[FOM] Wildberger on Foundations

[elliott mendelson]

Dear Professor Shipman:                                    Another reference book on the foundations of analysis is my book Number Systems and the Foundations of Analysis, now published by Dover Publications, 2008, ISBN-13: 978-0-486-45792-5 and ISBN-10: 0-48645792-3.Sincerely yours,Elliott Mendelsonemenqc at msn.com

From: JoeShipman at aol.com
Date: Thu, 19 Jul 2012 22:01:50 -0400
To: fom at cs.nyu.edu
Subject: Re: [FOM] Wildberger on Foundations

I look forward to your upcoming Treatise on the Binomial Theorem; should we anticipate a sequel on the Dynamics of an Asteroid? -- JS

Sent from my iPhone
On Jul 19, 2012, at 1:29 PM, Craig Smorynski <smorynski at sbcglobal.net> wrote:

When I was a student there was a very rigorous calculus textbook by Johnson and Kiokemeister. More to my liking were the pairCalculus with Analytic GeometryVector Calculus and Differential Equationsby Albert G. Fadell, both published by van Nostrand (1964 and 1968, respectively).
On the matter of the foundations of the real number line, I might note that I give an exhaustive treatment in my Adventures in Formalism, discussing treatments by Bolzano, Weierstrass (not quite so exhaustive), Dedekind (overly detailed), and Heine-Cantor-Meray using Cauchy sequences. (End of advertisement.)
Also by way of an advertisement, I might mention my upcoming A Treatise on the Binomial Theorem in which I discuss the development of rigour as it was needed to provide a genuine proof of Newton's binomial theorem, first with complete rigour by Bolzano and then Cauchy and finally almost completely rigorously by Abel.
On Jul 17, 2012, at 6:05 AM, Arnon Avron wrote:On Wed, Jul 11, 2012 at 12:51:02PM -0400, joeshipman at aol.com wrote:

In his discussion with me, he asks for examples of texts where the
modern framework of Analysis is developed completely rigorously from
first principles. 

Can anyone suggest some source books that might satisfy his request?

Here are two books which were used as the main textbooks in undergrduate 
courses I took about 40 years ago in Tel-Aviv university, and come
close to this ideal:

G. M. Fikhtengol'ts: The fundamentals of Mathematical Analysis 

  This is the book from which I have learned Analysis. It
  starts with a rigorous  introduction of the real numbers as 
  Dedekind cuts, and continue to provide rigorous definitions and proofs 
  in both of its two comprehensive volumes. It does not provide a list of 
  "basic principles", though.

J. Dugundji: Topology   

   This book is not a book in analysis. However, it is relevant here
  because it is almost fully self-contained. It starts from elementary 
  set theorys, and it  even provides a full list of axioms (GB in an 
  informal form).

And I should mention of course also Feferman's classic book on the
number systems.


Arnon Avron


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Craig


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