Goedel's Incompleteness Theorems are among the most significant
results in the foundation of mathematics. These results have a
positive consequence: any system of axioms for mathematics that we
recognize as correct can be properly extended by adding as a new
axiom a formal statement expressing that the original system is
consistent. This suggests that our mathematical knowledge is
inexhaustible, an essentially philosophical topic to which this
book is devoted. Basic material in predicate logic, set theory and
recursion theory is presented, leading to a proof of incompleteness
theorems. The inexhaustibility of mathematical knowledge is treated
based on the concept of transfinite progressions of theories as
conceived by Turing and Feferman. All concepts and results
necessary to understand the arguments are introduced as needed,
making the presentation self-contained and thorough.
General
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