. Try to come up with some mo dels for this axiomatic system. 2. Find a mo del containing exactly 15 p oints. 3. Solve the following famous puzzle prop osed byT.P. Kirkman in 1847: A school-mistress is in the habit of taking her girls for a daily walk. The girls are fteen in number, and are arrangedin verows of threeeach, so that each girl might have two companions. The problem is to dispose them so that for seven consecutive days no girl wil l walk with any of her school-fel lows in any triplet more than once. 5 1.4 Finite Pro jective Planes The axioms describ ed in the previous section de ne structures called nite projective planes.I have a game called Con gurations that is designed to intro duce the players to the existence, construction, and prop erties of nite pro jective planes. When I checked in August 1997 the game was available from WFF 'N PROOF Learning Games Asso ciates, 1490 South Boulevard, Ann Arb or, MI 48104, phone: 313 665-2269, fax: 313 764-9339, for a cost of $12.50. Here are examples of some problems from this game: 1. In eachbox b elowwriteanumb er from 1 to 7, sub ject to the two rules: 1 The three numb ers in each column must b e di erent; 2 the same pair of numb ers must not occurintwo di erent columns. Col 1 Col 2 Col 3 Col 4 Col 5 Col 6 Col 7 Row1 Row2 Row3 6 2. Use the solution to the ab ove problem to lab el the seven p oints of the following diagram with the numb ers 1 through 7 so that the columns of the ab ove problem corresp ond to the triples of p oints in the diagram b elow that lie on a common line or circle. This is called the Fano plane. 7.