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.
a label for an object or action that is left undefined
a statement (usually involving primitive terms or terms defined in terms of primitive terms) that is held to be true without proof
a collection of primitive terms and axioms
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.
In the axiomatic system of Axiom \(\PageIndex{1}\), Axiom 1 is redundant as we may infer from Axiom 4 that there exist three distinct woozles. But there is no harm in including this axiom for clarity. As well, we will later investigate the effect of altering it.
The axiomatic system of Example \(\PageIndex{1}\) seems like nonsense, but we can actually prove things from it.
There exist at least three distinct dorples.
- Proof
-
(In this proof, all references to axioms refer to the axioms of \(\PageIndex{1}\).)
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.
a label for an object or action that is defined in terms of primitive terms, axioms, and/or other defined terms
an formal explanation of the meaning of a defined term
Here is a definition relative to the axiomatic system of Example \(\PageIndex{1}\).
Solution
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.
A pair of snarf buddies snarf a unique woozle in common.
- 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.
Suppose we replace Axiom 1 in the system of Example \(\PageIndex{1}\) with the following.
- There exist exactly three distinct woozles.
In the new, modified axiomatic system, our previous two theorems (Theorem \(\PageIndex{1}\) and Theorem \(\PageIndex{2}\)) remain true, because it is still true that there exist at least three distinct woozles. But we can now also prove the following.
In the axiomatic system of Example \(\PageIndex{1}\) 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 \(\PageIndex{1}\) is just that — nonsense — and not much use unless there are actual examples to which the developed theory can be applied.
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.
Again consider the axiomatic system of Example \(\PageIndex{1}\), 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 \(\PageIndex{3}\) now says that there exist exactly three distinct lines in the plane which fit into our axiomatic system; can you find their equations?
Using nonsense terms like woozle , dorple , and snarf for the primitive terms in an axiomatic system is usually not a good idea, as it takes all intuition out of the process of discovering statements that can be deduced from the axioms. It would have been much better if we had used the words point instead of woozle , line instead of dorple , and lies on instead of snarfs as our primitive terms, to be able to use our intuition about how such objects interact. In such a case, the axioms we choose should be a reflection of our idea of the simplest possible properties about the primitive terms, properties that everyone could reasonably agree are “true” without proof. However, for the theorems deduced from such an axiomatic system to have the widest possible applicability, we should leave the words point and line as truly primitive , undefined terms — that is, point and line should not be taken to mean point in the plane and line in the plane , as in the example above, but rather just left as some abstract, intuitive idea of point and line .