{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

StrictRevMath110709.08

# StrictRevMath110709.08 - STRICT REVERSE MATHEMATICS by...

This preview shows pages 1–4. Sign up to view the full content.

STRICT REVERSE MATHEMATICS by Harvey M. Friedman Ohio State University November 7, 2009 Reverse Mathematics Workshop University of Chicago November 6-8, 2009

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

View Full Document
The Inevitability of Logical Strength: strict reverse mathematics. Logic Colloquium '06, ASL. October, 2009. Cambridge University Press. An extreme kind of logic skeptic claims that "the present formal systems used for the foundations of mathematics are artificially strong, thereby causing unnecessary headaches such as the Gödel incompleteness phenomena". The skeptic continues by claiming that "logician's systems always contain overly general assertions, and/or assertions about overly general notions, that are not used in any significant way in normal mathematics. For example, induction for all statements, or even all statements of certain restricted forms, is far too general - mathematicians only use induction for natural statements that actually arise. If logicians would tailor their formal systems to conform to the naturalness of normal mathematics, then various logical difficulties would disappear, and the story of the foundations of mathematics would look radically different than it does today. In particular, it should be possible to give a convincing model of actual mathematical practice that can be proved to be free of contradiction using methods that lie within what Hilbert had in mind in connection with his program”. Here we present some specific results in the direction of refuting this point of view, and introduce the Strict Reverse Mathematics (SRM) program. 1. Can we get logical strength out of strictly mathematical statements? 2. Can we build appropriate base theories out of strictly mathematical statements? 3. Can we give strictly mathematical versions of my five most basic RM systems?
I basically tried to do this in the unpublished but widely circulated papers leading up to my founding papers of the RM enterprise - the publication of the ICM address given in 1974 and the two JSL abstracts of 1976. These earlier papers were quite elaborate and evangelical. But the development was premature. They referred to some previous developments (reversals of mine over ACA) that date back to 1969. I had realized after writing these unpublished papers that matters were not in concise and attractive enough form to be founding a viable subject. So I consolidated matters and wrote those founding papers - the ICM paper used sets, as is common now, but the ASL abstracts used the most mathematical axiomatizations I could formulate at the time. More specifically, the most commonly used forms are logically equivalent to both of these early formulations (we now most commonly use sets as opposed to functions). They are considerably less mathematical than my original formulations.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}