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**.
%%ANKI
Basic
In set theory, what is a class?
Back: A collection defined using set theoretical notation that isn't a set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
Which two alternatives are usually employed when speaking of classes?
Back: The Zermelo-Fraenkel alternative and the von Neumann-Bernays alternative.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
What does the Zermelo-Fraenkel alternative say about classes?
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
What does the von Neumann-Bernays alternative say about classes?
Back: It refers to objects defined using set theory but that aren't actually sets.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Cloze
The {1:Zermelo}-{2:Fraenkel} alternative is a separate approach from the {2:von Neumann}-{1:Bernays} alternative.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
Which set theory alternative avoids the term "class"?
Back: The Zermelo-Fraenkel alternative.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
Which set theory alternative embraces the term "class"?
Back: The von Neumann-Bernays alternative.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
What kind of mathematical object is $\{x \mid x \neq x\}$?
Back: A set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
What kind of mathematical object is $\{x \mid x = x\}$?
Back: A class.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
Are sets or classes more general?
Back: Classes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
Is every set a class?
Back: Yes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
Is every class a set?
Back: No.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
Assuming entrance requirement $\_\_\_$, what kind of mathematical object is $\{x \mid \_\_\_\}$?
Back: A class.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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## Russell's Paradox
Let $R = \{x \mid x \not\in x\}$. Then $R \in R \Leftrightarrow R \not\in R$.
%%ANKI
Basic
What simpler set is $\{x \mid x \neq x\}$ equivalent to?
Back: The empty set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
Is $\{x \mid x \neq x\}$ a set?
Back: Yes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
What simpler set is $\{x \mid x = x\}$ equivalent to?
Back: N/A. This is a class.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
Is $\{x \mid x = x\}$ a set?
Back: No.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
What simpler set is $\{x \mid x \in x\}$ equivalent to?
Back: The empty set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
Is $\{x \mid x \in x\}$ a set?
Back: Yes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
What simpler set is $\{x \mid x \not\in x\}$ equivalent to?
Back: N/A. This is a class.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
Is $\{x \mid x \not\in x\}$ a set?
Back: No.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
Let $R = \{x \mid x \not\in x\}$. What biconditional demonstrates a paradox?
Back: $R \in R \Leftrightarrow R \not\in R$
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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END%%
%%ANKI
Basic
Given $R = \{x \mid x \not\in x\}$, what contradiction arises when we assume $R \in R$?
Back: The entrance requirement says $R \not\in R$.
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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%%ANKI
Basic
Given $R = \{x \mid x \not\in x\}$, what contradiction arises when we assume $R \not\in R$?
Back: $R$ satisfies the entrance requirement meaning $R \in R$.
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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%%ANKI
Basic
What special name is given to class $\{x \mid x \not\in x\}$?
Back: The Russell set.
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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%%ANKI
Basic
Explain how the Russell set is defined in plain English.
Back: It is the "set" of all sets that do not contain themselves.
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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%%ANKI
Basic
What is the entrance requirement of the Russell set?
Back: $x \not\in x$
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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%%ANKI
Basic
The barber paradox is a variant of what other paradox?
Back: Russell's paradox.
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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%%ANKI
Basic
What does the barber paradox assume existence of?
Back: A barber who shaves all those, and those only, who do not shave themselves.
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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%%ANKI
Basic
What question is posed within the barber paradox?
Back: Does the barber shave himself?
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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END%%
%%ANKI
Basic
In the barber paradox, what contradiction arises when we assume the barber shaves himself?
Back: The barber *only* shaves those who do not shave themselves.
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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END%%
%%ANKI
Basic
In the barber paradox, what contradiction arises when we assume the barber does not shave himself?
Back: The barber shaves *all* men who do not shave themselves.
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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END%%
## Bibliography
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
* “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).
