ERRATA LIST for Set Theory

(A PDF Errata List is also available.)

The following is the list of errata for the book Set Theory: With an Introduction to Real Point Sets, by Abhijit Dasgupta, ISBN 978-1-4614-8853-8 (hardcover), Birkhauser/Springer, 2014.

• p. ix: Add the following sentence at the end of the top paragraph: “In particular, the postscript chapters fall outside the main development followed in the book.”
• p. 9: Line before Problem 16: “\(xRu\) and \(uRy\)” should be “\(xRu\) and \(uSy\)”.
• pp. 42--43: The author wishes to thank Dr. Christoph Lamm for noticing this error. The rigorous definition of addition given in Section 2.10 (using primitive recursion) is based ultimately on Theorem 146 (Basic Principle of Recursive Definition). The proof of Theorem 146 assumes that the ordering relation \(<\) on \(\mathbf{N}\) has already been defined. But the definition of ordering on \(\mathbf{N}\) given earlier (Chapter 2, Definition 61) had presupposed addition itself, resulting in circular reasoning.

  The correction is discussed in this page, and a corrected complete replacement proof for Theorem 146 is in this PDF file.

• p. 49: After the figure, remove the extra space (indentation) preceding the phrase “if both legs …”.
• p. 60: Top line of text: Two union symbols after \(\{0\}\) are missing: In each case, “\(\{0\}\{\)” should become “\(\{0\}\cup\{\)”.
• p. 83: Item 2 in “Informal discussion”: “called bounded if we have \(-a < x < a\) for some real number \(a\)” should be changed to: “called bounded if there is a real number \(a\) such that \(-a < x < a\) for all \(x \in E\).”
• p. 92: Paragraph after Problem 290, second line: “congruence and similarity mappings” should be “congruence and similarity for geometric figures”.
• p. 97: Problem 313: \(\frac{1}{2}(m+n)(m+n-1) + m\) should be \(\frac{1}{2}(m+n-2)(m+n-1) + m\), and similarly, the displayed formula should become \[ \langle m,n \rangle \mapsto \frac{(m+n-2)(m+n-1)}{2} + m \]
• p. 111: After Theorem 350, add the following sentence: “By Theorem 350, the cardinal comparison symbols \(\leqslant\) and \(\leq\) become equivalent, and so we can (and henceforth will) use them interchangeably.”
• p. 124: Problem 418, last clause in the casewise definition for \(\boldsymbol{\Phi}\): Instead of \(x\) it should be \(\mathbf{h}(x)\).
• p. 166: The displayed sentence “The Suslin Problem. Is a CCC …” is all in italics, but the first three words should be upright bold, as in: “The Suslin Problem. Is a CCC …”.
• p. 207: In Theorem 714 part 1, change “but \(X^{(\alpha)} \subsetneq X^{(\mu)}\) for \(\alpha < \mu\)” to “but \(X^{(\alpha)} \supsetneq X^{(\mu)}\) for all \(\alpha < \mu\)”.
• p. 249: Proof (of Theorem 877, Outline): In 5th line of the proof, “with \((c,d) \cap E_\alpha \not= \emptyset\) …” should be “with \((c,d) \cap E_\alpha = \emptyset\) …”.
• p. 315: Section 17.2, third paragraph, line 3: “homeomorphism of \(F_1\) onto itself which interchanges …” should be “homeomorphism of \(\mathbf{K}_1\) onto itself which interchanges …”
• p. 323: In Problem 1126, Hint: “\(B_E := \cup_{n \in E} A_n {\small\setminus} \cup_{n\not\in E} A_n\) …” should be “\(B_E := \cap_{n \in E} A_n {\small\setminus} \cup_{n\not\in E} A_n\) …”
• p. 346: Beginning of 4th line of second paragraph of the proof of Theorem 1180: “with \(\mu(G {\,\small\setminus\,} A) \lt \mu(A) \)” should be “with \(\mu(G {\,\small\setminus\,} B) \lt \mu(A) \)”.
• p. 347: Definition 1185: \(p \not= q\) should be \(x \not= y\).
• p. 420: End of proof of Proposition 1335: “\(\geq m^*(A) + m^*(B)\)” should be “\(\geq m^*(A) + m^*(B) - \epsilon\)”.
• p. 431:
  • Change index entry “absolutism, 67” to “absolutism, 67, 70”
  • Under the index entry for “axiom of”:
    • Change subentry “choice, 13, 77, 90–94, 208–210” to “choice, 13, 90–94, 208–210, 224–225”
    • Change subentry “dependent choice (DC), 77, 101” to “dependent choice (DC), 101”
• p. 432:
  • Under the index entry for “axiom of”:
    • Change subentry “foundation, 393–395, 421” to “foundation, 393–394, 421”
    • Change subentry “replacement, 376–377, 421” to “replacement, 376–377, 409, 421”
  • Change index entry “Bernstein sets, 298–299, 335, 345–346, 408” to “Bernstein sets, 298–299, 335, 346, 402, 408”

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Last updated: 2016 January 22