Le 06-oct.-05, à 19:43, [EMAIL PROTECTED] a écrit :
I've been looking a little into what there is on-line about
descriptive set theory, a relatively new field.
It seems that with the questions about cardinality and descriptions on
this list, that descriptive set theory (Polish spaces being an
important element) would be useful, if not essential.
A search of this list doesn't turn up any references to it. Does
anyone have enough knowledge of it to give a brief note on how it ties
in with this list's discussion?
Descriptive set theory can be used in the foundations of analysis. The
idea consists in using some nice subsets of the reals so as to avoid
conceptual difficulties and keeping powerful tools in analysis.
Actually I have used descriptive set theory in my first attempts to
tackle the measure problem pertaining on the first person observer
moments (where Kripke models fails). Some people have used it also in
computational learning theory. I have worked hard to eliminate the use
of descriptive set theory if only because to use them in comp you need
some stronger from of Church thesis (but this makes them fruitful in
some non-comp approach). Now, honestly, from I can judge about the
knowledge of logic in this list, descriptive theory (which quantifies
on both the natural numbers and the reals) is far too technical a
subject so that it can be use easily.
I'm a bit busy to say much more, but perhaps you have a good intuition
because if you describe directly the set of infinite path (histories)
on which the 1-measure pertains, you cannot escape the "analytical
hierarchy", the "hyperarithmetic sets", etc. But then I am happy of
having find a way to single out the logic of comp-certainty without
addressing the need to classify mathematically those infinite path.
To sum up, the use of descriptive set theory seems to me premature,
although unavoidable for future work on the measure and probability
questions on OMs.
If you are interested, a good book on the subject is the Oxford Logic
Guides 11: "Recursive Aspects of Descriptive Set Theory" by Richard
Mansfield and Galen Weitkamp, 1985.
Prerequisites: the whole of Rogers' book (ref in my thesis). For my
thesis you need to understand about the half of Rogers book (the
easiest part I would say).
But, you know, with comp, we can expect that the whole of mathematical
logic can be of some use soon or later. Mathematical Logic is the
"philosophical logic" of the Platonists!
(But please don't repeat this to a mathematical logician!).