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=-0.3 required=5.0 tests=BAYES_00,FREEMAIL_FROM, LOTS_OF_MONEY,REPLYTO_WITHOUT_TO_CC autolearn=no autolearn_force=no version=3.4.4 X-Google-Language: ENGLISH,ASCII-7-bit X-Google-Thread: 103376,103b407e8b68350b X-Google-Attributes: gid103376,public X-Google-ArrivalTime: 2003-02-04 08:10:04 PST Path: archiver1.google.com!news1.google.com!newsfeed.stanford.edu!logbridge.uoregon.edu!msunews!not-for-mail From: "Chad R. Meiners" Newsgroups: comp.lang.ada Subject: Re: Anybody in US using ADA ? One silly idea.. Date: Tue, 4 Feb 2003 11:05:10 -0500 Organization: Michigan State University Message-ID: References: Reply-To: "Chad R. Meiners" NNTP-Posting-Host: arctic.cse.msu.edu X-Priority: 3 X-MSMail-Priority: Normal X-Newsreader: Microsoft Outlook Express 6.00.2720.3000 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2600.0000 Xref: archiver1.google.com comp.lang.ada:33775 Date: 2003-02-04T11:05:10-05:00 List-Id: "AG" wrote in message news:W1L%9.68847$F63.1401723@news.xtra.co.nz... > > "Grein, Christoph" wrote in message > news:mailman.10.1044267682.3911.comp.lang.ada@ada.eu.org... > > > > Do you know, an integer is definitely a real number. > > > > > > You sure? How about a simple theorem that any two distinct > > > real numbers have another real number between them? Either > > > this theorem doesn't hold (and the whole math goes out > > > the window) or the integers are most definitely not reals. > > > > > > Which will it be? > > > > Where is there the contradiction? Integers can be embedded in reals as a > subset. > > Operations available for the type and how they behave. > The theorem above was an example. Let's see: > > F(X, Y) = A > > such that for any given, finite X and Y where X < Y it > produces A that satisfies the requirement that X < A < Y > > Trivial to do for real numbers. Impossible for integers. > Which means that, at least in this respect, integers are > not a subset of reals. To be more precise, you could > argue that they are a subset or real *values* but that's > not the same thing. > > You could also define an operation F that takes two > integers and returns a generic real but that would also > mean that integers do not posess a property critical > to the definition of the real numbers. > Of course integers are a proper subset of reals. You cannot simply invent a new definition for subset ;) You are letting language confuse the issue. To reformulate your statements above Does there exist a function F(X,Y) with the signature F:R x R -> R such that for all X and Y there exists an A such that X < A < Y? Yes. and for the integers Does there exist a function F(X,Y) with the signature F:Z x Z -> Z such that for all X and Y there exists an A such that X < A < Y? No. but let's get tricky here and ask a similar question about integer numbers Does there exist a function F(X,Y) with the signature F:Z x Z -> R such that for all X and Y there exists an A such that X < A < Y? Yes. But wait those numbers are integers how dare there exist a number in-between them that belongs to the reals! ;) A number by itself has very few properties. Sets on the other hand have most of the properties you care about ;) Subsets do not necessarily have the same properties as their supersets. -CRM