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From: Bruno Marchal <marchal.domain.name.hidden>

Date: Mon, 10 Oct 2005 16:51:28 +0200

Hi Tom,

Le 06-oct.-05, à 19:43, daddycaylor.domain.name.hidden 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!).

Bruno

http://iridia.ulb.ac.be/~marchal/

Received on Mon Oct 10 2005 - 10:54:07 PDT

Date: Mon, 10 Oct 2005 16:51:28 +0200

Hi Tom,

Le 06-oct.-05, à 19:43, daddycaylor.domain.name.hidden a écrit :

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!).

Bruno

http://iridia.ulb.ac.be/~marchal/

Received on Mon Oct 10 2005 - 10:54:07 PDT

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