Abstract
One of the goals of this text is to get you proving mathematical statements in real analysis. Set theory provides a safe environment in which to learn about math statements, “if ... then,” “if and only if,” etc., and to learn the logic behind proofs. Since this is an introductory book on analysis, our treatment of sets is “very naive,” in the sense that we actually don’t define sets rigorously, only informally; we are mostly interested in how “they work,” not really what they are.
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Notes
- 1.
“The errors of definitions multiply themselves according as the reckoning proceeds; and lead men into absurdities, which at last they see but cannot avoid, without reckoning anew from the beginning.” Thomas Hobbes (1588–1679) [172].
- 2.
Later in your math career you will find some “neither true nor false” statements (perhaps a better wording is “neither provable nor refutable”) such as, e.g., the continuum hypothesis \(\ldots \) but that is another story! There is no such thing as a “neither statement” in this book.
- 3.
“Finally, two days ago, I succeeded—not on account of my hard efforts, but by the grace of the Lord. Like a sudden flash of lightning, the riddle was solved. I am unable to say what was the conducting thread that connected what I previously knew with what made my success possible.” Carl Friedrich Gauss (1777–1855) [72].
- 4.
“The point of philosophy is to start with something so simple as not to seem worth stating, and to end with something so paradoxical that no one will believe it.” Bertrand Russell (1872–1970).
- 5.
“Another advantage of a mathematical statement is that it is so definite that it might be definitely wrong; and if it is found to be wrong, there is a plenteous choice of amendments ready in the mathematicians’ stock of formulae. Some verbal statements have not this merit; they are so vague that they could hardly be wrong, and are correspondingly useless.” Lewis Richardson (1881–1953). Mathematics of War and Foreign Politics.
- 6.
P implies Q is sometimes translated as “P is sufficient for Q” in the sense that the truth of P is sufficient or enough or ample to imply that Q is also true. Moreover, P implies Q is also translated “Q is necessary for P” because Q is necessarily true given that P is true. However, we shall not use this language in this book.
- 7.
An integer exceeding 1 that is not prime.
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© 2017 Paul Loya
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Loya, P. (2017). Very Naive Set Theory, Functions, and Proofs. In: Amazing and Aesthetic Aspects of Analysis. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6795-7_1
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