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Gödel's shock

This is not the view of most mathematicians. Gödel's result was and remains a shock to the mathematical community that sees mathematical truth as the one absolute certainty in a confusing world. Many mathematicians believe intuition about infinite sets borders on the mystical. By asserting the existence of complex infinite sets one can indirectly define levels in the hierarchy of mathematical truth that are difficult to approach in other ways. This suggests to some that mathematical intuition can transcend the limits Gödel's theorem imposes on any single path approach to extending mathematics. In Chapters 3 and 4 we describe the structure of the mathematical hierarchy of self reflecting structures and possible approaches to extending it. We argue that there are more powerful approaches to exploring mathematical truth that have no need to transcend the limits of Gödel's theorem with mystical intuition.

Chapters 3 and 4 develop in an intuitive and semi-formal way the basics of formal set theory. This is done in terms of properties of logically determined sequence of events in a potentially infinite universe. Computer programs serve as an effective model of such processes. Developing mathematics in this way makes it more concrete and intuitive. Section 3.8 contains a sketch of a proof of a limited version Gödel's Incompleteness Theorem called the Halting Problem. Chapter 4 speculates about extending mathematics in light of Gödel's result.

Completed second draft of this book

PDF version of this book