Module 3

• In mathematics, the rules are called axioms.
• The new results that have evolved from the undefined terms, defined terms and axioms (rules) are called theorems.
• Axioms form the basis of mathematical proofs that are written in order to establish theorems. In order to formally prove conjectures, we must start with some assumed information. Axioms often supply us with this given information.
• The study of axiomatic systems

  • As early as 600 B.C., the Greeks began to study the logical connections among mathematical facts.
  • Around 300 B.C., Euclid organized most of the known mathematics of his time.
  • All theorems were proved from a small collection of definitions and axioms, and thus the axiomatic method was born.
  • Today, the axiomatic method is the distinctive structure of mathematics
• Here are the four essential components of an axiomatic system:

  1. defined terms
  2. undefined terms
  3. axioms
  4. theorems
• Definition (Definition)

  A definition is a statement of a single, unambiguous idea that the word, phrase, or symbol being defined represents.

  The single, unambiguous idea is called the characteristic property of the definition. It is a condition such that, given any object:

  1. we can determine whether or not that object satisfies the condition; and
  2. the term being defined is used to label everything that satisfies the condition and is not used to label anything else.
• Definition (Circular Definition)

  Circular Definition, is a type of definition where its defining condition either uses the term itself or uses terms that are themselves defined using the term being defined.
• A good definition must not be circular.
• Defining a word by simply giving a synonym can lead to a problem of circularity.
• Definition (Undefined (Primitive) Terms) Undefined (primitive) terms in an axiomatic system are used to form a fundamental vocabulary with which other terms can be defined. They acquire their meanings from the context.
• A term may be undefined, but this does not mean that it is meaningless.
• Two Types of Undefined Terms • Elements are undefined terms that imply objects. • Relations are undefined terms that imply relationships between objects
• Axioms

  From the axiomatic point of view, the undefined terms are implicitly

  defined by basic propositions that involve these terms. Such propositions

  are called axioms.
• Definition (Axioms) Axioms are statements that are accepted as true without proof.
• Definition (Proof) A proof is a logically sound argument that progresses from ideas you accept to the statement you are wondering about. It is a mixture of everyday language and strict logic
• Sound argument is argument that is valid and whose premises are all true.
• Experiments are performed in Mathematics:

  In mathematics, as in the physical sciences, we may run an experiment or check a few cases to come up with a conjecture1 for a theorem. However, in mathematics, experiment cannot replace a proof no matter how natural and obvious the conjecture is that they support
• Mathematical proof is fundamentally a matter of rigor. This means that theorems follow from axioms by means of systematic reasoning.
• An axiomatic system is consistent if there is no statement such that both the statement and its negation are axioms or theorems of the system. That is, these axioms must not contradict one another.
• Interpretation of Axiomatic System

  • An interpretation of an axiomatic system is any assignment of specific meanings to the undefined terms of that system.
  • An interpretation satisfies the axiom if an axiom becomes a true statement with the specific interpretation of the undefined terms.
  • A model of an axiomatic system is an interpretation that satisfies all the axioms of the system.
• Two Types of Axiomatic System Models

  • A axiomatic system model is a Concrete Model if the meanings assigned to the undefined terms are objects and relations adopted from the real world.
  • A axiomatic system model is an Abstract Model if the meanings assigned to the undefined terms are objects and relations adopted from another axiomatic development.
• A model for an axiomatic system makes its ideas more realistic, just as an architect’s model of a building makes the design ideas more concrete and visible.
• There are usually many things not specified by the axioms, so a variety of models may be possible.
• Consistent and Inconsistent Axiomatic System

  • An axiomatic system is Consistent if we can find a model for the axioms - a choice of objects that satisfy the axioms. Any system containing contradictory axioms is Inconsistent and is of no practical value at all.
• Definition (Theorems)

  • A statement that is derived from the axioms by strict logical proof is called theorem.
• To indicate that the proof is complete, mathematicians use this filled-in square at the end of the proof. This is called a halmos for Paul Halmos who introduced it.
• In the past, mathematicians like Euclid used to write Q.E.D. at the end of a proof to say job is done – it’s an abbreviation for the Latin quod erat demonstrandum (which was to be demonstrated).
• Sometimes they place a small rectangle with its shorter side horizontal. They call it a tombstone, meaning the death of suspicion of the validity of the statement that was to be proved. Nowadays, they use a filled-in square ■ . This is called a halmos for Paul Halmos who introduced it.