Abstract
After a review of basic set theory and countability, there is a discussion of Simon Stevin’s method for the construction of real numbers. This approach fits in well with the methods used in chapter “Basic Properties of Real Numbers, Sequences and Continuous Functions” for the proof of foundational results on bounded sets of real numbers and continuous functions.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Strictly speaking, strictly positive numbers; the concepts of negative and zero numbers were developed later in Indian and Arabian mathematics.
References
R. Abraham, J.W. Robbin, Transversal Mappings and Flows (Benjamin, New York, 1967)
M. Barnsley, Fractals Everywhere (Academic, New York, 1988)
P. Błaszczyk, M.G. Katz, D. Sherry, Ten misconceptions from the history of analysis and their debunking. Found. Sci. 18(1), 43–74 (2013)
A.V. Borovnik, Mathematics under the Microscope. Notes on Cognitive Aspects of Mathematical Practice (American Mathematical Society, Providence, RI, 2009)
T.J.I’A. Bromwich, Theory of Infinite Series, 2nd edn. (Macmillan and Co., London, 1959)
J. Dieudonné, Foundations of Modern Analysis (Academic, New York, 1960)
M. Faà di Bruno, Note sur une nouvelle formule de calcul differentiel. Q. J. Pure Appl. Math. 1, 359–360 (1857)
K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. (Wiley, New York, 2003)
M. Field, Differential Calculus and Its Applications (Van Nostrand Reinhold, New York, 1976)
M.J. Field, Stratification of equivariant varieties. Bull. Aust. Math. Soc. 16, 279–296 (1977)
M. Field, M. Golubitsky, Symmetry in Chaos: A Search for Pattern in Mathematics, Art and Nature, 2nd edn. (Society for Industrial and Applied Mathematics, Philadelphia, 2009)
A. Fraenkel, Abstract Set Theory (North Holland, Amsterdam, 1953)
A. Fraenkel, Y. Bar-Hillel, A. Levy, Foundations of Set Theory (North Holland, Amsterdam, 1958)
M.W. Hirsch, S. Smale, R. Devanney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, 3rd edn. (Academic, New York, 2013)
J.E. Hutchinson, Fractals and self similarity. Indiana Univ. Math. J. 30, 713–747 (1981)
M.C. Irwin, Smooth Dynamical Systems. Advanced Series in Nonlinear Dynamics, vol. 17 (World Scientific, Singapore, 2001). The original book, published by Academic press, appeared in 1980
W.P. Johnson, The curious history of Faà di Bruno’s formula. Am. Math. Mon. 109, 217–234 (2002)
J.L. Kelley, General Topology. Graduate Texts in Mathematics, vol. 27 (Springer, New York, 1975). Originally published 1955, Van Nostrand Reinhold
S.G. Krantz, Real Analysis and Foundations, 2nd edn. (Chapman and Hall/CRC, Boca Raton, 2004)
S.G. Krantz, H.R. Parks, A Primer of Real Analytic Functions. Basler Lehrbücher, vol. 4 (Birkhäuser, Basel, 1992)
L. Kuipers, H. Niederreiter, Uniform Distribution of Sequences (Dover, New York, 2006)
J.W. Lamperti, Probability, 2nd edn. (Wiley, New York, 1996)
D. Liberzon, Switching in Systems and Control. Systems and Control: Foundations and Applications (Birkhäuser, Basel, 2003)
P. Mandelbrot, The Fractal Geometry of Nature (W.H. Freeman and Co., New York, 1982)
J.W. Milnor, Topology from the Differentiable Viewpoint (Princeton University Press, Princeton, NJ, 1965)
H.-O. Peitgen, P.H. Richter, The Beauty of Fractals (Springer, New York, 1988)
W. Rudin, Principles of Mathematical Analysis, 3rd edn. (McGraw-Hill, New York, 1976)
S.H. Strogatz, Nonlinear Dynamics and Chaos (Studies in Nonlinearity), 2nd edn. (Westview Press, Boulder, 2015)
A.N. Whitehead, Science and the Modern World, Paperback edn. (Macmillan Company, New York, 1925; Cambridge University Press, Cambridge, 2011)
S. Willard, General Topology (Dover, New York, 2004). Originally Published by Addison-Wesley, Reading, MA, 1970
W.H. Young, On the distinction of right and left at points of discontinuity. Q. J. Math. 39, 67–83 (1908)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Field, M. (2017). Sets, Functions and the Real Numbers. In: Essential Real Analysis. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-67546-6_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-67546-6_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-67545-9
Online ISBN: 978-3-319-67546-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)