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a b Jorgenson, Jay; Krantz, Steven G., eds. (May 2006). "Serge Lang, 1927–2005" (PDF). Notices of the American Mathematical Society. 53 (5): 536–553. Even then, one might say that full rigor hasn't been achieved until one has presented a way to formalize the concept of proof to such an extent that the correctness of any given proof can be verified by an algorithm. (Such formal proofs are almost unreadable to humans because they are full of symbols.) This is accomplished in books on logic.

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Lang, Serge (1999). Fundamentals of differential geometry. Graduate Texts in Mathematics. Vol.191. New York: Springer-Verlag. doi: 10.1007/978-1-4612-0541-8. ISBN 0-387-98593-X. MR 1666820. This book is the fourth edition, previously published under the different titles of Introduction to Differentiable Manifolds (1962), Differential Manifolds (1972), and Differential and Riemannian Manifolds (1995). Lang also published a distinct second edition (preserving the title of the 1962 original) so as to provide a companion volume to Fundamentals of Differential Geometry which covers a portion of the same material, but with the more elementary exposition confined to finite-dimensional manifolds: Fulton, William; Lang, Serge (1985). Riemann–Roch algebra. Grundlehren der mathematischen Wissenschaften. Vol.277. New York: Springer-Verlag. doi: 10.1007/978-1-4757-1858-4. ISBN 978-1-4419-3073-6. MR 0801033.Jorgenson, Jay; Lang, Serge (2005). Posn(R) and Eisenstein Series. Lecture Notes in Mathematics. Vol.1868. Berlin: Springer-Verlag. doi: 10.1007/b136063. ISBN 978-3-540-25787-5. MR 2166237. On the question of whether Lang will prepare you adequately for Spivak, the answer is maybe. It's better than most books for this, because it does devote considerable attention to proofs. On the other hand, it doesn't develop a high level of computational skill in important areas. That's one reason why in my answer here

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Lang, Serge (1996). Topics in Cohomology of Groups. Lecture Notes in Mathematics. Vol.1625. Chapter X based on letters written by John Tate (Translated from the 1967 French originaled.). Berlin: Springer-Verlag. doi: 10.1007/BFb0092624. ISBN 3-540-61181-9. MR 1439508. [27] Lang, Serge (1986). A first course in calculus. Undergraduate Texts in Mathematics (Fifth edition of 1964 originaled.). New York: Springer-Verlag. doi: 10.1007/978-1-4419-8532-3. ISBN 978-1-4612-6428-6. The 1964 first edition was reprinted as: Lang, Serge (1987). Elliptic functions. Graduate Texts in Mathematics. Vol.112. With an appendix by J. Tate (Second edition of 1973 originaled.). New York: Springer-Verlag. doi: 10.1007/978-1-4612-4752-4. ISBN 0-387-96508-4. MR 0890960. [22] Lang spent much of his professional time engaged in political activism. He was a staunch socialist and active in opposition to the Vietnam War, volunteering for the 1966 anti-war campaign of Robert Scheer (the subject of his book The Scheer Campaign). [ citation needed] Lang later quit his position at Columbia in 1971 in protest over the university's treatment of anti-war protesters. With all of this being said, if you're eager to start with Spivak, there's no problem in just starting and seeing how successful you are on the exercises in the first few chapters.Terras, Audrey (1980). "Review: Introduction to modular forms, by Serge Lang" (PDF). Bull. Amer. Math. Soc. (N.S.). 2 (1): 206–214. doi: 10.1090/s0273-0979-1980-14722-9. I went through most of this book due to MicroMass's recommendation a little while ago, and really enjoyed my first introduction to real mathematics, and not the garbage I was/am being taught in middle/high school. Lang, Serge (1994). Algebraic number theory. Graduate Texts in Mathematics. Vol.110 (Second edition of 1970 originaled.). New York: Springer-Verlag. doi: 10.1007/978-1-4612-0853-2. ISBN 0-387-94225-4. MR 1282723. [25] The first edition was itself the second edition of Algebraic Numbers (1964) Kubert, Daniel S.; Lang, Serge (1981). Modular units. Grundlehren der Mathematischen Wissenschaften. Vol.244. New York–Berlin: Springer-Verlag. doi: 10.1007/978-1-4757-1741-9. ISBN 0-387-90517-0. MR 0648603. Lang, Serge; Murrow, Gene (1988). Geometry: a high school course. New York: Springer-Verlag. doi: 10.1007/978-1-4757-2022-8. ISBN 978-0-387-96654-0.

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Lang, Serge (1995). Introduction to Diophantine approximations (Second edition of 1966 originaled.). New York: Springer-Verlag. doi: 10.1007/978-1-4612-4220-8. ISBN 0-387-94456-7. MR 1348400.Lang, Serge (1993). Real and functional analysis. Graduate Texts in Mathematics. Vol.142 (Thirded.). New York: Springer-Verlag. doi: 10.1007/978-1-4612-0897-6. ISBN 0-387-94001-4. MR 1216137. This book is the third edition, previously published under the different titles of Analysis II (1968) and Real Analysis (1983) Shakarchi, Rami (1996). Solutions manual for Lang's "Linear Algebra". New York: Springer-Verlag. doi: 10.1007/978-1-4612-0755-9. ISBN 0-387-94760-4. MR 1415837.

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Lang, Serge (2000). Collected papers. I. 1952–1970. Springer Collected Works in Mathematics. New York: Springer. ISBN 0-387-98802-5. MR 1772967. Lang, Serge (1997). Undergraduate analysis. Undergraduate Texts in Mathematics (Seconded.). New York: Springer-Verlag. doi: 10.1007/978-1-4757-2698-5. ISBN 0-387-94841-4. MR 1476913. The first edition (1983) of this title was itself the second edition of Analysis I (1968) Edit: I forgot to answer the part about trigonometry. Plane trigonometry can be divided roughly into two parts: (1) geometric applications to triangles, quadrilaterals, etc.; (2) analytic trigonometry, which involves various algebraic manipulations of trigonometric functions. For the more advanced aspects of (1), good knowledge of (2) is necessary. a b Change, Kenneth; Warren Leary (September 25, 2005). "Serge Lang, 78, a Gadfly and Mathematical Theorist, Dies". New York Times . Retrieved August 13, 2010. Lang, Serge (1987). Linear algebra. Undergraduate Texts in Mathematics (Third edition of 1966 originaled.). New York: Springer-Verlag. doi: 10.1007/978-1-4757-1949-9. ISBN 0-387-96412-6. MR 0874113.If you're exceptionally concerned about rigor, you can get the full story on the set-theoretical foundations of mathematics from a book like Introduction to Set Theory, by Jech and Hrbacek. This builds up from the axioms of set theory to a construction of the natural numbers, and later of the integers, rational numbers and real numbers. The problem is that while such a program is pre-Spivak in purely logical terms, it is post-Spivak in the demands it places on readers' mathematical maturity. For comparison, Spivak constructs the real numbers in the last part of his book, but he takes the rational numbers and their properties as intuitively known. My opinion is that few people would benefit from working through a set theory book before Spivak, but it may be helpful to have a general idea of what the steps are in providing a firm logical foundation for mathematics.