Any book that revisits the foundations of analysis has to reckon with the formidable precedent of Edmund Landau's Grundlagen der Analysis (Foundations of Analysis) of 1930. Indeed, the influence of Landau's book is probably the reason that so few books since 1930 have even been attempted to include the construction of the real numbers in an introduction to analysis. On the other hand, Landau's account is virtually the last word in rigor. The only way to be more rigorous would be to rewrite Landau's proofs in computer-checkable form—which has in fact been done recently. On the other hand Landau's book is almost pathologically reader-unfriendly. ...While memories of Landau still linger, so too does fear of the real numbers. In my opinion, the problem with Landau's book is not so much the rigor (though it is excessive), but the lack of background, history, examples, and explanatory remarks. Also, the fact that he does nothing with the real numbers except construct them. In short, it could be an entirely different story if it were explained that the real numbers are interesting! That is what I have tried to do....

What It Means

This is a placeholder explanation generated by the low-overhead model. It interprets "Any book that revisits the foundations of analysis has to reckon with the formidable precedent of Edmund Landau's Grundlagen der Analysis (Foundations of Analysis) of 1930. Indeed, the influence of Landau's book is probably the reason that so few books since 1930 have even been attempted to include the construction of the real numbers in an introduction to analysis. On the other hand, Landau's account is virtually the last word in rigor. The only way to be more rigorous would be to rewrite Landau's proofs in computer-checkable form—which has in fact been done recently. On the other hand Landau's book is almost pathologically reader-unfriendly. ...While memories of Landau still linger, so too does fear of the real numbers. In my opinion, the problem with Landau's book is not so much the rigor (though it is excessive), but the lack of background, history, examples, and explanatory remarks. Also, the fact that he does nothing with the real numbers except construct them. In short, it could be an entirely different story if it were explained that the real numbers are interesting! That is what I have tried to do...." in the context of "John Stillwell, The Real Numbers: An Introduction to Set Theory and Analysis (2013)" to mean that wisdom is timeless.