{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Fortyyears052407 - 1 MY FORTY YEARS ON HIS SHOULDERS by...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
1 MY FORTY YEARS ON HIS SHOULDERS by Harvey M. Friedman* Ohio State University Gödel Centenary Delivered: April 29, 2007 Expanded: May 24, 2007 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 Graphs. 12. Incompleteness in the Future. 1. GENERAL REMARKS. Gödel's legacy is still very much in evidence. We will not attempt to properly discuss the full impact of his 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, [Wa87], [Wa96], [Da05], and the historically comprehensive five volume set [Go,86-03]. In sections 2-7 we briefly discuss some research projects that are suggested by some of his most famous contributions. 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 phenomenon in a critical direction. The incompleteness phenomenon lies at the heart of the Gödel legacy. Some careful formulations, informed by some post Gödelian developments, are presented in sections 3,4,5.
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
2 One particular issue that arises with regard to incompleteness has been a driving force for a considerable portion of my work over the last forty years. This has been the ongoing search for necessary uses of set theoretic methods in normal mathematics. By way of background, Gödel’s first incompleteness theorem is an existence theorem not intended to provide a mathematically intelligible example of an unprovable sentence. Gödel’s second incompleteness theorem does provide an entirely intelligible example of an unprovable sentence - specifically, the crucially important consistency statement. (Remarkably, Gödel demonstrates by a brief semiformal argument, that the sentence he constructs for his first incompleteness theorem is demonstrably equivalent to the consistency statement - hence the consistency statement is not provable.) Nevertheless, the consistency statement is obviously of a logical nature rather than of a mathematical nature. This is a distinction that is readily noticed by the general mathematical community, which naturally resists the notion that the incompleteness theorem will have practical consequences for their own research. Genuinely mathematical examples of incompleteness from substantial set theoretic systems had to wait until the well known work on the axiom of choice and the continuum hypothesis by Kurt Gödel and Paul Cohen. See [Go40], [Co63].
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern