Mathematical logic grew out of philosophical
questions regarding the foundations of mathematics, but
logic has now outgrown its philosophical roots, and has
become an integral part of mathematics in general. This
book is designed for students who plan to specialize in
logic, as well as for those who are interested in the
applications of logic to other areas of mathematics.
Used as a text, it could form the basis of a beginning
graduate-level course. There are three main chapters:
Set Theory, Model Theory, and Recursion Theory. The Set
Theory chapter describes the set-theoretic foundations
of all of mathematics, based on the ZFC axioms. It also
covers technical results about the Axiom of Choice,
well-orderings, and the theory of uncountable cardinals.
The Model Theory chapter discusses predicate logic and
formal proofs, and covers the Completeness, Compactness,
and L wenheim-Skolem Theorems, elementary submodels,
model completeness, and applications to algebra. This
chapter also continues the foundational issues begun in
the set theory chapter. Mathematics can now be viewed as
formal proofs from ZFC. Also, model theory leads to
models of set theory. This includes a discussion of
absoluteness, and an analysis of models such as H( ) and
R( ). The Recursion Theory chapter develops some basic
facts about computable functions, and uses them to prove
a number of results of foundational importance; in
particular, Church's theorem on the undecidability of
logical consequence, the incompleteness theorems of G
del, and Tarski's theorem on the non-definability of
truth. |
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