Day 1 in my "Into to Pure Maths" class...

I'd like to have a very simple set of axioms defining something, not necessarily a useful thing, but a system that is suitable for making short deductions.

I just want to convey the idea of proving something using the axioms and then revealing an actual instance and confirming that the newly-proved fact does indeed hold.

I seem to remember hearing something years (decades) ago that used something geometric - perhaps even finite projective planes - for this purpose, but I cannot find it online..

  • $\begingroup$ you could find one of the many derivations of the Lorentz transformations from Einstein's Axioms. This would allow a contrast between how axioms are held in math vs. physics. $\endgroup$ – James S. Cook Jan 23 '17 at 11:37

This may go beyond what you are asking for, but there is a wonderful book called Introduction to the Foundations of Mathematics by Raymond L. Wilder. I provided its axioms and an example of how they could be satisfied using SET cards in MESE 2528.

Here is again a list of the axioms:

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The full book is available online (open access) and I have derived quite some joy over the years from proving the various theorems that follow the above-mentioned axioms. Wilder himself begins his discussion of these five axioms by showing that they can be "satisfied" with an interpretation around people as members of clubs, and remarks that it is achievable with just four people; so, he continues, this cannot be a full description of (e.g.) Euclidean geometry, since we know there are many more than four points in the plane!

The theorems that Wilder uses begin as innocuous; according to my own notes, the first one is:

Theorem 1 Every point is on two distinct lines

and become increasingly more difficult; e.g., given: $n$ total points; let $m$ be the number of lines containing a given point - this has already been shown as well-defined in an earlier theorem; and let $r$ be the number of points on a line - also shown well-defined in an earlier theorem. Then one has:

Theorem 10 The total number of lines is $m \cdot r$.

The book is from 1952 (the link above goes to its reprinted second edition from 1965) but I think it holds up quite well. Even if the full book turns out not to be ideal for your present purposes, I would recommend this as a reference; there are some nice problems and explanations scattered throughout.

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    $\begingroup$ Thanks for this answer. I think it probably is too large in scope for my immediate purpose, but I'll certainly read it to see if I can extract a more substantial example for multi-week use. $\endgroup$ – Gordon Royle Dec 28 '16 at 9:10
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    $\begingroup$ I have accepted this answer because the Wilder book looks like just what I need for a whole range of examples. $\endgroup$ – Gordon Royle Jan 16 '17 at 9:03
  • $\begingroup$ @GordonRoyle If at some point you want a copy of proofs of these theorems, then I think I can dig up / locate the ones I wrote out... $\endgroup$ – Benjamin Dickman Jan 16 '17 at 17:40

Howard Eves has a nice example in Great Moments in Mathematics (After 1650), Lecture 36: a set $K$ of elements and a relation $R$, with the following 4 postulates:

  1. If $a \ne b$, then either $aRb$ or $bRa$.
  2. If $aRb$, then $a \ne b$.
  3. If $aRb$ and $bRc$, then $aRc$.
  4. $K$ consists of exactly four distinct elements.

He then proves seven specific theorems (with more in the exercise list), and also discusses changes if some postulates are altered. Then he provides applications of those theorems in a variety of fields (arithmetic, geometry, and genealogy).

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    $\begingroup$ Thanks for this suggestion. My kids won't formally know what a relation is (yet), but this sounds exactly the right sort of thing. $\endgroup$ – Gordon Royle Dec 28 '16 at 9:08

For my money the best possible "toy" examples are the MU-System and the pq-System introduced and defined in Chapters 1 and 2 of Douglas Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid. The latter, in particular, is extremely useful for stressing the delicate balance between syntax and semantics: on the one hand, having an interpretation of the words and axioms in your system can play an important role in helping anticipate which theorems are likely to be true and which ones false; on the other hand, one's reasoning should not depend on that interpretation, because there may be other interpretations that are equally valid.

I like this example better than some of the geometric ones people are suggesting because it's hard for beginners to suppress their intuition about what the words "point", "line" and "plane" mean. (Hilbert's apocryphal quote about "tables, chairs and beer mugs" fits in well here.) In Hofstadter's pq-System, the reader genuinely doesn't know what p and q mean (or might mean) until after they have figured out some of the theorems and non-theorems of the system.

  • $\begingroup$ I was about to post the same! $\endgroup$ – Dirk Jan 3 '17 at 15:23
  • $\begingroup$ This sounds interesting. I'll have to pull my copy of GEB out of storage to check the details. Thanks for the suggestion. $\endgroup$ – Gordon Royle Jan 5 '17 at 13:50

This approach below does not work equally well in all countries, but examples based on changing the rules of a familiar sport/game I think can help some students with regard to ideas related to axiomatics. In this column (basically about the taxicab plane) that I wrote some years ago for the Feature Column of the American Mathematical Society I have a brief discussion of this idea in the context of American baseball.

Scroll down to the section on "axomatics."


Usual baseball - (3,4,9)-baseball - has 3 strikes for an out, 4 balls for a walk, and 9 innings. What happens when one changes these parameters?

  • $\begingroup$ Interesting and relevant, but I think slightly different from the point I'm trying to make. I'm trying to illustrate the idea of reasoning purely from axioms with no particular knowledge of an actual model for those axioms, or at least the necessity of only using a model for general guidance that requires formal proof. The baseball example is more of a parametrised system where the reasoning is to describe how the model varies with th parameters. And of course I'd need a parametrised version of cricket not baseball! $\endgroup$ – Gordon Royle Jan 5 '17 at 14:05

One example from Euclidean geometry in planes. Consider the following three axioms:

1) There exist three non-collinear and pairwise different points in the plane.

2) Every 2 distinct points can be connected by a unique line.

3) For every line $g$ in the plane and every point $A$ not on $g$ there exists a unique line $h$ through $A$ (i.e. $A \in h$) with $g \cap h =\emptyset$.

If you assume these axioms to be true, then you can easily prove that the plane consists of at least 4 points in the plane:

Suppose you have a plane. Axiom 1) yields that there are at least 3 non-collinear pairwise different points $A$, $B$ and $C$. According to axiom 2) there is a unique line $g$ through $A$ and $B$. Yet, $C \not \in g$ since $A$, $B$ and $C$ are not collinear due to 1). Now axiom 3) tells you that there is a line $h$ through $C$ with $g \cap h =\emptyset$. According to 2) $h$ is a unique line through (at least) 2 points with one being $C$ by construction. Hence, there has to be a fourth point $D$ in the plane with $D \in h$.

I hope this example helps a bit. I tried to keep it simple and did not use any definitions for a line for example. In a more scientific context this should, however, be done. Since you asked for a toy example the above example should be sufficient.


To follow up Pere's answer and address your comment, you could work on some simple finite field or group in disguise, giving arbitrary names to elements. You take as undefined terms one or two operations (maybe denoted by $\square$ and and $\triangle$) and the objects (say $A,B,C$ if you work on $\mathbb{Z}/3\mathbb{Z}$), and as axioms the defining property of a group or field and part of the operation's tables. Then you seek to complete the tables.

This might be a bit tedious, though. Maybe the most interesting case is with $4$ elements, so that you can ultimately relate this introduction with the fact that there are two (isomorphism classes of) groups with $4$ elements, but only one field.

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    $\begingroup$ I've never found a way of dealing satisfactorily with the associative and distributive laws "from scratch" (i.e. how to check them in reasonable time for, say, a 4-element ring/group/field). Later in the class, when we actually get to polynomial rings amd then fields, these laws are always just inherited from the underlying ring, which in my case is just the integers, so no case-checking is required. $\endgroup$ – Gordon Royle Jan 5 '17 at 13:56

In the vein of other answers that would prefer a "real" axiomatic system, my suggestion would be...point-set topology. I know, crazy. Hear me out though...

  • Only three axioms to give!

    -- arbitrary unions of open sets are open

    -- finite intersections of open sets are open

    -- the empty set and universal set are open

  • There are some background definitions (what is a set? intersection/union of sets? what is the complement of a set?), but students taking a "pure maths" course likely have a reasonable intuition of what these are.

  • The start of point-set topology, with its three axioms, is probably simple enough to come up with a "toy system" for if you like the bait and switch idea. Replace "open/closed" with any other pair of descriptors you like...good/bad, left/right, red/blue, ...
  • It gives an example of a "real" bit of mathematics where axiomatization is vitally important. Axiomatizing arithmetic/numbers/planar geometry can feel a bit hair-splitting at times.
  • Within short order, you get to discuss the inevitable claim that "since this set is not open, it is closed!" In my mind, this is one of the primary reasons that we need axioms: the need to have a level playing field, so that we can separate logical truths from intuition.
  • You can axiomatize point-set topology in terms of closed sets instead: finite unions and arbitrary intersections of closed sets are closed, the empty set and universal set are closed. You can then show how these two collections of axioms are equivalent and lead to the same mathematical universe.

There are certainly some cons, which are largely connected to the mathematical sophistication of the class. Keep in mind that calculus students will be familiar with open and closed sets while multivariable calculus students will likely have seen simply connected domains.

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    $\begingroup$ Actually I like this idea, although my kids find it hard to reason about infinity in a sensible way. When I eventually try to convince them that the cofinite subsets of an infinite set satisfy the requirements for a topology they mostly don't like it. But they should have a reasonably clear intuitive idea of what an open interval is from calculus, though they won't know the formal definition unless they've done the "Extension Exercises" in 1st year. $\endgroup$ – Gordon Royle Jan 23 '17 at 8:32
  • $\begingroup$ Yep, there are definitely some pitfalls and it does require some mathematical sophistication on the part of the students. I would certainly want a good read on the group of the students before embarking. Personally, I might stay away from the cofinite topology (I can attest to being initially confused by it as a grad student myself) and stick with the standard topology, discrete topology, half-open topology, or even just finite point sets. $\endgroup$ – erfink Jan 24 '17 at 5:03

Instead of making a toy set of axioms, I would go to some axioms taken from an established set and prove something trivial.

I would try with the field axioms of real numbers (or just some of them) and then prove that 2+2=4 or 3*2=6 - of course you would need to define 2, 3, 4 and 6 first.

Anyway, although this approach would fit a mathematical analysis course, it might be not the best for a "pure math" course.

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    $\begingroup$ Actually, I have found this approach to be problematic. For example, it is standard procedure to give the axioms for a vector space and then prove, say, that there is a unique zero vector. But (too many of) the students just don't get it. They don't even grasp the issue, because the outcome seems obvious to them, and they can't even see why a proof is needed. $\endgroup$ – Gordon Royle Dec 28 '16 at 15:41

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