380 lines
13 KiB
Markdown
380 lines
13 KiB
Markdown
---
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title: Classes
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TARGET DECK: Obsidian::STEM
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FILE TAGS: set::class
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tags:
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- class
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- set
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---
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## Overview
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The **Zermelo-Fraenkel alternative** avoids speaking of collections defined using set theoretical notation that are not sets. The **von Neumann-Bernays** alternative calls these **classes**.
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%%ANKI
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Basic
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In set theory, what is a class?
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Back: A collection defined using set theoretical notation that isn't a set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715970576758-->
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END%%
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%%ANKI
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Basic
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Which two alternatives are usually employed when speaking of classes?
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Back: The Zermelo-Fraenkel alternative and the von Neumann-Bernays alternative.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715970576761-->
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END%%
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%%ANKI
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Basic
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What does the Zermelo-Fraenkel alternative say about classes?
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Back: It gives them no ontological status at all.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715970576763-->
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END%%
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%%ANKI
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Basic
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What does the von Neumann-Bernays alternative say about classes?
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Back: It refers to objects defined using set theory but that aren't actually sets.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715970576765-->
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END%%
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%%ANKI
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Cloze
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The {1:Zermelo}-{2:Fraenkel} alternative is a separate approach from the {2:von Neumann}-{1:Bernays} alternative.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715970576766-->
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END%%
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%%ANKI
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Basic
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Which set theory alternative avoids the term "class"?
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Back: The Zermelo-Fraenkel alternative.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715970576768-->
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END%%
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%%ANKI
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Basic
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Which set theory alternative embraces the term "class"?
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Back: The von Neumann-Bernays alternative.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715970576769-->
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END%%
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%%ANKI
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Basic
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What kind of mathematical object is $\{x \mid x \neq x\}$?
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Back: A set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715970576771-->
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END%%
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%%ANKI
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Basic
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What kind of mathematical object is $\{x \mid x = x\}$?
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Back: A class.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715970576774-->
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END%%
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%%ANKI
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Basic
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Are sets or classes more general?
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Back: Classes.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715970576777-->
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END%%
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%%ANKI
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Basic
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Is every set a class?
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Back: Yes.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715970576779-->
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END%%
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%%ANKI
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Basic
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Is every class a set?
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Back: No.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715970576781-->
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END%%
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%%ANKI
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Basic
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Assuming entrance requirement $\_\_\_$, what kind of mathematical object is $\{x \mid \_\_\_\}$?
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Back: A class.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715970576782-->
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END%%
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%%ANKI
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Basic
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What kind of formulas are permitted in the entrance requirement of set-builder notation?
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Back: Well-formed formulas in predicate logic.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716237736452-->
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END%%
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%%ANKI
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Basic
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*Why* do we require first-order logic in the entrance requirement of set-builder notation?
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Back: To avoid any ambiguity introduced by plain English.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716237736482-->
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END%%
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%%ANKI
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Basic
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When is plain English permitted in the entrance requirement of set-builder notation?
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Back: When the English can be translated into predicate logic.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716237736487-->
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END%%
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%%ANKI
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Basic
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Why is "$x$ is an int definable in one line of type" an invalid entrance requirement?
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Back: Because this sentence cannot be expressed in predicate logic.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716309007883-->
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END%%
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## Russell's Paradox
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Let $R = \{x \mid x \not\in x\}$. Then $R \in R \Leftrightarrow R \not\in R$.
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%%ANKI
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Basic
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What simpler set is $\{x \mid x \neq x\}$ equivalent to?
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Back: The empty set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715970576772-->
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END%%
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%%ANKI
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Basic
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Is $\{x \mid x \neq x\}$ a set?
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Back: Yes.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716074591194-->
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END%%
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%%ANKI
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Basic
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What simpler set is $\{x \mid x = x\}$ equivalent to?
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Back: N/A. This is a class.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715970576775-->
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END%%
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%%ANKI
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Basic
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Is $\{x \mid x = x\}$ a set?
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Back: No.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716074591199-->
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END%%
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%%ANKI
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Basic
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What simpler set is $\{x \mid x \in x\}$ equivalent to?
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Back: The empty set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716074591202-->
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END%%
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%%ANKI
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Basic
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Is $\{x \mid x \in x\}$ a set?
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Back: Yes.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716074610694-->
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END%%
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%%ANKI
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Basic
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What simpler set is $\{x \mid x \not\in x\}$ equivalent to?
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Back: N/A. This is a class.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716074591205-->
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END%%
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%%ANKI
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Basic
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Is $\{x \mid x \not\in x\}$ a set?
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Back: No.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716074610697-->
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END%%
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%%ANKI
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Basic
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Let $R = \{x \mid x \not\in x\}$. What biconditional demonstrates a paradox?
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Back: $R \in R \Leftrightarrow R \not\in R$
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Reference: “Russell’s Paradox,” in *Wikipedia*, April 18, 2024, [https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437](https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437).
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<!--ID: 1716075743527-->
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END%%
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%%ANKI
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Basic
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Given $R = \{x \mid x \not\in x\}$, what contradiction arises when we assume $R \in R$?
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Back: The entrance requirement says $R \not\in R$.
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Reference: “Russell’s Paradox,” in *Wikipedia*, April 18, 2024, [https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437](https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437).
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<!--ID: 1716075811572-->
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END%%
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%%ANKI
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Basic
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Given $R = \{x \mid x \not\in x\}$, what contradiction arises when we assume $R \not\in R$?
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Back: $R$ satisfies the entrance requirement meaning $R \in R$.
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Reference: “Russell’s Paradox,” in *Wikipedia*, April 18, 2024, [https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437](https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437).
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<!--ID: 1716075811577-->
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END%%
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%%ANKI
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Basic
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What special name is given to class $\{x \mid x \not\in x\}$?
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Back: The Russell set.
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Reference: “Russell’s Paradox,” in *Wikipedia*, April 18, 2024, [https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437](https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437).
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<!--ID: 1716075743531-->
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END%%
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%%ANKI
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Basic
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Explain how the Russell set is defined in plain English.
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Back: It is the "set" of all sets that do not contain themselves.
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Reference: “Russell’s Paradox,” in *Wikipedia*, April 18, 2024, [https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437](https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437).
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<!--ID: 1716075743534-->
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END%%
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%%ANKI
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Basic
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What is the entrance requirement of the Russell set?
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Back: $x \not\in x$
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Reference: “Russell’s Paradox,” in *Wikipedia*, April 18, 2024, [https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437](https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437).
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<!--ID: 1716075743537-->
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END%%
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%%ANKI
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Basic
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The barber paradox is a variant of what other paradox?
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Back: Russell's paradox.
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Reference: “Russell’s Paradox,” in *Wikipedia*, April 18, 2024, [https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437](https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437).
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<!--ID: 1716075743540-->
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END%%
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%%ANKI
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Basic
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What does the barber paradox assume existence of?
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Back: A barber who shaves all those, and those only, who do not shave themselves.
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Reference: “Russell’s Paradox,” in *Wikipedia*, April 18, 2024, [https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437](https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437).
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<!--ID: 1716075743544-->
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END%%
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%%ANKI
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Basic
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What question is posed within the barber paradox?
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Back: Does the barber shave himself?
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Reference: “Russell’s Paradox,” in *Wikipedia*, April 18, 2024, [https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437](https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437).
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<!--ID: 1716075743547-->
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END%%
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%%ANKI
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Basic
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In the barber paradox, what contradiction arises when we assume the barber shaves himself?
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Back: The barber *only* shaves those who do not shave themselves.
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Reference: “Russell’s Paradox,” in *Wikipedia*, April 18, 2024, [https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437](https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437).
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<!--ID: 1716075743551-->
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END%%
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%%ANKI
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Basic
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In the barber paradox, what contradiction arises when we assume the barber does not shave himself?
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Back: The barber shaves *all* men who do not shave themselves.
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Reference: “Russell’s Paradox,” in *Wikipedia*, April 18, 2024, [https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437](https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437).
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<!--ID: 1716075743555-->
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END%%
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%%ANKI
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Basic
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Let $A$ be a set. What does $\{x \in A \mid x \not\in x\}$ evaluate to?
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Back: $A$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716237736492-->
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END%%
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%%ANKI
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Basic
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*Why* does $A = \{x \in A \mid x \not\in x\}$?
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Back: No set is a member of itself.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716237736497-->
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END%%
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%%ANKI
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Basic
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What two sets are used in the proof of "no set contains all sets"?
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Back: An arbitrary set $A$ and $\{x \in A \mid x \not\in x\}$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716237736501-->
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END%%
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%%ANKI
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Basic
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Let $B = \{x \in A \mid x \not\in x\}$. What biconditional proves $B \not\in A$?
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Back: $B \in B \Leftrightarrow B \in A \land B \not\in B$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716237736505-->
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END%%
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%%ANKI
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Basic
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Which paradox is used in the proof of "no set contains all sets"?
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Back: Russell's paradox.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716237736514-->
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END%%
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%%ANKI
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Basic
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Which axiom is used in the proof of "no set contains all sets"?
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Back: The subset axiom(s).
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716237736518-->
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END%%
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%%ANKI
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Basic
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Let $B = \{x \in A \mid x \not\in x\}$. What contradiction arises when $B \in B$?
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Back: $B \in A \land B \not\in B$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716237736522-->
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END%%
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%%ANKI
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Basic
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Let $B = \{x \in A \mid x \not\in x\}$. What contradiction arises when $B \not\in A \land B \not\in B$?
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Back: N/A.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716237736526-->
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END%%
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%%ANKI
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Basic
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Let $B = \{x \in A \mid x \not\in x\}$. What contradiction arises when $B \in A \land B \not\in B$?
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Back: $B \in B$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716237736529-->
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END%%
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## Bibliography
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* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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* “Russell’s Paradox,” in *Wikipedia*, April 18, 2024, [https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437](https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437). |