Russell's paradox arises when we consider those sets that do not
belong to themselves. The collection of such sets cannot constitute
a set. Step back a bit. Logical formulas define sets (in a standard
model). Formulas, being mathematical objects, can be thought of as
sets themselves-mathematics reduces to set theory. Consider those
formulas that do not belong to the set they define. The collection
of such formulas is not definable by a formula, by the same
argument that Russell used. This quickly gives Tarski's result on
the undefinability of truth. Variations on the same idea yield the
famous results of Godel, Church, Rosser, and Post. This book gives
a full presentation of the basic incompleteness and undecidability
theorems of mathematical logic in the framework of set theory.
Corresponding results for arithmetic follow easily, and are also
given. Godel numbering is generally avoided, except when an
explicit connection is made between set theory and arithmetic. The
book assumes little technical background from the reader. One needs
mathematical ability, a general familiarity with formal logic, and
an understanding of the completeness theorem, though not its proof.
All else is developed and formally proved, from Tarski's Theorem to
Godel's Second Incompleteness Theorem. Exercises are scattered
throughout.
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