Chapter 1
Logic and Set Theory
To criticize mathematics for its abstraction is to miss the point entirely.
Abstraction is what makes mathematics work. If you concentrate too
closely on too limited an application of a mathematical idea
,yourob
the mathematician of his most important tools: analogy, generality, and
simplicity.
– Ian Stewart
Does God play dice? The mathematics of chaos
In mathematics, a
proof
is a demonstration that, assuming certain axioms, some
statement is necessarily true. That is, a proof is a logical argument, not an empir
ical one. One must demonstrate that a proposition is true in all cases before it is
considered a theorem of mathematics. An unproven proposition for which there is
some sort of empirical evidence is known as a
conjecture
.M
a
th
em
a
t
i
c
a
llog
i
ci
s
the framework upon which rigorous proofs are built. It is the study of the principles
and criteria of valid inference and demonstrations.
Logicians have analyzed set theory in great details, formulating a collection of
axioms that affords a broad enough and strong enough foundation to mathematical
reasoning. The standard form of axiomatic set theory is the ZermeloFraenkel set
theory, together with the axiom of choice. Each of the axioms included in this the
ory expresses a property of sets that is widely accepted by mathematicians. It is
unfortunately true that careless use of set theory can lead tocontradictions
.Avoid
ing such contradictions was one of the original motivations for the axiomatization
of set theory.
1
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CHAPTER 1. LOGIC AND SET THEORY
Arigorousanalysisofsettheorybelongstothefoundations of mathematics and
mathematical logic. The study of these topics is, in itself,aform
idab
letask
. For
our purposes, it will suf±ce to approach basic logical concepts informally. That is,
we adopt a naive point of view regarding set theory and assume that the meaning of
a set as a collection of objects is intuitively clear. While informal logic is not itself
rigorous, it provides the underpinning for rigorous proofs
.T
h
er
u
l
e
sw
ef
o
l
l
ow
in dealing with sets are derived from established axioms. At some point of your
academic career, you may wish to study set theory and logic in greater detail. Our
main purpose here is to learn how to state mathematical results clearly and how to
prove them.
1.1
Statements
Aproofinmathematicsdemonstratesthetruthofcertain
statement
.I
tistherefore
natural to begin with a brief discussion of statements. A statement, or
proposition
,
is the content of an assertion. It is either true or false, but cannot be both true and
false at the same time. For example, the expression “There arenoclassesatTexas
A&M University today” is a statement since it is either true orfalse.Theexpression
“Do not cheat and do not tolerate those who do” is not a statement. Note that an
expression being a statement does not depend on whether we personally can verify
its validity. The expression “The base of the natural logarithm, denoted
e
,isan
irrational number” is a statement that most of us cannot prove.
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 Fall '08
 Staff
 Logic, Set Theory, Mathematical logic

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