ljdickey@water.waterloo.edu (Lee Dickey) (01/20/89)
In article <1088@l.cc.purdue.edu> cik@l.cc.purdue.edu (Herman Rubin) writes a lengthy article in which he makes many points about education and other things. But, in the middle of it, he has this: >The Peano postulates (really due to Dedekind) contain the essence of the >structure of the integers. ... [ much more deleted ] Really? I thought that the Peano Postulates pertained to counting numbers and not to integers, and that they really had been done by Peano. Can anybody confirm, deny, or supply a reference for this? (I think that Dedekind *did* contribute to the theory of real numbers through the notion that we call "Dedekind Cut". But again this is neither the counting numbers nor the integers.) -- L. J. Dickey, Faculty of Mathematics, University of Waterloo. ljdickey@WATDCS.UWaterloo.ca ljdickey@water.BITNET ljdickey@water.UUCP ..!uunet!watmath!water!ljdickey ljdickey@water.waterloo.edu
dross@umn-d-ub.D.UMN.EDU (david ross) (01/23/89)
In article <2035@water.waterloo.edu> ljdickey@water.waterloo.edu (Lee Dickey) writes: >In article <1088@l.cc.purdue.edu> cik@l.cc.purdue.edu (Herman Rubin) writes > >>The Peano postulates (really due to Dedekind) contain the essence of the >>structure of the integers. ... [ much more deleted ] > >Really? > >I thought that the Peano Postulates pertained to counting numbers >and not to integers, and that they really had been done by Peano. > Once you have the natural numbers, the integers are a triviality (in a sense that can be made mathematically precise) Peano: Sul concetto di numero, 1891 Dedekind: Stetigkeit und irrationale Zahlen, 1872 Was sind und was sollen die Zahlen?, 1888 Dedekind was just less explicit. -- _ _ _ David A. Ross (Dept.Math.&Stat.,U.ofMN,Duluth) / \/ \/ \ BITNET: dross@umndul THISNET: dross@ub.d.umn.edu / /--/--/ (...all the opinions expressed herein are facts, /__/ / \ hence they belong to nobody, least of all me...)
flash@cs.qmc.ac.uk (Flash Sheridan) (01/25/89)
In article <2035@water.waterloo.edu> ljdickey@water.waterloo.edu (Lee Dickey) writes: >In article <1088@l.cc.purdue.edu> cik@l.cc.purdue.edu (Herman Rubin) writes >>The Peano postulates (really due to Dedekind) contain the essence of the >>structure of the integers. ... [ much more deleted ] > >Really? Yes. Peano deserves zilch in the way of credit. Nothing. He'd read Was SInd und Was Zollen die Zahlen; the only difference is that he uses a weaker logic (first order, Dedekind uses 2nd). -- From: flash@cs.qmc.ac.uk (Flash Sheridan) Reply-To: sheridan@nss.cs.ucl.ac.uk Portal,MacNet: FlashsMom