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AXIOMATICS THROUGH THE METRIC SPACE AXIOMS
JAMES FENNELL, UNIVERSITY COLLEGE CORK
In this article we discuss some elementary concepts in axiomatics, which is the
study of axioms. The axiomatic system we use is that of a metric space, which
will be familiar to many readers. Metric spaces were introduced by the French
mathematician Maurice Fréchet in 1906 as a generalization of the real numbers
and the “distance function” f (x, y) = |x − y|. The aim is to construct limits,
continuity, openness and eventually differential calculus on this generalized base
alone; the results will then apply to a wide class of spaces such as R2 without
little particular work. Here we are only interested in metric spaces as an example
of an axiomatic system; that is, a system which behaviors according to clear, welldefined rules.
What do we desire in an axiom set? Intuitively, we might want our axiom
sets to be as small as possible. In particular, if one axiom in an axiom set can be
logically deduced from the others we might be inclined to remove it, as it serves
no purpose and just makes our set less “pure” than it could be. If a specific axiom
can’t be deduced from the others then we say that that axiom is independent.
If every axiom in a set is independent, we say the whole set is independent.
Another, more important, requirement of our set of axioms is that it does imply
contradictory statement. Such an axiom set would be effectively useless. An
axiom set that isn’t inherently contradictory is called consistent.
The traditional definition of a metric space provides suitable grounds to discuss these and other concepts, and to illustrate techniques for proving properties
of axiom sets.
1.
Independence and equivalence
The standard definition of a metric space, as given in [2] for example, is as
follows.
Definition 1. A metric space M = {S, d} consists of a non-empty set S together
with a map d : S × S → R such that for all x, y, z ∈ S
A1 d(x, y) = 0 if and only if x = y
A2 d(x, y) ≥ 0
A3 d(x, y) = d( y, x)
A4 d(x, y) ≤ d(x, z) + d(z, y)
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All of these axioms are intuitive. A1 states that the distance from a point
to itself is 0; A2 that the distance from two points can’t be negative; A3 that
the distance is the same irrespective of the direction it is measured; and A4 is
a generalized statement of the triangle inequality: if you take a detour on your
way, the journey can’t be shorter.
It can be easily demonstrated that the axiom set {A1 , A2 , A3 , A4 } is not independent. We do this by showing that one of the axioms (in this case A2 ) can be
logically deduced from the others.
Proposition 2. The axiom set {A1 , A2 , A3 , A4 } is not independent.
Proof. If we assume only A1 , A3 and A4 we have
d(x, y) ≤ d(x, z) + d(z, y)
by A4
d(x, x) ≤ d(x, z) + d(z, x)
letting x = y
0 ≤ d(x, z) + d(z, x)
by A1
0 ≤ 2d(x, z)
by A2
0 ≤ d(x, z)
which is precisely A2 .
This form of proof is common in the study of axiomatics. As opposed to
normal large-scale proofs, these involve tiny, very precise manipulations that
make clear exactly which axioms are being employed.
Now, the fact that the axiom set in definition 2 is not independent is motivation to find (a presumably smaller) one that is. In [1] Howard Eves provides
alternative axioms for the distance function of a metric space which replace those
described above.
Definition 2. A metric space M = {S, d} consists of a non-empty set S together
with a map d : S × S → R such that for all x, y, z ∈ S
M1 d(x, y) = 0 if and only if x = y
M2 d(x, y) ≤ d( y, z) + d(z, x)
The reader will observe that M1 is exactly A1 . M2 is a restatement of the
triangle inequality that is different to A4 in a subtle but important way. Our first
task is to demonstrate that the axiom set {M1 , M2 } is independent; after that, we
will show that the set {A2 , A3 , A4 } can be deduced from it.
To prove that a set of axioms is independent we must show that no axiom is
implied by the others. This can be achieved by inventing systems in which all
of the axioms hold but one. Thus the axiom we are evaluating can’t be implied
by the others as if it were implied it would be true in our invented system! We
illustrate the technique in the following theorem.
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Theorem 1. The axiom set {M1 , M2 } is independent.
Proof. First, we will show that M1 ; M2 . Consider the space R, d1 with
d1 (x, y) = x − y. Clearly d1 (x, y) = 0 if and only if x = y, and so d1 satisfies M1 . Suppose it satisfies M2 . Then
d(x, y) ≤ d( y, z) + d(z, x)
x − y ≤ y −z+z− x
x−y≤ y−x
Setting, for example, x = 2 and y = 1 leads to an absurdity, so d1 does not
satisfy M2 . Thus M1 cannot imply M2 .
Now consider the space R, d2 with d2 (x, y) = x − y + 1. We have
|x − y| + 1 = |x − z + z − y| + 1
≤ |x − z| + |z − y| + 2
≤ | y − z| + 1 + (|z − x| + 1)
d(x, y) ≤ d( y, z) + d(z, x)
so d2 satisfies M2 . But we have d2 (x, x) = 1 6= 0 so it does not satisfy M1 . Thus
M2 cannot imply M1 .
Now that we have proved that the set {M1 , M2 } is independent and so satisfies our aesthetic demands, we must show that it implies properties A2 , A3 and
A4 and then that definition 2 can be used as an alternative to definition 1.
Theorem 2. Given the axioms M1 and M2 we have
(i) d(x, y) ≥ 0
(ii) d(x, y) = d( y, x)
(iii) d(x, y) ≤ d(x, z) + d(z, y)
which are precisely axioms A2 , A3 and A4 respectively.
Proof of (ii).
d(x, y) ≤ d( y, z) + d(z, x)
d(x, y) ≤ d( y, x) + d(x, x)
d(x, y) ≤ d( y, x)
Similarly, we have
d( y, x) ≤ d(x, y)
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And so d(x, y) = d( y, x) as required.
The proof of (i) is that of proposition 2 and the proof of (iii) is an obvious
application of (ii) to axiom M2 .
If we go in the other direction – assume axioms {A1 , A2 , A3 , A4 } and deduce
axioms {M1 , M2 } – then we will have proved that the two axiom sets are equivalent. In this case either set can be used with the exact same results. The proof is
left as a first exercise. The second exercise is to prove that the axioms A4 and M2
are independent. So though both are statements of the triangle inequality they
are fundamentally different.
2.
Consistency
Having considered independence we now turn to the second concept mentioned
in the introduction: consistency. Recall that an axiom set is consistent if it does
not imply contradictory statements. The standard means for proving an axiom
set consistent is the method of models. A model is a specific instance of the
general object described by the axiom set. The real numbers and the distance
function f (x, y) = |x − y| is thus a model of the metric space axioms. The
aim is to furnish a particular model that we know (or hypothesise) to be free of
contradictions; if the model satisfies our axioms we will know that the axiom set
cannot inherently imply contradictory statements, as those contradictions would
appear in the model.
There are two kinds of model one can have: ideal and concrete. Ideal models
are those that are based on another axiomatic system. The real number model
is ideal as the real numbers are themselves governed by axioms. Ideal models
seem the natural choice for mathematicians, but they come with a caveat: if the
axiomatic system the model is based on hasn’t been proved consistent then the
axiom set under consideration isn’t necessarily consistent either. The best we can
hope for in such a situation is relative consistency where we prove that if system
A is consistent then so is system B (and perhaps vice-versa). The consistency of
the real numbers is a famous open question given as one of the 23 problems in
David Hilbert’s 1902 lecture. The consistency of other sets, such as the natural
numbers, is dependent on that of the real numbers (and vice-versa, in this case).
On the other hand, concrete models are taken from the real world. If we accept that the real world contains no contradictions (at the macro-level of human
experience at least, and excepting the speeches of politicians) then providing a
concrete model will establish the absolute consistency of the axiom system in
question. Again, if the set were inconsistent then the contradictions would appear in the real world. We now devise a concrete model of metric space axioms,
and thus establish the set’s absolute consistency.
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Let I be the set of Irish-associated mathematicians William Rowan Hamilton
(WRH) George Boole (GB) and George Stokes (GS). Their date of births are:
William Rowan Hamilton
George Boole
George Stokes
4th of August 1805
2nd of November 1815
13th of August 1818
Now define the “distance function” B to be the number of years between the
births of any two members of I, rounded up to the nearest integer. Thus
B(WRH, GB) = B(GB, WRH) = 11
B(WRH, GS) = B(GS, WRH) = 14
B(GB, GS) = B(GS, GB) = 3
and
B(WRH, WRH) = B(GB, GB) = B(GS, GS) = 0
Now to prove consistency we must show that each possible occurrence M2 is
satisfied (M1 being immediate). For example,
B(WRH, GS) = 14
≤ 3 + 11
≤ B(GS, GB) + B(GB, WRH)
The other 5 cases follow similarly, and we have then established
Theorem 2.1. The axiom set {M1 , M2 } is consistent.
3.
Conclusion
We have seen that in the standard definition of a metric space, axiom A2 is redundant (proposition 2), that a small change in the statement of the triangle
inequality makes A3 redundant (theorem 2), and that once we have dispensed
with A2 and A3 we can go no further (theorem 1). We have also established the
consistency of the second axiom set, which by equivalence applies to the first
axiom set, too.
The fact that a standard definition in the literature is not independent may
be surprising, but this occurs for other systems too. The standard definitions of
a field, a vector space and an equivalence relation all contain redundant axioms;
the reader might find and fix these as an exercise. In practise, what we typically
call axioms might be better stylised as properties. A lecturer giving a course on
topology will not want to enter an effectively tangential discussion of axiomatics,
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and so presents the definition 1 as the properties of the distance function as
opposed to taking a strict axiomatic line.
Further discussion of axiomatics and the philosophy of mathematics can be
found in Howard Eves’ excellent book [1].
References
[1] Howard Eves. Foundations and fundamental concepts of mathematics, pages
232–233. Dover Publications, Mineola, NY, 1997.
[2] W. A. Sutherland. Introduction to metric and topological spaces, page 21.
Clarendon Press, Oxford, 1975.
AN INTRODUCTION TO UNIVERSAL ALGEBRA, AND CATEGORIES OF
ALGEBRAS OF SOME TYPE F
JACK KELLY, TRINITY COLLEGE DUBLIN
The goal of this article is to provide a brief introduction to the subject of universal
algebra, and also, to prove some basic but important properties of the category of
algebras of some type F . Universal algebra, particularly the theory of varieties,
provides a generalization of objects such as groups and rings, and only knowledge of these areas should be required to follow the first few sections. Obviously,
for the last section, wherein we discuss the category of algebras of some type,
some knowledge of category theory is required. Any propositions whose proofs
are available in one of the references will not be provided here (as [1] and [2],
the books wherein the proofs are available, have been made available online by
the authors). The proofs that are present are our own. Hence, it is possible
that there are mistakes. Should you find any errors, feel free to get in contact
at kellyj1129@gmail.com Any comments and criticisms are greatly appreciated.
We would also like to thank Dr. Rupert Levene (UCD) who kindly agreed to
supervise a project on this topic as part of the Trinity College Dublin Summer
Internship Programme 2011, and whose help and guidance was invaluable.
1.
Basic Definitions and Constructions
Definition Let A be a set and let ω be a cardinal. An ω-ary operation on A is
a map f : Aω → A. For the purposes of this article, ω ∈ N0 . For ω = 0, A0 is