Section 8.1 Basics and examples
Any mathematical system must have a starting point; we cannot create something out of nothing. The starting point of a mathematical system (or any logical system, for that matter) is a collection of basic terminology accompanied by a collection of assumed facts about the things the terminology describes.
- primitive term
a label for an object or action that is left undefined
- axiom
a statement (usually involving primitive terms or terms defined in terms of primitive terms) that is held to be true without proof
- axiomatic system
a collection of primitive terms and axioms
Example 8.1.1. An axiomatic system.
Primitive terms.
woozle (noun),
dorple (noun),
snarf (verb).
Axioms.
There exist at least three distinct woozles.
A woozle snarfs a dorple if and only if the dorple snarfs the woozle.
Each pair of distinct woozles snarfs exactly one dorple in common.
There is at least one trio of distinct woozles that snarf no dorple in common.
Each dorple is snarfed by at least two distinct woozles.
The axiomatic system of
Example 8.1.1 seems like nonsense, but we can actually prove things from it.
Theorem 8.1.3.
There exist at least three distinct dorples.
Proof.
(In this proof, all references to axioms refer to the axioms of
Example 8.1.1.)
By
Axiom 4, there exists a trio
\(w_1,w_2,w_3\) of distinct woozles that snarf no dorple in common. Breaking this trio into various pairs and applying
Axiom 3, we see that there exists a dorple
\(d_1\) that
\(w_1\) and
\(w_2\) both snarf in common, there also exists a dorple
\(d_2\) that
\(w_1\) and
\(w_3\) both snarf in common, and there also exists a dorple
\(d_3\) that
\(w_2\) and
\(w_3\) both snarf in common. These snarfing relationships are illustrated in the diagram below.
Now, suppose \(d_1\) and \(d_2\) were actually the same dorple — then all three woozles would snarf it in common.
As this would contradict our initial assumption, it must be the case that \(d_1\) and \(d_2\) are distinct. Similar arguments allow us to also conclude that \(d_1 \ne d_3\) and \(d_2 \ne d_3\text{.}\)
It is often useful to give names to important properties of objects.
- defined term
a label for an object or action that is defined in terms of primitive terms, axioms, and/or other defined terms
- definition
an formal explanation of the meaning of a defined term
Example 8.1.4. Making a definition.
Here is a definition relative to the axiomatic system of
Example 8.1.1.
- snarf buddies
two distinct dorples that snarf a common woozle
A definition allows us to more succinctly communicate ideas and facts about the objects of an axiomatic system.
Theorem 8.1.5.
Proof.
Suppose
\(d_1,d_2\) are snarf buddies. By contradiction, suppose they snarf
more than one woozle in common: let
\(w_1,w_2\) be distinct woozles both snarfed by
\(d_1\) and
\(d_2\text{.}\) By
Axiom 2, each of
\(w_1,w_2\) snarfs each of
\(d_1,d_2\text{.}\) But this contradicts
Axiom 3, as two distinct woozles cannot snarf more than one dorple in common.
There exist exactly three distinct woozles.
In the new, modified axiomatic system, our previous two theorems (
Theorem 8.1.3 and
Theorem 8.1.5) remain true, because it is still true that there exist at least three distinct woozles. But we can now also prove the following.
Theorem 8.1.6.
In the axiomatic system of
Example 8.1.1 with the above modified version of
Axiom 1, there exist exactly three distinct dorples.
Proof.
You are asked to prove this in the exercises.
A nonsense system like the one in
Example 8.1.1 is just that — nonsense — and not much use unless there are actual examples to which the developed theory can be applied.
- model
a system obtained by replacing the primitive terms in an axiomatic system with more “concrete” terms in such a way that all the axioms are true statements about the new terms
If we agree that the axiom statements are still all true with the new terms, then any theorems proved under the abstract system are still valid in the new model system.
Example 8.1.7. A model for the woozel-dorple system.
Again consider the axiomatic system of
Example 8.1.1, still using the modified version of
Axiom 1. Let the three distinct woozles be the points
\((0,0)\text{,}\) \((1,1)\text{,}\) and
\((2,0)\) in the Cartesian plane. Let
dorple now mean
line in the plane, and let
snarf now mean
lies on. Convince yourself that the axioms of the system are all true with this interpretation of the primitive terms.
Theorem 8.1.6 now says that there exist exactly three distinct lines in the plane which fit into our axiomatic system; can you find their equations?
