The set of natural numbers can show up in the oddest places! Suppose, for example, that *f* is *any* function defined on *any* non-empty set *S*, subject only to the following conditions:

*1. f* is injective (one-to-one)

*2. f* is not surjective (not onto) *

Since *f* is not surjective, by definition, there must exist at least one element of *S* that has no pre-image under* f*. Each such element can be shown to be the starting point (the 0 or 1) of its own distinct number system satisfying Peano’s Axioms for the natural numbers, with* f* as the required successor function.

See the formal proof (112 lines in the DC Proof format) at Constructing the Natural Numbers

**Follow-up **(2016-01-16)

Interestingly, if *f* is just an arbitrary function on *S*, then, for every element *x* in *S*, we can construct a subset of *S* on which induction will hold, using *f* as the successor function and *x* as the “first element” of that subset.

See the formal proof (89 lines in the DC Proof format) at Minimum Requirements for Induction.

The Cretan poet, Epimenides (circa 600 BC), famously wrote that Cretans are “always liars.”

Paradoxically, it would seem that if he was telling the truth, then he was lying. And if he was lying, he only confirmed that he was telling the truth! Well, not exactly.

It turns out that there are many possible narratives that would be logically consistent with the original scenario. Epimenides’ famous rant could, for example, have been the *only* lie ever told by a Cretan. Or, at the other extreme, *all but one* statement ever made by a Cretan was lie. All that is required is that Epimenides’ statement be a lie and that at least one Cretan once told the truth.

For a formal proof, see The Original Cretan Liar Paradox.

Here, it is argued that formal logic and set theory be taught, not necessarily as ends in themselves, but as perhaps the best way to introduce the various methods of proof to mathematics undergraduates and advanced high school students, methods that are applicable in all branches of mathematics, in both formal and informal proofs.

*The Problem with a Geometric Approach*

There has been much discussion on the most effective way to introduce the methods of proof to mathematics undergraduates and advanced high school students. The traditional approach is one based on Euclidean geometry, one that, it is hoped, would build on the student’s spatial sense developed over the years since childhood. Studies have shown, however, that proof-writing skills learned in one branch of mathematics such as geometry may not be easily transferred to other branches such as abstract algebra and analysis. F. A. Ersoz [1] (2009) suggests that the many informal “axioms” of Euclidean geometry, as usually taught, are based largely on personal intuition and imagination (p. 163). While this may serve as a productive basis for some discussion, it can blur the boundary between the formal and informal, and lead to confusion as to what constitutes a legitimate proof in other domains (branches) of mathematics. Ersoz also suggests that introductory geometry courses seldom present many of the methods of proof used in more abstract courses ─ methods like proofs by induction, contrapositive or contradiction (p. 164).

Then, why not teach a formal theory of geometry? First, the long list of often counter-intuitive axioms required to formalize even the geometry of the plane can be overwhelming. (See, for example, the work of Hilbert or Tarski in this area.) It has also been the author’s experience that even the simplest geometric result can explode into a formal proof hundreds of lines in length.

*Simplified Rules and Axioms*

It has also been said that the axioms of formal logic and set theory are beyond the average undergraduate who might be struggling with proofs. This is certainly the case with the standard axioms ─ e.g. standard first-order logic, the ZFC axioms of set theory. In DC Proof, however, the axioms of logic and set theory are based on the simplifying assumption that all mathematical theory is based on one more underlying sets ─ the set of natural numbers in number theory, the set of points in the plane in Euclidean geometry, etc. The resulting rules and axioms are fewer and more intuitive, without sacrificing any expressive power.

*The Simplest Possible Domain*

When first introducing the various methods of proof, it would seem reasonable to use illustrations from the simplest possible domains, that is, from systems with a*minimum* number of rules and axioms. The simplest such domains are, of course, *logic and set theory*. An approach based on formal logic and set theory may then be the best way to introduce the methods of proof in a way that can be widely applied in *every* branch of mathematics, in both formal proofs, *and* in the more *informal* proofs that you will find in most mathematics textbooks.

While it may be impractical to present most proofs formally, the rules and axioms of formal logic and set theory presented here must be understood by every mathematician. Every geometer, for example, must understand the law of the contrapositive. Every number theorist must understand De Morgan’s Law, and so on. Teaching formal logic and set theory then should not necessarily be seen as an end in itself. These are just the simplest domains from which to draw examples of the various methods of proof. These examples would be not only simpler, but more rigorous and much more widely applicable than, say, examples from informal Euclidean geometry.

*An Alternative Approach*

Included in DC Proof is a tutorial that not only introduces the features of the program, but can also serve as an introductory, self-study course in formal logic and the methods of proof at the undergraduate or advanced high school levels. The tutorial includes several worked examples, plus exercises with hints and full solutions. Departing from the traditional geometric approach, it draws on examples on from logic, set theory and some elements of number theory that are common to all branches of mathematics.

*Reference*

1. F.A. Ersoz, “Proof in different mathematical domains,” *Proceedings of the ICMI Study 19 Conference, Volume 1*, 2009 http://140.122.140.1/~icmi19/files/Volume_1.pdf