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2015-2: Analysis Zero
by Anthony G. O'Farrell.

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Anthony G. O'Farrell was born in Dublin, Ireland, and grew up in County Tipperary before returning to Dublin. He worked in the Irish Meteorological Service from 1964 to 1968. For most of that time he was a student of Mathematical Science at University College, Dublin, where he was particularly influenced by P.G. Gormley, who turned him decisively to analysis, particularly complex analysis. Resigning from the Meteorological Service to devote himself to Mathematics, he was further influenced by T. J. Laffey and E. C. Schlesinger, and decided to continue his studies in the United States. At Brown University, his horizons were expanded by H. Federer, B. Harris, A. Browder, J. Wermer, B. Cole, W. Fulton and A. Landman, among others, and his subsequent work focussed mainly on algebraic and geometric aspects of real and complex analysis. In 1975, after two years at the University of California at Los Angeles, he was appointed Professor of Mathematics at Maynooth College, He was elected to the Royal Irish Academy in 1980. While engaged in teaching, research and administration at Maynooth, he contributed to the mathematical and scientific community in Ireland and abroad, and visited research institutes and universities in Canada, France, Germany, India, Israel, Japan, Russia, Sweden, Spain, the UK and USA, as well as speaking at many international conferences and collaborating with many researchers.

Now Professor Emeritus of Maynooth University, he devotes himself to research, writing and pro-bono activities, as well as a modicum of outdoor activity and cultural pursuits. He is married, with three surviving children and seven grandchildren.

From the Preface:

This book leads up to the starting-point of a rigorous course in basic real analysis. It is designed to satisfy a student who wants to start 'further back' than the axioms of a complete ordered field. Typically, such a student will be in the second or higher year at University, and will have attained some level of mathematical maturity.

We present a foundation in set theory, and build up through the natural numbers, integers, rational numbers, real and complex numbers, and we establish their properties on the basis of some more basic axioms.

We try to avoid various extremes: The theory is not a 'formal theory', in the strict, logical sense. There are no axioms for the logic used. It is thus a rigorous theory for actual people, not for automata.

It does not try to do the impossible. You have to understand that, because of the work of twentieth-century logicians, we know that we cannot place mathematics on the kind of foundation of which David Hilbert dreamed. That is, it is impossible to give an account of analysis (or of any other sufficiently-rich mathematical theory) which is provably consistent. The only comfort I can offer is that if the theory is inconsistent, then it is already inconsistent by the end of the chapter about the natural numbers,

We do not get involved in the technical discussion of mathematical logic.

We avoid doing 'clever' stuff that violates common-sense notions. For instance, we do not insist that every object under discussion be a set (or even a class).

Some of the ideas and concepts introduced are purely auxiliary, and you can forget about them once you have worked through this book. Indeed, the entire book is designed to be read in a week or two, and then forgotten, unless you are one of the people who want to dig deeper still, and get really serious about logic.

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Updated 11-7-2017
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