[FOM] Wildberger on Foundations
Joe Shipman
JoeShipman at aol.com
Thu Jul 19 22:01:50 EDT 2012
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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