[FOM] Wildberger on Foundations

[Joe Shipman]

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 pair
> Calculus with Analytic Geometry
> Vector Calculus and Differential Equations
> by 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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