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Section 2.2: Axiomatic Systems. Math 333 – Euclidean and Non-Euclidean Geometry Dr. Hamblin. What is an Axiomatic System?. An axiomatic system is a list of undefined terms together with a list of axioms. A theorem is any statement that can be proved from the axioms.

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section 2 2 axiomatic systems

Section 2.2: Axiomatic Systems

Math 333 – Euclidean and Non-Euclidean Geometry

Dr. Hamblin

what is an axiomatic system
What is an Axiomatic System?
  • An axiomatic system is a list of undefined terms together with a list of axioms.
  • A theorem is any statement that can be proved from the axioms.
example 1 committees
Example 1: Committees
  • Undefined terms: committee, member
  • Axiom 1: Each committee is a set of three members.
  • Axiom 2: Each member is on exactly two committees.
  • Axiom 3: No two members may be together on more than one committee.
  • Axiom 4: There is at least one committee.
example 2 monoid
Example 2: Monoid
  • Undefined terms: element, product
  • Axiom 1: Given two elements, x and y, the product of x and y, denoted x * y, is a uniquely defined element.
  • Axiom 2: Given elements x, y, and z, the equation x * (y * z) = (x * y) * z is always true.
  • Axiom 3: There is an element e, called the identity, such that x * e = x and e * x = x for all elements x.
example 3 silliness
Example 3: Silliness
  • Undefined terms: silly, dilly.
  • Axiom 1: Each silly is a set of exactly three dillies.
  • Axiom 2: There are exactly four dillies.
  • Axiom 3: Each dilly is contained in a silly.
  • Axiom 4: No dilly is contained in more than one silly.
models
Models
  • A model for an axiomatic system is a way to define the undefined terms so that the axioms are true.
  • A given axiomatic system can have many different models.
models of the monoid system
Models of the Monoid System
  • The elements are real numbers, and the product is multiplication of numbers.
  • The elements are 2x2 matrices of integers, and the product is the product of matrices.
  • The elements are integers, the product is addition of numbers.
  • Discussion: Can we add an axiom so that the first two examples are still models, but the third is not?
a model of committees
A Model of Committees
  • Members: Alan, Beth, Chris, Dave, Elena, Fred
  • Committees: {A, B, C}, {A, D, E}, {B, D, F}, {C, E, F}
  • We need to check each axiom to make sure this is really a model.
axiom 1 each committee is a set of three members
Axiom 1: Each committee is a set of three members.
  • Members: Alan, Beth, Chris, Dave, Elena, Fred
  • Committees: {A, B, C}, {A, D, E}, {B, D, F}, {C, E, F}
  • We can see from the list of committees that this axiom is true.
axiom 2 each member is on exactly two committees
Axiom 2: Each member is on exactly two committees.
  • Members: Alan, Beth, Chris, Dave, Elena, Fred
  • Committees: {A, B, C}, {A, D, E}, {B, D, F}, {C, E, F}
  • We need to check each member:
    • Alan: {A, B, C}, {A, D, E}
    • Beth: {A, B, C}, {B, D, F}
    • Chris: {A, B, C}, {C, E, F}
    • Dave: {A, D, E}, {B, D, F}
    • Elena: {A, D, E}, {C, E, F}
    • Fred: {B, D, F}, {C, E, F}
axiom 3 no two members may be together on more than one committee
Axiom 3: No two members may be together on more than one committee
  • Members: Alan, Beth, Chris, Dave, Elena, Fred
  • Committees: {A, B, C}, {A, D, E}, {B, D, F}, {C, E, F}
  • We need to check each pair of members. There are 15 pairs, but only a few are listed here.
    • A&B: {A, B, C}
    • A&C: {A, B, C}
    • A&F: none
    • E&F: {C, E, F}
axiom 4 there is at least one committee
Axiom 4: There is at least one committee
  • Members: Alan, Beth, Chris, Dave, Elena, Fred
  • Committees: {A, B, C}, {A, D, E}, {B, D, F}, {C, E, F}
  • This axiom is obviously true.
independence
Independence
  • An axiom is independent from the other axioms if it cannot be proven from the other axioms.
  • Independent axioms need to be included: they can’t be proved as theorems.
  • To show that an axiom is independent, find a model where it is not true, but all of the other axioms are.
the logic of independence
The Logic of Independence
  • If Axiom 1 could be proven as a theorem from Axioms 2-4, then the statement “If Axioms 2-4, then Axiom 1” would be true.
  • Consider this truth table, where P = Axioms 2-4 and Q = Axiom 1:

Finding a model where Axioms 2-4 are true and Axiom 1 is false shows that the if-then statement is false!

committees example
Committees Example
  • Members: Adam, Brian, Carla, Dana
  • Committees: {A, B}, {B, C, D}, {A, C}, {D}
  • In this model, Axioms 2-4 are true, but Axiom 1 is false.
  • This shows that Axiom 1 is independent from the other axioms.
consistency
Consistency
  • The axioms of an axiomatic system are consistent if there are no internal contradictions among them.
  • We can show that an axiomatic system is consistent simply by finding a model in which all of the axioms are true.
  • Since we found a way to make all of the axioms true, there can’t be any internal contradictions!
inconsistency
Inconsistency
  • To show that an axiomatic system is inconsistent, we need to somehow prove that there can’t be a model for a system. This is much harder!
  • There is a proof in the printed packet that the “silliness” system is inconsistent.
completeness
Completeness
  • An axiomatic system is complete if every true statement can be proven from the axioms.
  • There are many conjectures in mathematics that have not been proven. Are there statements that are true but cannot be proven?
david hilbert 1862 1943
David Hilbert (1862-1943)
  • In 1900, Hilbert posed a list of 23 unsolved mathematical problems he hoped would be solved during the next 100 years.
  • Some of these problems remain unsolved!
  • Hilbert’s Second Problem challenged mathematicians to prove that mathematics itself could be reduced to a consistent, complete set of independent axioms.
principia mathematica 1910 1913
Principia Mathematica (1910-1913)
  • Two mathematicians, Alfred North Whitehead (1861-1947) and Bertrand Russell (1872-1970) published a series of books known as the Principia Mathematica.
  • This was partially in an attempt to solve Hilbert’s Second Problem.
  • The Principia is a landmark in the 20th century drive to formalize and unify mathematics.
kurt g del 1906 1978
Kurt Gödel (1906-1978)
  • After the Principia was published, the question remained of whether the axioms presented were consistent and complete.
  • In 1931, Gödel proved his famous Incompleteness Theorems that stated that any “sufficiently complex” axiomatic system cannot be both consistent and complete.
implications for geometry
Implications for Geometry
  • As we develop our axiomatic system for geometry, we will want to have a consistent set of independent axioms.
  • We will investigate many models of our geometric system, and include new axioms over time as necessary.
  • The models we construct will show that the axioms are consistent and independent, but as Gödel proved, we cannot hope to have a complete axiomatic system.