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Re: FW: data structure vs. mathematical structure (was: [Axiomdeveloper
From: 
Gabriel Dos Reis 
Subject: 
Re: FW: data structure vs. mathematical structure (was: [Axiomdeveloper] Graph theory) 
Date: 
15 Nov 2006 02:26:28 +0100 
"Page, Bill" <address@hidden> writes:
 On November 14, 2006 12:01 AM Gaby wrote:
 > 
 >  > From constructive mathematics point of view, the only things
 >  > that are required for a set are:
 >  >
 >  > (1) say how to build element of a set
 >  > (2) equality test.
 >  >
 > Bill Page wrote:
 >  No, there is a lot more to the mathematics of set than that.
 >  It would mean that all sets are finite and that is quite far
 >  from the case.

 On Tuesday, November 14, 2006 1:20 PM Gaby wrote:
 >
 > How do you arrive to that conclusion?
 >

 I thought I was stating something obvious. Are you asking why I
 think all sets would be finite if the only operation you require
 to form sets is the ability to "build an element"?
Why all sets would be finite. I hoenestly don't see how it is obvious.
 Ok, maybe there
 is also induction so that for example we could define the set of
 positive integers (natural numbers) as consisting of the element 1
 and the operation +1 that "builds a new element". So depending on
 what else you assume about the "environment", I would have to
 withdraw my claim of finiteness. But considering complexity issues
 in constructing these numbers we would be better off with at least
 one more element forming operation  doubling  leading to binary
 representation.

 However it is not clear to me that there always exists the
 possibility to define a set by it's elements. For example is this
 possible if we wish to define the set of exact real numbers such
 as suggested by your reference below?

 >  Rather, sets should be strongly related to types.
 > 
 >  I think the "constructive mathematics" that is most suitable
 >  to Axiom is probably is probably Intuitionist type theory
 >  (MartinLöf). See:
 >
 > In fact, I prefer the definition given by Erret Bishop. See
 > chapters 1 and 2 "his" book
 >
 > "Constructive Analysis",
 >
 > Erret Bishop
 > Douglas Bridges
 >
 > From my perspective, it shows a much deeper impact.
 >

 When it comes to the concept of exact real numbers I think you
 are absolutely right but is not clear to me why you would prefer
 this text in the context of set theory and Axiom. Can you explain?
I don't think Bishop is dealing with exact realy numbers, as opposed
to constructive mathametics  with analysis as an example.
I don't believe Axiom has a clear constructive notion of set theory,
so I'm not preferring anything. I'm just ignoring what Axiom does :)
Reflecting my view that the requirement of "hash" is nonmathematical.
 Exact reals for Axiom is a subject that we have previously
 discussed on this list. See:

 http://wiki.axiomdeveloper.org/RealNumbers

 Here is a recent article on this subject.

 http://www.cs.ru.nl/~herman/PUBS/exactreals.pdf

 by Herman Geuvers et al. http://www.cs.ru.nl/~herman
Thanks, I'll have a deeper look.
 Gaby
 [Axiomdeveloper] Re: FW: data structure vs. mathematical structure, (continued)
 [Axiomdeveloper] Re: FW: data structure vs. mathematical structure, Ralf Hemmecke, 2006/11/15
 Re: [Axiomdeveloper] Re: FW: data structure vs. mathematical structure, Martin Rubey, 2006/11/15
 [Axiomdeveloper] Re: FW: data structure vs. mathematical structure, Gabriel Dos Reis, 2006/11/15
 [Axiomdeveloper] Re: FW: data structure vs. mathematical structure, Ralf Hemmecke, 2006/11/15
 [Axiomdeveloper] Re: FW: data structure vs. mathematical structure, Gabriel Dos Reis, 2006/11/15
 [Axiomdeveloper] Re: FW: data structure vs. mathematical structure, Ralf Hemmecke, 2006/11/15
 [Axiomdeveloper] Re: FW: data structure vs. mathematical structure, Gabriel Dos Reis, 2006/11/15
 [Axiomdeveloper] RE: FW: data structure vs. mathematical structure, Page, Bill, 2006/11/15
 [Axiomdeveloper] Re: FW: data structure vs. mathematical structure, Gabriel Dos Reis, 2006/11/15
 Re: [Axiomdeveloper] RE: FW: data structure vs. mathematical structure, Martin Rubey, 2006/11/15
 Re: FW: data structure vs. mathematical structure (was: [Axiomdeveloper] Graph theory),
Gabriel Dos Reis <=
 Re: FW: data structure vs. mathematical structure (was: [Axiomdeveloper] Graph theory), Vanuxem Grégory, 2006/11/14