<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN"> <HTML><HEAD> <META http-equiv=Content-Type content="text/html; charset=iso-8859-1"> <META content="MSHTML 6.00.6000.16809" name=GENERATOR> <STYLE></STYLE> </HEAD> <BODY bgColor=#ffffff> <DIV><FONT face="Courier New" size=2>The following formulation of the axioms of Peano is (almost exactly) copied from Hofstadter's book Goedel, Escher, Bach:</FONT></DIV> <DIV><FONT face="Courier New" size=2></FONT> </DIV> <DIV><FONT face="Courier New" size=2>1) Genie is a djinn.</FONT></DIV> <DIV><FONT face="Courier New" size=2>2) Every djinn has a meta, which is also a djinn.</FONT></DIV> <DIV><FONT face="Courier New" size=2>3) Genie is not the meta of any djinn.</FONT></DIV> <DIV><FONT face="Courier New" size=2>4) Different djinns have different metas.</FONT></DIV> <DIV><FONT face="Courier New" size=2>5) If genie has property X and each djinn with property X relays the property to its meta, then all djinns have property X.</FONT></DIV> <DIV><FONT face="Courier New" size=2></FONT> </DIV> <DIV><FONT face="Courier New" size=2>Read "djinn" as "natural number" and "meta of djinn" as "djinn + 1".</FONT></DIV> <DIV><FONT face="Courier New" size=2>You can read "Genie" as any arbitrary number, for example pi, which give us pi, pi+1, pi+2, etc.</FONT></DIV> <DIV><FONT face="Courier New" size=2></FONT> </DIV> <DIV><FONT face="Courier New" size=2>Sometimes an additional restriction is formulated: The set of djinns is the smallest set that satisfies the axioms.</FONT></DIV> <DIV><FONT face="Courier New" size=2></FONT> </DIV> <DIV><FONT face="Courier New" size=2>When using natural numbers for counting, I agree with Matthias that the most natural starting point is 0. You get odd arithmetics when not starting with 0. For example in music a quart (4) *plus* a quint (5) is an octave (8). You dont want 4+5=8, I think (it should be 3+4=7). Apparently the nomenclature of intervals in music were formulated before the notion of number zero was discovered/invented. </FONT></DIV> <DIV><FONT face="Courier New" size=2>(For who it concerns: I have put 'plus' between asterixes, because when dealing with frequencies, you must multiply ratios of frequencies, not add them :)</FONT></DIV> <DIV><FONT face="Courier New" size=2></FONT> </DIV> <DIV><FONT face="Courier New" size=2>Jos</FONT></DIV> <BLOCKQUOTE style="PADDING-RIGHT: 0px; PADDING-LEFT: 5px; MARGIN-LEFT: 5px; BORDER-LEFT: #000000 2px solid; MARGIN-RIGHT: 0px"> <DIV style="FONT: 10pt arial">----- Original Message ----- </DIV> <DIV style="BACKGROUND: #e4e4e4; FONT: 10pt arial; font-color: black"><B>From:</B> <A title=adityashukla1983@gmail.com href="mailto:adityashukla1983@gmail.com">aditya shukla</A> </DIV> <DIV style="FONT: 10pt arial"><B>To:</B> <A title=plt-scheme@list.cs.brown.edu href="mailto:plt-scheme@list.cs.brown.edu">PLT Scheme ML</A> </DIV> <DIV style="FONT: 10pt arial"><B>Sent:</B> Thursday, March 12, 2009 12:19 AM</DIV> <DIV style="FONT: 10pt arial"><B>Subject:</B> [plt-scheme] Natural numbers</DIV> <DIV><BR></DIV>Can someone please explain why natural numbers in chapter 11 of htdp begin with 0 ? <BR><BR>Aditya<BR> <P> <HR> <P></P>_________________________________________________<BR>  For list-related administrative tasks:<BR>  <A href="http://list.cs.brown.edu/mailman/listinfo/plt-scheme">http://list.cs.brown.edu/mailman/listinfo/plt-scheme</A><BR></BLOCKQUOTE></BODY></HTML>