Preface
My main aim is to make accessible to readers without any
specialist training in mathematics, and with only an elementary
knowledge of modern logic, complete proofs of the fundamental
metatheorems of standard (i.e. basically truthfunctional) first
order logic, including a complete proof of the undecidability of
a system of first order predicate logic with identity.
Many elementary logic books stop just where the subject gets
interesting. This book starts at that point and goes through the
interesting parts, as far as and including
a
proof that
it is impossible to program a computer to give the right
answer (and no wrong answer) to each question of the form
'Is

a truth of pure logic?'
The book is intended for nonmathematicians, and concepts
of mathematics and set theory
are
explained
as
they
are needed.
The main contents are: Proofs of the consistency, complete
ness and decidability of a formal system of standard truth
functional propositional logic. The same for first order monadic
predicate logic. Proofs of the consistency and completeness of a
formal system of first order predicate logic. Proofs of the con
sistency, completeness and undecidability of a formal system of
first order predicate logic with identity.
A
proof of the existence
of a nonstandard model of a formal system of arithmetic.
The reader will be assumed to have an elementary knowledge
of truthfunctional connectives, truth tables and quantifiers. For
the reader with no knowledge of set theory, here
very brief
explanations of some notations and ideas that
will
be taken for
granted
later on:
1
.
mr!~
brakt
nsteti~;;
fii


'(Fido, Joe)' means 'The set whose sole members
are
Fido and
Joe'. '{3,2, 1,3,2)' means 'The set whose sole members
are
the
numbers 3,2,1,3,2' (and this last set is the same set
{1,2,3),
i.e. the
set
whose sole members are the numbers 1.2 and
3).
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METALOGIC
2.
The
epsilon notation for setmembershb
'n E
X' means
'n
is a member of the set X'.
3.
T;he
criterion of identity for sets
A
set
A
is the same set
as
a set B
if
and only
A
and B have
exactly the same members. Nothing else matters for set identity.
4.
empty set,
0
By
the criterion of identity for
sets
[(3) above], if
A
is a set
with no members and B is a set with no members, then
A
is the
same set
as B;
so if there is a set with no members, there is just
one such set. We shall assume that there is such a set.
Further
introductory material on set theory
can
be found in,
for example, chap. 9 of Suppes (1957) or chap. 1 of Fraenkel
(1961).
The book deals only
with
(1) standard (i.e. basically truth
fuctional) logic, and
(2)
axiomatic systems.
(1)
Standard first order logic, with its metatheory, is now a
secure field of knowledge; it is not the whole of logic, but it is
important, and it
is
a jumpingoff point for most other develop
ments in modern logic. There seemed to me to be no book that
tried
to make accessible
to
nonmathematicians complete proofs
of the basic metatheory of standard logic: hence this one.
Axiomless systems (socalled 'natural deduction systems')
are
nowadays getting more popular
than
axiomatic systems, for
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 Spring '07
 FITELSON
 Set Theory, The Land, Natural number, Countable set

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