Fortyyears110508

Fortyyears110508 - 1 MY FORTY YEARS ON HIS SHOULDERS by...

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1 MY FORTY YEARS ON HIS SHOULDERS by Harvey M. Friedman* Ohio State University Gödel Centenary Delivered: April 29, 2006 Revised November 5, 2008 1. General Remarks. 2. The Completeness Theorem. 3. The First Incompleteness Theorem. 4. The Second Incompleteness Theorem. 5. Lengths of Proofs. 6. The Negative Interpretation. 7. The Axiom of Choice and the Continuum Hypothesis. 8. Wqo Theory. 9. Borel Selection. 10. Boolean Relation Theory. 11. Finite Relations. 12. Incompleteness in the Future. We wish to thank Warren Goldfarb and Hilary Putnam for help with several historical points. 1. GENERAL REMARKS Gödel's legacy is still very much in evidence. His legacy is overwhelmingly decisive, particularly in the arena of general mathematical and philosophical inquiry.
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2 The extent of Gödel's impact in the more restricted domain of mathematical practice is more open to question. In fact, there is an in depth assessment of this impact in Macintyre 2009. But even in this comparatively specialized domain, Gödel's impact is seen to be substantial. As indicated here, particularly in section 12, we believe that the potential impact of Gödel's work on mathematical practice is also overwhelming. However, the full realization of this potential impact will have to wait for some new breakthroughs. We have every confidence that these breakthroughs will materialize. Generally speaking, current mathematical practice has now become very far removed from general mathematical and philosophical inquiry, where Gödel's legacy is most decisively overwhelming. However, there are some signs that some of our most distinguished mathematicians recognize the need for some sort of reconciliation. Here is a quote from Atiyah M. 2008b: "Mathematicians took the role of philosophers, but I want to bring the philosophers back in. I hope someday we will be able to explain mathematics in a philosophical way using philosophical methods". We will not attempt to properly discuss the full impact of Gödel's work and all of the ongoing important research programs that it suggests. This would require a book length manuscript. Indeed, there are several books discussing the Gödel legacy from many points of view, including, for example, (Wang 1987, 1996), (Dawson 2005), and the historically comprehensive five volume set (Gödel 1986-2003). In sections 2-7 we briefly discuss some research projects that are suggested by some of his most famous contributions.
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3 In sections 8-11 we discuss some highlights of a main recurrent theme in our own research, which amounts to an expansion of the Gödel incompleteness phenomena in new critical directions. The incompleteness phenomena lie at the heart of the Gödel legacy. Some careful formulations, informed by some post Gödelian developments, are presented in sections 3,4,5. One particular issue that arises with regard to incompleteness has been a
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Fortyyears110508 - 1 MY FORTY YEARS ON HIS SHOULDERS by...

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