notebook/notes/set/classes.md

---
title: Classes
TARGET DECK: Obsidian::STEM
FILE TAGS: set::class
tags:
  - class
  - set
---

## Overview

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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END%%

%%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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END%%

%%ANKI
Basic
What does the Zermelo-Fraenkel alternative say about classes?
Back: It gives them no ontological status at all.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715970576763-->
END%%

%%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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END%%

%%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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END%%

%%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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END%%

%%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).
<!--ID: 1715970576769-->
END%%

%%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).
<!--ID: 1715970576771-->
END%%

%%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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END%%

%%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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END%%

%%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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END%%

%%ANKI
Basic
Is every class a set?
Back: No.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715970576781-->
END%%

%%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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END%%

%%ANKI
Basic
What kind of formulas are permitted in the entrance requirement of set-builder notation?
Back: Well-formed formulas in predicate logic.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

%%ANKI
Basic
*Why* do we require first-order logic in the entrance requirement of set-builder notation?
Back: To avoid any ambiguity introduced by plain English.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716237736482-->
END%%

%%ANKI
Basic
When is plain English permitted in the entrance requirement of set-builder notation?
Back: When the English can be translated into predicate logic.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

%%ANKI
Basic
Why is "$x$ is an int definable in one line of type" an invalid entrance requirement?
Back: Because this sentence cannot be expressed in predicate logic.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

## 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).
<!--ID: 1715970576772-->
END%%

%%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).
<!--ID: 1716074591194-->
END%%

%%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).
<!--ID: 1715970576775-->
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).
<!--ID: 1716074591199-->
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).
<!--ID: 1716074591202-->
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).
<!--ID: 1716074610694-->
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).
<!--ID: 1716074591205-->
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).
<!--ID: 1716074610697-->
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).
<!--ID: 1716075811572-->
END%%

%%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).
<!--ID: 1716075811577-->
END%%

%%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).
<!--ID: 1716075743531-->
END%%

%%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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END%%

%%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).
<!--ID: 1716075743537-->
END%%

%%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).
<!--ID: 1716075743540-->
END%%

%%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).
<!--ID: 1716075743544-->
END%%

%%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).
<!--ID: 1716075743547-->
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).
<!--ID: 1716075743551-->
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).
<!--ID: 1716075743555-->
END%%

%%ANKI
Basic
Let $A$ be a set. What does $\{x \in A \mid x \not\in x\}$ evaluate to?
Back: $A$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716237736492-->
END%%

%%ANKI
Basic
*Why* does $A = \{x \in A \mid x \not\in x\}$?
Back: No set is a member of itself.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716237736497-->
END%%

%%ANKI
Basic
What two sets are used in the proof of "no set contains all sets"?
Back: An arbitrary set $A$ and $\{x \in A \mid x \not\in x\}$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716237736501-->
END%%

%%ANKI
Basic
Let $B = \{x \in A \mid x \not\in x\}$. What biconditional proves $B \not\in A$?
Back: $B \in B \Leftrightarrow B \in A \land B \not\in B$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716237736505-->
END%%

%%ANKI
Basic
Which paradox is used in the proof of "no set contains all sets"?
Back: Russell's paradox.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716237736514-->
END%%

%%ANKI
Basic
Which axiom is used in the proof of "no set contains all sets"?
Back: The subset axiom(s).
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716237736518-->
END%%

%%ANKI
Basic
Let $B = \{x \in A \mid x \not\in x\}$. What contradiction arises when $B \in B$?
Back: $B \in A \land B \not\in B$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716237736522-->
END%%

%%ANKI
Basic
Let $B = \{x \in A \mid x \not\in x\}$. What contradiction arises when $B \not\in A \land B \not\in B$?
Back: N/A.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716237736526-->
END%%

%%ANKI
Basic
Let $B = \{x \in A \mid x \not\in x\}$. What contradiction arises when $B \in A \land B \not\in B$?
Back: $B \in B$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716237736529-->
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).