--- 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). 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). 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). 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). 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). 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). 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). 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). 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). END%% %%ANKI Basic Are sets or classes more general? Back: Classes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is every set a class? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is every class a set? Back: No. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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).