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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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