{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

GodelLegPanelNij080906

# GodelLegPanelNij080906 - 1 GODELS LEGACY IN MATHEMATICAL...

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

1 GODEL’S LEGACY IN MATHEMATICAL PHILOSOPHY by Harvey M. Friedman Ohio State University LC ’06 Gödel Legacy Panel Nijmegen, Netherlands Presented August 2, 2006 Revised August 9, 2006 NOTE: This is an edited version of my 20 minute lecture at the Gödel legacy panel of the LC ‘06. Gödel's definitive results and his essays leave us with a rich legacy of philosophical programs that promise to be subject to mathematical treatment. After surveying some of these, we focus attention on the program of circumventing his demonstrated impossibility of a consistency proof for mathematics by means of extramathematical concepts. This program has seen substantial progress through our new Concept Calculus. 1. INCOMPLETENESS THEOREMS. Gödel’s First Incompleteness theorem asserts that any consistent formal system obeying rather weak conditions, such as PA = Peano Arithmetic, is incomplete. Gödel’s Second Incompleteness theorem asserts that any consistent formal system obeying rather weak conditions, equipped with a “standard” formalized proof predicate with which to formulate its own consistency, cannot prove its own consistency. This suggests a number of obviously important programs, which are, to various degrees, implicit and explicit in Gödel’s writings, and certainly well known to him. 1a. What mathematics can be formalized in a way that is not subject to the First Incompleteness theorem, so that one has completeness? Perhaps the most well known examples of real depth are the complete axiomatizations of the ordered group of integers

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

View Full Document
2 (Presburger) and the ordered field of reals (Tarski) and Euclidean geometry (Tarski). I think that there is room for new dramatic results where complete axiomatizations of natural significant portions of mathematics are given. However, many of these will likely involve rather imaginative delineations that are not simply the full first order theory of some fundamental structures, as was the case with Presburger and Tarski.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern