[FOM] Wildberger on Foundations
Craig Smorynski
smorynski at sbcglobal.net
Thu Jul 19 13:29:55 EDT 2012
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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