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Creative mathematics

Gödel's result led to the discovery of a hierarchy of
unsolvable problems. There is no general way to solve *all*
problems even at the lowest level of this hierarchy. Yet there is
some axiom of mathematics that can solve any individual problem at
any level in the hierarchy. Biological evolution has created the
human mind which is capable of developing a set of mathematical
axioms that are very powerful and that seem intuitively obvious to
most educated mathematical minds. These axioms are at the core of
contemporary mathematics and science.

Gödel's result implies that any finite set of axioms is an infintessimal fragment of objectively true mathematics. This chapter develops the hierarchy of unsolvable problems and gives the remaining axioms of set theory that solve a significant, albeit infintessimal, fragment of these problems. The chapter ends with a philosophy of mathematical truth that connects biological evolution with the creative nature of mathematics. One crucial result is mathematical boundary conditions that would support unlimited creativity in future biological evolution and unlimited expansion of mathematics. This is vitally important because it is almost inevitable that future biological evolution will be, in large measure, consciously directed. Knowing the boundary conditions for unbounded creativity is essential to the wise use of the enormous power that science is providing us.

- Arithmetical Hierarchy
- Ordinal induction
- Axiom scheme of replacement
- Ordinal numbers
- Searching all possible paths
- Power set axiom
- Axiom of Choice
- Trees of trees
- Extending mathematics
- Is the cardinality of the reals 0 or ?
- A philosophy of mathematical truth

PDF version
of this book

**Next:** Arithmetical Hierarchy **Up:** whatrh
**Previous:** The Halting Problem **Contents**

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