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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. Some 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 5 and 6 we describe the structure of the mathematical hierarchy of self reflecting structures and possible approaches to extending it.

Chapters 5 and 6 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 5.8 contains a sketch of a proof of a limited version of Gödel's Incompleteness Theorem called the Halting Problem. Chapter 6 speculates about extending mathematics in light of Gödel's result.

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