From mboxrd@z Thu Jan 1 00:00:00 1970 X-Spam-Checker-Version: SpamAssassin 3.4.4 (2020-01-24) on polar.synack.me X-Spam-Level: X-Spam-Status: No, score=-0.9 required=5.0 tests=BAYES_00,FORGED_GMAIL_RCVD, FREEMAIL_FROM autolearn=no autolearn_force=no version=3.4.4 X-Google-Thread: 103376,640b65cbfbab7216 X-Google-Attributes: gid103376,domainid0,public,usenet X-Google-Language: ENGLISH,ASCII-7-bit Path: g2news1.google.com!postnews.google.com!s13g2000prd.googlegroups.com!not-for-mail From: Eric Hughes Newsgroups: comp.lang.ada Subject: Re: Ada.Strings.Bounded Date: Mon, 14 Apr 2008 19:07:25 -0700 (PDT) Organization: http://groups.google.com Message-ID: <80c6fdca-1a89-4d98-b61d-9a405e57d8e5@s13g2000prd.googlegroups.com> References: <44d88b93-6a90-4c18-8785-2164934ba700@a9g2000prl.googlegroups.com> <47F652F7.9050502@obry.net> <47f7028d$1_6@news.bluewin.ch> <47F749CB.30806@obry.net> <96x8my4o4m7e.fskzcb6i31ty$.dlg@40tude.net> <276e98e3-3b3b-4cbf-b85c-dcae79f11ec5@b5g2000pri.googlegroups.com> <013e1d52-c25f-49ea-83ef-6ac4860858bf@s13g2000prd.googlegroups.com> <8g2rpvi2ahu0$.1ebsyq5yu1whf.dlg@40tude.net> <9a3ad8ca-9f44-42db-9f7c-c5f9e3ee60f3@w1g2000prd.googlegroups.com> <1jdzw15tbj376$.nyv9yml75wj4$.dlg@40tude.net> NNTP-Posting-Host: 166.70.57.218 Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: posting.google.com 1208225245 13762 127.0.0.1 (15 Apr 2008 02:07:25 GMT) X-Complaints-To: groups-abuse@google.com NNTP-Posting-Date: Tue, 15 Apr 2008 02:07:25 +0000 (UTC) Complaints-To: groups-abuse@google.com Injection-Info: s13g2000prd.googlegroups.com; posting-host=166.70.57.218; posting-account=5RIiTwoAAACt_Eu87gmPAJMoMTeMz-rn User-Agent: G2/1.0 X-HTTP-UserAgent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.8.1.13) Gecko/20080311 Firefox/2.0.0.13,gzip(gfe),gzip(gfe) Xref: g2news1.google.com comp.lang.ada:20949 Date: 2008-04-14T19:07:25-07:00 List-Id: On Apr 14, 12:52 pm, "Dmitry A. Kazakov" wrote: > I think it is wrong to consider N and universal_integer equivalent. Sure. It's {\bb Z} and universal_integer that are equivalent. Seriously, we just disagree about this. I can't take universal_integer seriously as a root class, because it's impossible to write down any representation of it. I believe the approach I've been thinking about could provide some reasonably solid grounding for what universal integer is. > Subseting is not a sufficient condition for a > successful modeling. In a discussion that's got a lot of formal logic in it, the word "model" already means something pretty specific, usually involving a Tarski structure. On the other hand, the informal use of the word "model", in this context, is basically beside the point, which is to get a precise definition of a hypothetical universal real type, amongst others. There are plenty of useful things that are not-quite- real numbers, such as the one- and two-point compactifications of the real line, but these are the same thing as real numbers. If you want them, fine; just don't try to claim that they are same thing and use a confusing name for them. > Actually it is the opposite, a perfect subset cannot > be a good model. There was paper from the fifties (sorry, no reference handy), which used Turing machines to compute a Dedekind cut. On input a rational number, it returned one of the two symbols "<=x" or ">=x". (You cannot compute exact trichotomy without solving the halting problem.) In any language describing these machines, there's a least one (Kleene minimization), so there's a unique representative of such a machine for every computable real number, which means there's a subset bijection. Addition, multiplication, and their inverses are defined in terms of the underlying operand-machine (it's pretty easy coding, actually). So there's pretty close to a perfect model of the real numbers, whose only real limitation is that run times are horrendously slow. But it's also completely exact, with no compromises but execution speed and representation size of a machine. The subset is every real number that's computable. About as good as you can do with computers, I'd say. > The best models of R aren't even close to subsets. For > example intervals with rational bounds. In the precise meaning of model, it's just not a model, because there's no total ordering on such intervals, so the ordering axioms are not satisfied. In an imprecise meaning, it's real numbers plus some other concept, which is more than {\bb R}. Eric