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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.
"Among the many expositions of G del's incompleteness theorems
written for non-specialists, this book stands apart. With
exceptional clarity, Franz n gives careful, non-technical
explanations both of what those theorems say and, more importantly,
what they do not. No other book aims, as his does, to address in
detail the misunderstandings and abuses of the incompleteness
theorems that are so rife in popular discussions of their
significance. As an antidote to the many spurious appeals to
incompleteness in theological, anti-mechanist and post-modernist
debates, it is a valuable addition to the literature." --- John W.
Dawson, author of "Logical Dilemmas: The Life and Work of Kurt G
del"
"Among the many expositions of Goedel's incompleteness theorems
written for non-specialists, this book stands apart. With
exceptional clarity, Franzen gives careful, non-technical
explanations both of what those theorems say and, more importantly,
what they do not. No other book aims, as his does, to address in
detail the misunderstandings and abuses of the incompleteness
theorems that are so rife in popular discussions of their
significance. As an antidote to the many spurious appeals to
incompleteness in theological, anti-mechanist and post-modernist
debates, it is a valuable addition to the literature." --- John W.
Dawson, author of Logical Dilemmas: The Life and Work of Kurt
Goedel
Godel'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."
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