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Finite Axiomatic Geometry

Geometry
Finite Axiomatic Geometry

Finite Axiomatic Geometry
This abstract geometry confuses me and I just can't follow this logic. Here is a set up of the question. Please help with some insight.

Consider the following axiom system.
The undefined terms are point, line, and on.
Definition: two lines are parallel if there is no point on both of them.

Axioms:
I. Given any two distinct points, there exactly one line on both of them.
II. Every line has at least one point on it.
III. Given any line, there is at least one other line that is parallel to it.
IV. There exists at least one line.

[The book defines each axiom an "undefined term" called a "interpretation of a system."]

[If, for a given interpretation of a system, all the axioms are "correct" statements, we call the interpretation a "MODEL"]

Questions:
(a) Show that this axiom system is consistent by finding a model for it.

(b) Using models, show that Axiom III is independent of the other axioms.

(c) Show that this axiom system is not complete.

Solution Preview

Geometry
Finite Axiomatic Geometry

Finite Axiomatic Geometry
This abstract geometry confuses me and I just can't follow this logic. Here is a set up of the question. Please help with some insight.

Consider the following ...

Solution Summary

This solution is comprised of a detailed explanation of the Finite Axiomatic Geometry.
It contains step-by-step explanation for the following problem:
Finite Axiomatic Geometry
This abstract geometry confuses me and I just can't follow this logic. Here is a set up of the question. Please help with some insight.

Consider the following axiom system.
The undefined terms are point, line, and on.
Definition: two lines are parallel if there is no point on both of them.

Axioms:
I. Given any two distinct points, there exactly one line on both of them.
II. Every line has at least one point on it.
III. Given any line, there is at least one other line that is parallel to it.
IV. There exists at least one line.

[The book defines each axiom an "undefined term" called a "interpretation of a system."]

[If, for a given interpretation of a system, all the axioms are "correct" statements, we call the interpretation a "MODEL"]

Questions:
(a) Show that this axiom system is consistent by finding a model for it.

(b) Using models, show that Axiom III is independent of the other axioms.

(c) Show that this axiom system is not complete.

Solution contains detailed step-by-step explanation.

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