Understanding Axiomatic Systems: Components, Types, and Examples, Study notes of Mathematics

Download Understanding Axiomatic Systems: Components, Types, and Examples and more Study notes Mathematics in PDF only on Docsity! The Axiomatic Method Axiomatic System Axiomatic Systems have four essential components: Undefined terms Defined Terms Axioms Theorems Axiom (Postulate) Just as one cannot expect to define every term, so one cannot expect to prove every mathematical statement. To have some statements to start one has to have some statements that are assumed without proof. These statements are called axioms. Theorem The statements that are derived from the axioms, undefined terms, defined terms, and previously derived theorems by strict logical proof are called theorems. Types of Axiomatic Systems A material axiomatic system considers its undefined terms to have meanings derived from reality. A formal axiomatic system considers its undefined terms to have no meaning at all and to behave much like algebraic symbols. Vocabulary of Axiomatic Systems The negation of a statement is another statement that is true when the original statement is false and false when the original statement is true. Vocabulary of Axiomatic Systems An axiomatic system is inconsistent if it is possible to prove both a statement and its negation from the axioms. A system is consistent if it is not inconsistent. Vocabulary of Axiomatic Systems An axiom A in a consistent axiomatic system is said to be independent if the axiomatic system formed by replacing A with its negation is also consistent. An axiom is dependent if it is not independent. An axiomatic system is independent if each of its axioms is independent. Example of a Material Axiomatic System Undefined terms: squirrel, tree, climb Axioms: 1. There are exactly three squirrels. 2. Every squirrel climbs at least two trees. 3. No tree is climbed by more than two squirrels. Example of a Formal Axiomatic System Undefined terms: X,Y related to Axioms: 1. There exist at least one X and one Y. 2. If a and b are distinct Xs then exactly one Y is related to both of them. 3. If c and d are distinct Ys then exactly one X is related to both of them. 4. At least three Xs are related to any Y. 5. Not all Xs are related to the same Y. Example of a Material Axiomatic System Undefined terms: brass ring, wire related to Axioms: 1. There exist at least one brass ring and one wire. 2. If a and b are distinct brass rings then exactly one wire is related to both of them. 3. If c and d are distinct wires then exactly one brass ring is related to both of them. 4. At least three brass rings are related to any wire. 5. Not all brass rings are related to the same wire. Example: Using a triangle to prove the squirrel-and-tree axiom system is consistent. Undefined terms: vertex, side, is on Axioms: 1. There are exactly three vertices. 2. Every vertex is on at least two sides. 3. No side is on more than two vertices. If we interpret “squirrels” as “vertices”, “trees” as “sides”, and “climb” as “on”, then it is clear that all axioms are satisfied. Examples of non-mathematical use of the axiomatic method Ethics by Spinoza in 1677 is in five parts, each consisting of a list of definitions, a set of axioms, and a number of propositions derived from them. Declaration of Independence, the U.S. Constitution (“We hold these truths to be self- evident . . .”) Examples of use of the axiomatic method Principia by Newton in 1677 which contained development of the laws of motion and universal graviation. Grundbegriffe der Wahrscheinlichkeitsrechnung by Andrei Kolmogorov set up the axiomatic basis for modern probability theory in 1933. Euclid’s Parallel Postulate If a straight line falling on two straight lines makes the sum of the interior angles on the same side less than two right angles, then the two straight lines, if extended indefinitely, meet on that side on which the angle sum is less than two right angles. Flaws in Euclid's Postulates Euclid takes existence of points for granted, never stating such existence as a postulate. In Hilbert's system these assumptions are stated in axiom I-2 and I-3. Euclid takes betweenness and line separation for granted, never stating the properties he uses in any axioms or postulates. In Hilbert's system these properties are stated in axioms II-1, II-2, II-3, and II-4. Flaws in Euclid's Postulates Euclid has a faulty proof of SAS where he assumes that certain motions are possible without stating in postulates or axioms that such motions are possible. Some modern treatments of geometry do assume motions in their axioms. However, Hilbert's does not, so in Hilbert's system SAS has to be taken as an axiom (axiom III-5). Euclid takes continuity properties for granted. For example, he assumes (without stating it as an axiom) that if two circles are sufficiently close together then they intersect in two points. Hilbert's Axioms Incidence Axioms The following axioms set out the basic incidence relations between lines, points and planes. They also characterize the concept of “dimension” that we associate with these notions. Postulate I-1. For every two points A, B there exists a line a that contains each of the points A, B. Postulate I-2. For every two points A, B there exists no more than one line that contains each of the points A, B. Hilbert's Axioms Incidence Axioms Postulate I-3. There exists at least two points on a line. There exist at least three points that do not lie on a line. Postulate I-4. For any three points A, B, C that do not lie on the same line there exists a plane α that contains each of the points A, B, C. For every plane there exists a point which it contains. Postulate I-5. For any three points A, B, C that do not lie on one and the same line there exists no more than one plane that contains each of the three points A, B, C. Hilbert's Axioms Incidence Axioms Postulate I-6. If two points A, B of a line a lie in a plane α then every point of a lies in the plane α. Postulate I-7. If two planes α, β have a point A in common then they have at least one more point B in common. Postulate I-8. There exist at least four points which do not lie in a plane. Hilbert's Axioms Order (Betweenness) Axioms Postulate II.3. Of any three points on a line there exists no more than one that lies between the other two. Postulate II.4. Let A, B, C be three points that do not lie on a line and let a be a line in the plane ABC which does not meet any of the points A, B, C. If the line a passes through a point of the segment AB, it also passes through a point of the segment AC, or through a point of the segment BC. Hilbert's Axioms Congruence Axioms Hilbert's resolution was to introduce axioms to define congruence. Instead of moving figures, he postulated the ability to construct an exact copy of a figure at any place on the plane, rotated through any angle, and oriented either way. The SAS condition is one of the axioms. Hilbert's Axioms Congruence Axioms Postulate III.1. If A, B are two points on a line a, and A' is a point on the same or on another line a' then it is always possible to find a point B' on a given side of the line a' such that AB and A'B' are congruent. Postulate III.2. If a segment A'B' and a segment A"B" are congruent to the same segment AB, then segments A'B' and A"B" are congruent to each other. Parallel Axiom The axiom in this section caused the most controversy and confusion of all. The axioms of parallels (which is also an incidence axiom) is Postulate IV.1. Let a be any line and A a point not on it. Then there is at most one line in the plane that contains a and A that passes through A and does not intersect a. Hilbert's Axioms Continuity Axioms These axioms are the axioms which give us our correspondence between the real line and a Euclidean line. These are necessary to guarantee that our Euclidean plane is complete. Hilbert's Axioms Continuity Axioms Postulate V.1. (Archimedes Axiom) If AB and CD are any segments, then there exists a number n such that n copies of CD constructed contiguously from A along the ray AB will pass beyond the point B.