276 lines
9.4 KiB
Markdown
276 lines
9.4 KiB
Markdown
---
|
||
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).
|
||
<!--ID: 1715970576758-->
|
||
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).
|
||
<!--ID: 1715970576761-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What does the Zermelo-Fraenkel alternative say about classes?
|
||
Back: It gives it 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).
|
||
<!--ID: 1715970576765-->
|
||
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).
|
||
<!--ID: 1715970576766-->
|
||
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).
|
||
<!--ID: 1715970576768-->
|
||
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).
|
||
<!--ID: 1715970576774-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Are sets or classes more general?
|
||
Back: Classes.
|
||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||
<!--ID: 1715970576777-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Is every set a class?
|
||
Back: Yes.
|
||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||
<!--ID: 1715970576779-->
|
||
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).
|
||
<!--ID: 1715970576782-->
|
||
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).
|
||
<!--ID: 1716075743527-->
|
||
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).
|
||
<!--ID: 1716075743534-->
|
||
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%%
|
||
|
||
## 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). |