Here we consider simple variations of the famous Barber Paradox (3 versions) and demonstrate the need for set theory in a very simple and direct way. We will prove using set theory that the fabled village barber can actually shave those and only those men in that village who do not shave themselves *if and only if* that barber is *not* a man in the village. This would seem to be impossible to prove using only first-order predicate logic.

Last updated Feb. 10, 2017

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Last updated: Nov. 18, 2015

Here we turn Hilbert’s Hotel on its head. Instead of frantically

shuffling infinitely many guests for one room to the next in

Hilbert’s mythical infinite hotel, we start with a leisurely walk

through an ordinary and quite finite village.

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Last updated: Oct. 12, 2015

*The Drinker’s Theorem: Consider the set of all drinkers in the world, and the set of all people in a given pub. Then there exists a person who, if he or she is drinking, then everyone in that pub is drinking. *

It doesn’t matter how many people are there. Or how many are drinking. Or how few. No one needs to be taking their cues from some “lead drinker,” but in every pub, in every town and village, it just happens! How is this possible?

There are several possible approaches to this problem. Here, we will turn to British philosopher and mathematician, Bertrand Russell (1872 – 1970). His famous Paradox is the key.

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Last updated: Nov. 1, 2016

The Definition of the set of natural numbers

is given by nothing more or less than

Peano’s Axioms.

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Last updated: Jan. 12, 2015

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Now featuring a formal construction of the

partial function for exponentiation on *N*x*N.*

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The Cantor-Bernstein-Schroeder Theorem (CBST), is one of the most important and widely applied results in set theory:

If set

Xcan be mapped one-to-one into setY(an injection), and setYcan be mapped one-to-one into set X (an injection), then X can be mapped one-to-oneonto Y(a bijection).

Though seemingly self-evident, some proofs of CBST can make your head spin! Every line of my machine-verified, formal proof (updated 2014-09-11) is justified by one of a very limited list of simple axioms and rules of inference (indicated in a grey font at the end of each line).

Such complete rigour does come at a price, however. While it is completely free of any of the sort of the hand-waving that plagues many informal versions of this proof, like most formal proofs of any complexity, it is *very* long. My commentary (indicated by a blue font) is inserted throughout using *DC Proof.* I believe this makes my proof considerably more readable than machine-generated formal proofs in other systems.

This proof was written to demonstrate he capabilities of the *DC Proof* system. Though designed for ease of use by the complete beginner, *DC Proof* is quite capable of some mathematical heavy lifting.

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Last updated: Oct. 21, 2016

According to the Pigeonhole Principle, if you have more pigeons than pigeonholes (as in photo), then at least two pigeons will be in the same hole. Here we present a *non-numeric* version.

Usually, *more* pigeons than pigeonholes is taken to mean that the *number* of pigeons is *greater than* the *number* of pigeonholes. Here, we take more pigeons than pigeonholes to mean that the set of pigeonholes cannot be mapped *onto* the set the pigeons.

In this sense then, we can prove that, if we put a non-empty set of pigeons into a set of pigeonholes and there are more pigeons than holes, then we will have put at least two pigeons in the same hole. Note that there is no requirement here that there be finitely many pigeons or pigeonholes.

See formal proof (70 lines) at The Pigeonhole Principle: A non-numeric version.

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*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.

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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.

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*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

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