[FOM] soliciting help with a precise definition
William.Piper@colorado.edu
William.Piper at colorado.edu
Tue Jun 22 17:14:25 EDT 2004
Hello all,
I am soliciting ideas and advice concerning a rigorous development of the
concept "absolutely undecidable proposition."
After many hours of research I've reached the conclusion that a more or less
definite understanding of the concept of absolutely undecidable proposition
does not yet exist. The lack of a more or less definite conception contrasts
the many, many references it; Post, Mostowski, Feferman and many others allude
to this concept and yet do not offer a definition of any sort, just some
informal hand-waving. Some of those just mentioned actually stress the
(philosphical) importance of establishing whether absolutely undecidable
propostions of mathematics exist.
A rigorous definition of this concept should not make appeal to higher-order
logics, yet correspound to our intuitions concerning the respective notions
of "undecidability" and "absolute." I want to stress that I am not asking for a
decision algorithm or effective procedure, just what it MEANS to say that p is a
proposition of mathematics which is absolutely undecided (of course, we want to
exempt trivial cases and a precise notion of triviality in this context is also
needed).
It has also been pointed out to me that the notion of "absolute" in the
context of human understanding may be the source for the lack of a precise
definition. Although I am skeptical that a purely logical definition can be
given, my aim is to approximate our intuitions about the notion of "absolutely
undecidable" relying as heavily as possible on a purely logical
characterization.
If anyone has any references concerning this particular aim, I would greatly
appreciate it. Also, if any of you has comments on the possibility of acheiving
this aim or helpful suggestions on more (or less) stringent requirements for a
rigorous definition, please feel free to contact me.
Everett
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