The Axiomatic System: Definition & Properties

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  • 0:01 The Axiomatic System
  • 2:11 Consistency
  • 2:55 Independence
  • 3:44 Completeness
  • 4:16 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Learn what kinds of things are included in an axiomatic system in this video lesson. Also learn why consistency, independence, and completeness are important in axiomatic systems.

The Axiomatic System

What exactly is an axiomatic system? I know it sounds like a big word for a complicated system, but it's actually not all that complicated. Defined, an axiomatic system is a set of axioms used to derive theorems. What this means is that for every theorem in math, there exists an axiomatic system that contains all the axioms needed to prove that theorem. An axiom is a statement that is considered true and does not require a proof. It is considered the starting point of reasoning. Axioms are used to prove other statements. They are basic truths. For example, the statement that all right angles are equal to each other is an axiom and does not require a proof. We know that all right angles are equal to each other and we do not argue that point. Instead, we use this information to prove other things. A collection of these basic, true statements forms an axiomatic system.

The subject that you are studying right now, geometry, is actually based on an axiomatic system known as Euclidean geometry. This system has only five axioms or basic truths that form the basis for all the theorems that you are learning. Everything can be traced back to these five axioms. What are they? Let me tell you.

1. A straight line can be drawn from any one point to any other point.

2. A line segment can be extended infinitely in both directions.

3. A circle can be described with a center and radius.

4. All right angles are equal to each other.

5. If a line intersecting two lines forms interior angles less than 90 degrees, then the two lines will intersect on the same side as the angles that are less than 90 degrees. The fifth axiom is also known as the parallel postulate.

Axiomatic systems also have three different properties.


The first property is called consistency. When an axiomatic system is consistent, then the system will NOT be able to prove both a statement and its negation. The consistent system will prove either the statement or its negative, but not both. If it did, then it would contradict itself. For example, if an axiomatic system was able to prove the statement 'squares are made from two triangles' as well as the statement 'squares are not made from two triangles,' then the system is not consistent. The system actually contradicts itself. You can't rely on the system. Because of this, this property is a requirement for an axiomatic system.

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