From nhj@buffalo.edu Wed Mar 21 07:49:13 2007 Received: from ares.cse.buffalo.edu (ares.cse.Buffalo.EDU [128.205.32.79]) by castor.cse.Buffalo.EDU (8.13.6/8.12.10) with ESMTP id l2LBnCxZ001274 for ; Wed, 21 Mar 2007 07:49:12 -0400 (EDT) Received: from front2.acsu.buffalo.edu (warmfront.acsu.buffalo.edu [128.205.6.88]) by ares.cse.buffalo.edu (8.13.6/8.13.6) with SMTP id l2LBnAli023001 for ; Wed, 21 Mar 2007 07:49:10 -0400 (EDT) Received: (qmail 9322 invoked from network); 21 Mar 2007 11:49:10 -0000 Received: from mailscan5.acsu.buffalo.edu (128.205.6.137) by front2.acsu.buffalo.edu with SMTP; 21 Mar 2007 11:49:10 -0000 Received: (qmail 10098 invoked by uid 60001); 21 Mar 2007 11:49:09 -0000 To: "William J. Rapaport" Subject: Re: UNCOMPUTABLE REAL NUMBERS MAY BE HAZARDOUS TO YOUR HEALTH! AND MIND! Date: Wed, 21 Mar 2007 07:49:09 -0400 From: nhj@buffalo.edu Cc: CSE584-SP07-LIST@LISTSERV.BUFFALO.EDU Message-ID: <1174477749.46011bb562f48@mail1.buffalo.edu> References: <200703210003.l2L03CQ9012173@castor.cse.Buffalo.EDU> In-Reply-To: <200703210003.l2L03CQ9012173@castor.cse.Buffalo.EDU> X-Mailer: University at Buffalo WebMail Cyrusoft SilkyMail v1.1.11 MIME-Version: 1.0 Content-Type: text/plain; charset=Big5-HKSCS Content-Transfer-Encoding: 8bit X-Originating-IP: 72.88.52.205 X-UB-Relay: (internal) X-PM-EL-Spam-Prob: X: 18% X-DCC-Buffalo.EDU-Metrics: castor.cse.Buffalo.EDU 1336; Body=0 Fuz1=0 Fuz2=0 X-Spam-Status: No, score=-1.3 required=5.0 tests=AWL,BAYES_00, MANY_EXCLAMATIONS,NO_REAL_NAME autolearn=no version=3.1.7 X-Spam-Checker-Version: SpamAssassin 3.1.7 (2006-10-05) on ares.cse.buffalo.edu X-Virus-Scanned: ClamAV 0.88.6/2891/Wed Mar 21 05:14:11 2007 on ares.cse.buffalo.edu X-Virus-Status: Clean Status: R Content-Length: 762 Sorry, I neglected to mention that d(i,j) gives the jth digit FROM THE RIGHT (so starting right next to the decimal point; d(1,2)=0; d(10,2)=1) of ith number in the list. And yes, I did mean the natural numbers, apologies again. I think the most important thing you said in your reply is that a natural number can't be infinite. I don't really see why, but if that is so (and it seems like an arbitrary statement) then one obviously couldn't have a natural number with a digit for every number in the list of natural numbers since it would have an infinite number of digits. Oh, and yes, I think my number does come out to be an infinite number of '3's which lends credence to your statement that you can't have a natural number with infinite digits. Nicolas From owner-cse584-sp07-list@LISTSERV.BUFFALO.EDU Wed Mar 21 08:54:23 2007 Received: from ares.cse.buffalo.edu (ares.cse.Buffalo.EDU [128.205.32.79]) by castor.cse.Buffalo.EDU (8.13.6/8.12.10) with ESMTP id l2LCsNNX003682 for ; Wed, 21 Mar 2007 08:54:23 -0400 (EDT) Received: from front3.acsu.buffalo.edu (coldfront.acsu.buffalo.edu [128.205.6.89]) by ares.cse.buffalo.edu (8.13.6/8.13.6) with SMTP id l2LCsK3v027171 for ; Wed, 21 Mar 2007 08:54:20 -0400 (EDT) Received: (qmail 11377 invoked from network); 21 Mar 2007 12:54:20 -0000 Received: from mailscan1.acsu.buffalo.edu (128.205.6.133) by front3.acsu.buffalo.edu with SMTP; 21 Mar 2007 12:54:20 -0000 Received: (qmail 6316 invoked from network); 21 Mar 2007 12:54:20 -0000 Received: from deliverance.acsu.buffalo.edu (128.205.7.57) by front1.acsu.buffalo.edu with SMTP; 21 Mar 2007 12:54:20 -0000 Received: (qmail 22132 invoked from network); 21 Mar 2007 12:54:09 -0000 Received: from listserv.buffalo.edu (128.205.7.35) by deliverance.acsu.buffalo.edu with SMTP; 21 Mar 2007 12:54:09 -0000 Received: by LISTSERV.BUFFALO.EDU (LISTSERV-TCP/IP release 14.5) with spool id 3987803 for CSE584-SP07-LIST@LISTSERV.BUFFALO.EDU; Wed, 21 Mar 2007 08:54:08 -0400 Delivered-To: cse584-sp07-list@listserv.buffalo.edu Received: (qmail 10712 invoked from network); 21 Mar 2007 12:54:08 -0000 Received: from mailscan1.acsu.buffalo.edu (128.205.6.133) by listserv.buffalo.edu with SMTP; 21 Mar 2007 12:54:08 -0000 Received: (qmail 19224 invoked from network); 21 Mar 2007 12:54:07 -0000 Received: from castor.cse.buffalo.edu (128.205.32.14) by smtp1.acsu.buffalo.edu with SMTP; 21 Mar 2007 12:54:07 -0000 Received: from castor.cse.Buffalo.EDU (rapaport@localhost [127.0.0.1]) by castor.cse.Buffalo.EDU (8.13.6/8.12.10) with ESMTP id l2LCs7wu003670 for ; Wed, 21 Mar 2007 08:54:07 -0400 (EDT) Received: (from rapaport@localhost) by castor.cse.Buffalo.EDU (8.13.6/8.12.9/Submit) id l2LCs7li003669 for cse584-sp07-list@listserv.buffalo.edu; Wed, 21 Mar 2007 08:54:07 -0400 (EDT) X-UB-Relay: (castor.cse.buffalo.edu) X-PM-EL-Spam-Prob: : 7% Message-ID: <200703211254.l2LCs7li003669@castor.cse.Buffalo.EDU> Date: Wed, 21 Mar 2007 08:54:07 -0400 Reply-To: "William J. Rapaport" Sender: "Philosophy of Computer Science, Spring 2007" From: "William J. Rapaport" Subject: Re: UNCOMPUTABLE REAL NUMBERS MAY BE HAZARDOUS TO YOUR HEALTH! AND MIND! To: CSE584-SP07-LIST@LISTSERV.BUFFALO.EDU Precedence: list List-Help: , List-Unsubscribe: List-Subscribe: List-Owner: List-Archive: X-UB-Relay: (castor.cse.buffalo.edu) X-DCC-Buffalo.EDU-Metrics: castor.cse.Buffalo.EDU 1336; Body=0 Fuz1=0 Fuz2=0 X-Spam-Status: No, score=-2.1 required=5.0 tests=AWL,BAYES_00, MANY_EXCLAMATIONS autolearn=ham version=3.1.7 X-Spam-Checker-Version: SpamAssassin 3.1.7 (2006-10-05) on ares.cse.buffalo.edu X-Virus-Scanned: ClamAV 0.88.6/2892/Wed Mar 21 06:40:09 2007 on ares.cse.buffalo.edu X-Virus-Status: Clean Status: R Content-Length: 1405 Nicolas writes: | Date: Wed, 21 Mar 2007 07:49:09 -0400 | From: nhj@buffalo.edu | Cc: CSE584-SP07-LIST@LISTSERV.BUFFALO.EDU | | Sorry, I neglected to mention that d(i,j) gives the jth digit FROM THE | RIGHT (so starting right next to the decimal point; d(1,2)=0; d(10,2)=1) | of ith number in the list. And yes, I did mean the natural numbers, | apologies again. I'm still a bit unclear. First of all, what decimal point? Natural numbers don't have them. (Unless you want to arbitrarily write the natural number n as n.00000 (with some finite? infinite? amount of 0s after the point).) Also, could you please give an example of how d(1,2)=0? | I think the most important thing you said in your reply is that a | natural number can't be infinite. I don't really see why,... Let me be a bit more precise. In some sense, there's no such thing as an "infinite number". There are infinite cardinal numbers: aleph-0, aleph-1, etc., which were introduced by Georg Cantor to represent, respectively, the number of natural numbers, the number of reals, etc.: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cantor.html http://mathworld.wolfram.com/Aleph-0.html http://www.alephzero.co.uk/what.htm And there are real numbers with aleph-0 digits to the right of the decimal point. What I meant when I said that natural numbers can't be infinite is that no natural number has aleph-0 digits. From ag33@cse.Buffalo.EDU Wed Mar 21 09:14:57 2007 Received: from hadar.cse.Buffalo.EDU (root@hadar.cse.Buffalo.EDU [128.205.32.1]) by castor.cse.Buffalo.EDU (8.13.6/8.12.10) with ESMTP id l2LDEvj6004690 for ; Wed, 21 Mar 2007 09:14:57 -0400 (EDT) Received: from hadar.cse.Buffalo.EDU (ag33@localhost [127.0.0.1]) by hadar.cse.Buffalo.EDU (8.13.6/8.12.10) with ESMTP id l2LDEv47003289; Wed, 21 Mar 2007 09:14:57 -0400 (EDT) Received: (from ag33@localhost) by hadar.cse.Buffalo.EDU (8.13.6/8.12.9/Submit) id l2LDEu0r003288; Wed, 21 Mar 2007 09:14:56 -0400 (EDT) Date: Wed, 21 Mar 2007 09:14:56 -0400 (EDT) From: Albert Goldfain To: "William J. Rapaport" cc: CSE584-SP07-LIST@LISTSERV.BUFFALO.EDU Subject: Re: UNCOMPUTABLE REAL NUMBERS MAY BE HAZARDOUS TO YOUR HEALTH! AND MIND! In-Reply-To: <200703211254.l2LCs7li003669@castor.cse.Buffalo.EDU> Message-ID: References: <200703211254.l2LCs7li003669@castor.cse.Buffalo.EDU> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-DCC-Buffalo.EDU-Metrics: castor.cse.Buffalo.EDU 1336; Body=0 Fuz1=0 Fuz2=0 Status: R Content-Length: 1306 > What I meant when I said that natural numbers can't be infinite is that no > natural number has aleph-0 digits. Dr. Rapaport is right in saying that any natural number can be written with finitely many digits. Since some reals are natural, there are two notational variants of natural numbers that have aleph-0 many digits. The first, as mentioned, is writing natural n as n.0000000.... The second, lesser-known way, is writing natural n as (n-1).9999999.... Don't believe me? :-) Here is a proof that 1 = 0.9999.... (read the number on the right as "point 9 repeating") First of all, 1/3 = 0.33333... Now multiply both sides by 3 3*(1/3) = 3*(0.33333...) 1 = 0.99999... QED Most people find this hard to believe...they "perceive" the number on the right as coming arbitrarily close to 1, or "approaching" 1...but these are just metaphors...the number 0.99999... is not "approaching" anything...it is "staying still"...numbers don't move :) (FYI...what people are thinking of is the series of numbers 0.9, 0.99, 0.999... This series of numbers, when plotted, approaches 1 as we move right on the x axis...however none of the numbers in this series *is* 0.99999... which is sitting happily *at* 1). This just says that 1 and 0.99999... are notational variants. Albert From owner-cse584-sp07-list@LISTSERV.BUFFALO.EDU Wed Mar 21 19:54:16 2007 Received: from ares.cse.buffalo.edu (ares.cse.Buffalo.EDU [128.205.32.79]) by castor.cse.Buffalo.EDU (8.13.6/8.12.10) with ESMTP id l2LNsGo9000740 for ; Wed, 21 Mar 2007 19:54:16 -0400 (EDT) Received: from front3.acsu.buffalo.edu (coldfront.acsu.buffalo.edu [128.205.6.89]) by ares.cse.buffalo.edu (8.13.6/8.13.6) with SMTP id l2LNsC4R074524 for ; Wed, 21 Mar 2007 19:54:12 -0400 (EDT) Received: (qmail 25357 invoked from network); 21 Mar 2007 23:54:12 -0000 Received: from mailscan5.acsu.buffalo.edu (128.205.6.137) by front3.acsu.buffalo.edu with SMTP; 21 Mar 2007 23:54:12 -0000 Received: (qmail 3798 invoked from network); 21 Mar 2007 23:54:11 -0000 Received: from deliverance.acsu.buffalo.edu (128.205.7.57) by front1.acsu.buffalo.edu with SMTP; 21 Mar 2007 23:54:11 -0000 Received: (qmail 20374 invoked from network); 21 Mar 2007 23:54:07 -0000 Received: from listserv.buffalo.edu (128.205.7.35) by deliverance.acsu.buffalo.edu with SMTP; 21 Mar 2007 23:54:07 -0000 Received: by LISTSERV.BUFFALO.EDU (LISTSERV-TCP/IP release 14.5) with spool id 4005374 for CSE584-SP07-LIST@LISTSERV.BUFFALO.EDU; Wed, 21 Mar 2007 19:54:07 -0400 Delivered-To: cse584-sp07-list@listserv.buffalo.edu Received: (qmail 10114 invoked from network); 21 Mar 2007 23:54:07 -0000 Received: from mailscan4.acsu.buffalo.edu (128.205.6.136) by listserv.buffalo.edu with SMTP; 21 Mar 2007 23:54:07 -0000 Received: (qmail 5501 invoked from network); 21 Mar 2007 23:54:06 -0000 Received: from castor.cse.buffalo.edu (128.205.32.14) by smtp4.acsu.buffalo.edu with SMTP; 21 Mar 2007 23:54:06 -0000 Received: from castor.cse.Buffalo.EDU (rapaport@localhost [127.0.0.1]) by castor.cse.Buffalo.EDU (8.13.6/8.12.10) with ESMTP id l2LNs6TF000734 for ; Wed, 21 Mar 2007 19:54:06 -0400 (EDT) Received: (from rapaport@localhost) by castor.cse.Buffalo.EDU (8.13.6/8.12.9/Submit) id l2LNs6Uv000733 for cse584-sp07-list@listserv.buffalo.edu; Wed, 21 Mar 2007 19:54:06 -0400 (EDT) X-UB-Relay: (castor.cse.buffalo.edu) X-PM-EL-Spam-Prob: : 7% Message-ID: <200703212354.l2LNs6Uv000733@castor.cse.Buffalo.EDU> Date: Wed, 21 Mar 2007 19:54:06 -0400 Reply-To: "William J. Rapaport" Sender: "Philosophy of Computer Science, Spring 2007" From: "William J. Rapaport" Subject: Peter Suber, "Logical Systems" To: CSE584-SP07-LIST@LISTSERV.BUFFALO.EDU Precedence: list List-Help: , List-Unsubscribe: List-Subscribe: List-Owner: List-Archive: X-UB-Relay: (castor.cse.buffalo.edu) X-DCC-Buffalo.EDU-Metrics: castor.cse.Buffalo.EDU 1336; Body=0 Fuz1=0 Fuz2=0 X-Spam-Status: No, score=-2.5 required=5.0 tests=AWL,BAYES_00 autolearn=unavailable version=3.1.7 X-Spam-Checker-Version: SpamAssassin 3.1.7 (2006-10-05) on ares.cse.buffalo.edu X-Virus-Scanned: ClamAV 0.88.6/2895/Wed Mar 21 19:46:31 2007 on ares.cse.buffalo.edu X-Virus-Status: Clean Status: R Content-Length: 364 ------------------------------------------------------------------------ Subject: Peter Suber, "Logical Systems" ------------------------------------------------------------------------ Here's a website that some of you might find useful and/or interesting: Peter Suber's "Logical Systems" course website http://www.earlham.edu/~peters/courses/logsys/lshome.htm