440 lines
14 KiB
Markdown
440 lines
14 KiB
Markdown
---
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title: Cardinality
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TARGET DECK: Obsidian::STEM
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FILE TAGS: set::cardinality
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tags:
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- set
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---
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## Equinumerosity
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We say set $A$ is **equinumerous** to set $B$, written ($A \approx B$) if and only if there exists a [[set/functions#Injections|one-to-one]] function from $A$ [[set/functions#Surjections|onto]] $B$.
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%%ANKI
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Basic
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Suppose $A$ is equinumerous to $B$. How does Enderton denote this?
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Back: $A \approx B$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060344-->
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END%%
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%%ANKI
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Basic
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What does it mean for $A$ to be equinumerous to $B$?
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Back: There exists a bijection between $A$ and $B$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060352-->
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END%%
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%%ANKI
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Basic
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Suppose $A \approx B$. Then what must exist?
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Back: A bijection between $A$ and $B$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060355-->
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END%%
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%%ANKI
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Basic
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Suppose there exists a one-to-one function $F$ from $A$ into $B$. When is $A \approx B$?
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Back: When $F$ is also onto $B$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060358-->
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END%%
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%%ANKI
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Basic
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Suppose there exists a function $F$ from $A$ onto $B$. When is $A \approx B$?
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Back: When $F$ is also one-to-one.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060362-->
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END%%
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%%ANKI
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Basic
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Suppose there exists a one-to-one function $F$ from $A$ onto $B$. When is $A \approx B$?
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Back: Always, by definition.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060366-->
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END%%
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### Power Sets
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No set is equinumerous to its [[set/index#Power Set Axiom|power set]]. This is typically shown using a diagonalization argument.
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%%ANKI
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Basic
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What basic set operation produces a new set the original isn't equinumerous to?
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Back: The power set operation.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732541309202-->
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END%%
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%%ANKI
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Basic
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What kind of argument is typically made to prove no set is equinumerous to its power set?
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Back: A diagonalization argument.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732541309208-->
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END%%
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%%ANKI
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Basic
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Who is attributed the discovery of the diagonalization argument?
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Back: Georg Cantor.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732541309212-->
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END%%
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%%ANKI
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Basic
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Let $g \colon A \rightarrow \mathscr{P}A$. Using a diagonalization argument, what set is *not* in $\mathop{\text{ran}}(g)$?
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Back: $\{ x \in A \mid x \not\in g(x) \}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732541309216-->
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END%%
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%%ANKI
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Basic
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Let $g \colon A \rightarrow \mathscr{P}A$. *Why* isn't $B = \{x \in A \mid x \not\in g(x) \}$ in $\mathop{\text{ran}}(g)$?
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Back: For all $x \in A$, $x \in B \Leftrightarrow x \not\in g(x)$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732541309221-->
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END%%
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### Equivalence Concept
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For any sets $A$, $B$, and $C$:
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* $A \approx A$;
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* if $A \approx B$, then $B \approx A$;
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* if $A \approx B$ and $B \approx C$, then $A \approx C$.
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Notice though that $\{ \langle A, B \rangle \mid A \approx B \}$ is *not* an equivalence relation since the equivalence concept of equinumerosity concerns *all* sets.
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%%ANKI
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Basic
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Concisely state the equivalence concept of equinumerosity in Zermelo-Fraenkel set theory.
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Back: For all sets $A$, $B$, and $C$:
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* $A \approx A$;
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* $A \approx B \Rightarrow B \approx A$;
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* $A \approx B \land B \approx C \Rightarrow A \approx C$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060370-->
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END%%
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%%ANKI
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Basic
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Concisely state the equivalence concept of equinumerosity in von Neumann-Bernays set theory.
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Back: Class $\{ \langle A, B \rangle \mid A \approx B \}$ is reflexive on the class of all sets, symmetric, and transitive.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060374-->
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END%%
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%%ANKI
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Basic
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What is the reflexive property of equinumerosity in FOL?
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Back: $\forall A, A \approx A$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060379-->
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END%%
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%%ANKI
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Basic
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What is the symmetric property of equinumerosity in FOL?
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Back: $\forall A, B, A \approx B \Rightarrow B \approx A$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060383-->
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END%%
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%%ANKI
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Basic
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What is the transitive property of equinumerosity in FOL?
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Back: $\forall A, B, C, A \approx B \land B \approx C \Rightarrow A \approx C$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060387-->
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END%%
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%%ANKI
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Basic
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Is $\{ \langle A, B \rangle \mid A \approx B \}$ a set?
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Back: No.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060390-->
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END%%
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%%ANKI
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Basic
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*Why* isn't $\{ \langle A, B \rangle \mid A \approx B \}$ a set?
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Back: Because then the field of this "relation" would be a set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060394-->
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END%%
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%%ANKI
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Basic
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Is $\{ \langle A, B \rangle \mid A \approx B \}$ an equivalence relation?
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Back: No.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060398-->
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END%%
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%%ANKI
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Basic
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*Why* isn't $\{ \langle A, B \rangle \mid A \approx B \}$ an equivalence relation?
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Back: Because then the field of this "relation" would be a set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732295060403-->
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END%%
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## Finiteness
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A set is **finite** if and only if it is equinumerous to a [[natural-numbers|natural number]]. Otherwise it is **infinite**.
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%%ANKI
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Basic
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How does Enderton define a finite set?
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Back: As a set equinumerous to some natural number.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231320-->
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END%%
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%%ANKI
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Basic
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How does Enderton define an infinite set?
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Back: As a set not equinumerous to any natural number.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231330-->
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END%%
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%%ANKI
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Basic
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Is $n \in \omega$ a finite set?
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Back: Yes.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231336-->
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END%%
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%%ANKI
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Basic
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*Why* isn't $n \in \omega$ a finite set?
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Back: N/A. It is.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231342-->
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END%%
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%%ANKI
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Basic
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Is $\omega$ a finite set?
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Back: No.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231347-->
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END%%
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%%ANKI
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Basic
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*Why* isn't $\omega$ a finite set?
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Back: There is no natural number equinumerous to $\omega$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231353-->
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END%%
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### Pigeonhole Principle
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No natural number is equinumerous to a proper subset of itself. More generally, no finite set is equinumerous to a proper subset of itself.
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Likewise, any set equinumerous to a proper subset of itself must be infinite.
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%%ANKI
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Basic
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How does Enderton state the pigeonhole principle for $\omega$?
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Back: No natural number is equinumerous to a proper subset of itself.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231358-->
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END%%
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%%ANKI
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Basic
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How does Enderton state the pigeonhole principle for finite sets?
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Back: No finite set is equinumerous to a proper subset of itself.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231364-->
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END%%
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%%ANKI
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Basic
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Let $m \in n \in \omega$. What principle precludes $m \approx n$?
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Back: The pigeonhole principle.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231369-->
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END%%
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%%ANKI
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Basic
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Let $S$ be a set and $n \in \omega$ such that $S \approx n$. For $m \in \omega$, when might $S \approx m$?
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Back: *Only* if $m = n$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231374-->
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END%%
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%%ANKI
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Basic
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What is the generalization of the pigeonhole principle for $\omega$?
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Back: The pigeonhole principle for finite sets.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231379-->
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END%%
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%%ANKI
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Basic
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What is the specialization of the pigeonhole principle for finite sets?
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Back: The pigeonhole principle for $\omega$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231385-->
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END%%
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%%ANKI
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Basic
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What name is given to the following theorem? $$\text{No finite set is equinumerous to a proper subset of itself.}$$
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Back: The pigeonhole principle.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231391-->
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END%%
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%%ANKI
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Basic
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Let $S$ be a finite set and $f \colon S \rightarrow S$ be injective. Is $f$ a bijection?
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Back: Yes.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231396-->
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END%%
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%%ANKI
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Basic
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Let $S$ be a finite set and $f \colon S \rightarrow S$ be injective. *Why* must $f$ be surjective?
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Back: Otherwise $f$ is a bijection between $S$ and a proper subset of $S$, a contradiction.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231401-->
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END%%
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%%ANKI
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Basic
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Let $S$ be a finite set and $f \colon S \rightarrow S$ be surjective. Is $f$ a bijection?
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Back: Yes.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231407-->
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END%%
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%%ANKI
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Basic
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Let $S$ be a finite set and $f \colon S \rightarrow S$ be surjective. *Why* must $f$ be injective?
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Back: Otherwise $f$ is a bijection between a proper subset of $S$ and $S$, a contradiction.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1732545231412-->
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END%%
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%%ANKI
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Basic
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What does the contrapositive of the pigeonhole principle state?
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Back: Any set equinumerous to a proper subset of itself is infinite.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1733407760079-->
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END%%
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%%ANKI
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Basic
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What general strategy is used to prove $\omega$ is an infinite set?
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Back: Prove $\omega$ is equinumerous to a proper subset of itself.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1733407760085-->
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END%%
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## Cardinal Numbers
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A **cardinal number** is a set that is $\mathop{\text{card}} A$ for some set $A$. The set $\mathop{\text{card}} A$ is defined such that
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* For any sets $A$ and $B$, $\mathop{\text{card}}A = \mathop{\text{card}}B$ iff $A \approx B$.
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* For a finite set $A$, $\mathop{\text{card}}A$ is the natural number $n$ for which $A \approx n$.
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%%ANKI
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Basic
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How is the cardinal number of set $A$ denoted?
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Back: As $\mathop{\text{card}} A$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1733407760088-->
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END%%
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%%ANKI
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Basic
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Suppose $A$ is finite. What does $\mathop{\text{card}} A$ evaluate to?
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Back: The unique $n \in \omega$ such that $A \approx n$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1733407760091-->
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END%%
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%%ANKI
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Basic
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Consider $n \in \omega$. What does $\mathop{\text{card}} n$ evaluate to?
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Back: $n$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1733407760094-->
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END%%
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%%ANKI
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Basic
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Suppose $a$, $b$, and $c$ are distinct objects. What does $\mathop{\text{card}} \{a, b, c\}$ evaluate to?
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Back: $3$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1733407760097-->
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END%%
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%%ANKI
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Basic
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What does Enderton refer to by the "process called 'counting'"?
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Back: Choosing a one-to-one correspondence between two sets.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1733407760099-->
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END%%
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%%ANKI
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Cloze
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A {cardinal number} is a set that is {$\mathop{\text{card}} A$} for some set $A$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1733407760102-->
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END%%
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%%ANKI
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Basic
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How do cardinal numbers relate to equinumerosity?
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Back: For any sets $A$ and $B$, $\mathop{\text{card}} A = \mathop{\text{card}} B$ iff $A \approx B$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1733407760105-->
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END%%
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%%ANKI
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Basic
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According to Enderton, what is the "essential demand" for defining numbers?
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Back: Defining cardinal numbers such that for any sets $A$ and $B$, $\mathop{\text{card}} A = \mathop{\text{card}} B$ iff $A \approx B$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1733407760108-->
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END%%
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%%ANKI
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Basic
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What name is given to $\mathop{\text{card}} \omega$?
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Back: $\aleph_0$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1733407760110-->
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END%%
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%%ANKI
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Basic
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Who is attributed the assignment $\mathop{\text{card}} \omega = \aleph_0$?
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Back: Georg Cantor.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1733407760113-->
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END%%
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## Bibliography
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* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). |