--- title: Cardinality TARGET DECK: Obsidian::STEM FILE TAGS: set::cardinality tags: - set --- ## Equinumerosity 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]] $B$. %%ANKI Basic Suppose $A$ is equinumerous to $B$. How does Enderton denote this? Back: $A \approx B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What does it mean for $A$ to be equinumerous to $B$? Back: There exists a bijection between $A$ and $B$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose $A \approx B$. Then what must exist? Back: A bijection between $A$ and $B$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose there exists a one-to-one function $F$ from $A$ into $B$. When does this imply $A \approx B$? Back: When $F$ is also onto $B$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose there exists a function $F$ from $A$ onto $B$. When does this imply $A \approx B$? Back: When $F$ is also one-to-one. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose there exists a one-to-one function $F$ from $A$ onto $B$. When does this imply $A \approx B$? Back: Always, by definition. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ### Power Sets No set is equinumerous to its [[set/index#Power Set Axiom|power set]]. This is typically shown using a diagonalization argument. %%ANKI Basic What basic set operation produces a new set the original isn't equinumerous to? Back: The power set operation. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What kind of argument is typically made to prove no set is equinumerous to its power set? Back: A diagonalization argument. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Who is attributed the discovery of the diagonalization argument? Back: Georg Cantor. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $g \colon A \rightarrow \mathscr{P}A$. Using a diagonalization argument, what set is *not* in $\mathop{\text{ran}}(g)$? Back: $\{ x \in A \mid x \not\in g(x) \}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $g \colon A \rightarrow \mathscr{P}A$. *Why* isn't $B = \{x \in A \mid x \not\in g(x) \}$ in $\mathop{\text{ran}}(g)$? Back: For all $x \in A$, $x \in B \Leftrightarrow x \not\in g(x)$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ### Equivalence Concept For any sets $A$, $B$, and $C$: * $A \approx A$; * if $A \approx B$, then $B \approx A$; * if $A \approx B$ and $B \approx C$, then $A \approx C$. 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. %%ANKI Basic Concisely state the equivalence concept of equinumerosity in Zermelo-Fraenkel set theory. Back: For all sets $A$, $B$, and $C$: * $A \approx A$; * $A \approx B \Rightarrow B \approx A$; * $A \approx B \land B \approx C \Rightarrow A \approx C$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Concisely state the equivalence concept of equinumerosity in von Neumann-Bernays set theory. Back: Class $\{ \langle A, B \rangle \mid A \approx B \}$ is reflexive on the class of all sets, symmetric, and transitive. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the reflexive property of equinumerosity in FOL? Back: $\forall A, A \approx A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the symmetric property of equinumerosity in FOL? Back: $\forall A, B, A \approx B \Rightarrow B \approx A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the transitive property of equinumerosity in FOL? Back: $\forall A, B, C, A \approx B \land B \approx C \Rightarrow A \approx C$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is $\{ \langle A, B \rangle \mid A \approx B \}$ a set? Back: No. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic *Why* isn't $\{ \langle A, B \rangle \mid A \approx B \}$ a set? Back: Because then the field of this "relation" would be a set. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is $\{ \langle A, B \rangle \mid A \approx B \}$ an equivalence relation? Back: No. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic *Why* isn't $\{ \langle A, B \rangle \mid A \approx B \}$ an equivalence relation? Back: Because then the field of this "relation" would be a set. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Finiteness A set is **finite** if and only if it is equinumerous to a [[natural-numbers|natural number]]. Otherwise it is **infinite**. %%ANKI Basic How does Enderton define a finite set? Back: As a set equinumerous to some natural number. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How does Enderton define an infinite set? Back: As a set not equinumerous to any natural number. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is $n \in \omega$ a finite set? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic *Why* isn't $n \in \omega$ a finite set? Back: N/A. It is. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is $\omega$ a finite set? Back: No. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic *Why* isn't $\omega$ a finite set? Back: There is no natural number equinumerous to $\omega$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ### Pigeonhole Principle No natural number is equinumerous to a proper subset of itself. More generally, no finite set is equinumerous to a proper subset of itself. Likewise, any set equinumerous to a proper subset of itself must be infinite. %%ANKI Basic How does Enderton state the pigeonhole principle for $\omega$? Back: No natural number is equinumerous to a proper subset of itself. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How does Enderton state the pigeonhole principle for finite sets? Back: No finite set is equinumerous to a proper subset of itself. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $m \in n \in \omega$. What principle precludes $m \approx n$? Back: The pigeonhole principle. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $S$ be a set and $n \in \omega$ such that $S \approx n$. For $m \in \omega$, when might $S \approx m$? Back: *Only* if $m = n$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the generalization of the pigeonhole principle for $\omega$? Back: The pigeonhole principle for finite sets. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the specialization of the pigeonhole principle for finite sets? Back: The pigeonhole principle for $\omega$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What name is given to the following theorem? $$\text{No finite set is equinumerous to a proper subset of itself.}$$ Back: The pigeonhole principle. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $S$ be a finite set and $f \colon S \rightarrow S$ be injective. Is $f$ a bijection? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $S$ be a finite set and $f \colon S \rightarrow S$ be injective. *Why* must $f$ be surjective? Back: Otherwise $f$ is a bijection between $S$ and a proper subset of $S$, a contradiction. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $S$ be a finite set and $f \colon S \rightarrow S$ be surjective. Is $f$ a bijection? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $S$ be a finite set and $f \colon S \rightarrow S$ be surjective. *Why* must $f$ be injective? Back: Otherwise $f$ is a bijection between a proper subset of $S$ and $S$, a contradiction. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What does the contrapositive of the pigeonhole principle state? Back: Any set equinumerous to a proper subset of itself is infinite. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What general strategy is used to prove $\omega$ is an infinite set? Back: Prove $\omega$ is equinumerous to a proper subset of itself. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Cardinal Numbers 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 * For any sets $A$ and $B$, $\mathop{\text{card}}A = \mathop{\text{card}}B$ iff $A \approx B$. * For a finite set $A$, $\mathop{\text{card}}A$ is the natural number $n$ for which $A \approx n$. %%ANKI Basic How is the cardinal number of set $A$ denoted? Back: As $\mathop{\text{card}} A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose $A$ is finite. What does $\mathop{\text{card}} A$ evaluate to? Back: The unique $n \in \omega$ such that $A \approx n$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider $n \in \omega$. What does $\mathop{\text{card}} n$ evaluate to? Back: $n$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose $a$, $b$, and $c$ are distinct objects. What does $\mathop{\text{card}} \{a, b, c\}$ evaluate to? Back: $3$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What does Enderton refer to by the "process called 'counting'"? Back: Choosing a one-to-one correspondence between two sets. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze A {cardinal number} is a set that is {$\mathop{\text{card} } A$} for some set $A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How do cardinal numbers relate to equinumerosity? Back: For any sets $A$ and $B$, $\mathop{\text{card}} A = \mathop{\text{card}} B$ iff $A \approx B$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic According to Enderton, what is the "essential demand" for defining cardinal numbers? Back: Defining cardinal numbers such that for any sets $A$ and $B$, $\mathop{\text{card}} A = \mathop{\text{card}} B$ iff $A \approx B$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What name is given to $\mathop{\text{card}} \omega$? Back: $\aleph_0$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Who is attributed the assignment $\mathop{\text{card}} \omega = \aleph_0$? Back: Georg Cantor. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% If one set $A$ of cardinality $\kappa$ is finite, then all of them are. In this case $\kappa$ is a **finite cardinal**. Otherwise $\kappa$ is an **infinite cardinal**. %%ANKI Basic How many sets $A$ exist such that $\mathop{\text{card}} A = 0$? Back: $1$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How many sets $A$ exist such that $\mathop{\text{card}} A = n^+$ for some $n \in \omega$? Back: An infinite many. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $n \in \omega$. When is $\{X \mid \mathop{\text{card}} X = n\}$ a set? Back: When $n = 0$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $n \in \omega$. When is $\{X \mid \mathop{\text{card}} X = n\}$ a class? Back: Always. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What class can we construct to prove $\{X \mid \mathop{\text{card}} X = 1\}$ is not a set? Back: $\bigcup\, \{\{X\} \mid X \text{ is a set} \}$, i.e. the union of all singleton sets. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is a finite cardinal? Back: A cardinal number equal to $\mathop{\text{card}} A$ for some finite set $A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is an infinite cardinal? Back: A cardinal number equal to $\mathop{\text{card}} A$ for some infinite set $A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic The finite cardinals are exactly what more basic set? Back: $\omega$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What set does $\aleph_0$ refer to? Back: $\mathop{\text{card}} \omega$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the "smallest" infinite cardinal? Back: $\aleph_0$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $C \subseteq A$ where $A \approx n$ for some $n \in \omega$. What does $\mathop{\text{card}} C$ evaluate to? Back: A natural number $m$ such that $m \underline{\in} n$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $C \subset A$ where $A \approx n$ for some $n \in \omega$. What does $\mathop{\text{card}} C$ evaluate to? Back: A natural number $m$ such that $m \in n$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is proposition "any subset of a finite set is finite" expressed in FOL? Back: $\forall n \in \omega, \forall A \approx n, \forall B \subseteq A, \exists m \in n, B \approx m$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the following more succinctly stated? $$\forall n \in \omega, \forall A \approx n, \forall B \subseteq A, \exists m \in n, B \approx m$$ Back: Any subset of a finite set is finite. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose sets $A$ and $B$ are finite. When is $A \cup B$ infinite? Back: The union of two finite sets is always finite. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $m, n \in \omega$ s.t. $A \approx m$ and $B \approx n$. What is the largest value $\mathop{\text{card}}(A \cup B)$ can evaluate to? Back: $m + n$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $m, n \in \omega$ s.t. $A \approx m$ and $B \approx n$. What is the smallest value $\mathop{\text{card}}(A \cup B)$ can evaluate to? Back: $\mathop{\text{max}}(m, n)$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $m, n \in \omega$ s.t. $A \approx m$ and $B \approx n$. When does $\mathop{\text{card}}(A \cup B) = m + n$? Back: When $A$ and $B$ are disjoint. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $m, n \in \omega$ s.t. $A \approx m$ and $B \approx n$. When does $\mathop{\text{card}}(A \cup B) = m$? Back: When $B \subseteq A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose sets $A$ and $B$ are finite. When is $A \cap B$ finite? Back: The intersection of two finite sets is always finite. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $m, n \in \omega$ s.t. $A \approx m$ and $B \approx n$. What is the largest value $\mathop{\text{card}}(A \cap B)$ can evaluate to? Back: $\mathop{\text{min}}(m, n)$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $m, n \in \omega$ s.t. $A \approx m$ and $B \approx n$. What is the smallest value $\mathop{\text{card}}(A \cap B)$ can evaluate to? Back: $0$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose sets $A$ and $B$ are finite. When is $A \times B$ finite? Back: The Cartesian product of two finite sets is always finite. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $m, n \in \omega$ such that $A \approx m$ and $B \approx n$. What does $\mathop{\text{card}}(A \times B)$ evaluate to? Back: $m \cdot n$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ### Addition Let $\kappa$ and $\lambda$ be any cardinal numbers. Then $\kappa + \lambda = \mathop{\text{card}}(K \cup L)$, where $K$ and $L$ are any disjoint sets of cardinality $\kappa$ and $\lambda$, respectively. %%ANKI Basic Let $\kappa$ and $\lambda$ be any cardinal numbers. How is $\kappa + \lambda$ defined? Back: As $\mathop{\text{card}}(K \cup L)$ where $K$ and $L$ are disjoint sets with cardinality $\kappa$ and $\lambda$, respectively. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $K$ and $L$ be sets. What does $\mathop{\text{card}}(K \cup L)$ evaluate to? Back: N/A. $K$ and $L$ must be disjoint sets for evaluation to make sense. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $K$ and $L$ be disjoint sets. What does $\mathop{\text{card}}(K \cup L)$ evaluate to? Back: As $\kappa + \lambda$ where $\kappa = \mathop{\text{card}} K$ and $\lambda = \mathop{\text{card}} L$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $K \approx \kappa$ and $L \approx \lambda$. What is necessary for $\mathop{\text{card}}(K \cup L) \approx \kappa + \lambda$? Back: That $K$ and $L$ are disjoint. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze {Addition} of cardinal numbers is defined in terms of the {union} of sets. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How do we prove $2 + 2 = 4$ using the recursion theorem? Back: By proving $A_2(2) = 2^{++} = 4$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How do we prove $2 + 2 = 4$ using cardinal numbers? Back: By proving for disjoint sets $K \approx 2$ and $L \approx 2$, that $K \cup L \approx 4$ holds. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $m, n \in \omega$. What does $m + n$ evaluate to in terms of cardinal numbers? Back: $\mathop{\text{card}}((m \times \{0\}) \cup (n \times \{1\}))$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ### Multiplication Let $\kappa$ and $\lambda$ be any cardinal numbers. Then $\kappa \cdot \lambda = \mathop{\text{card}}(K \times L)$, where $K$ and $L$ are any sets of cardinality $\kappa$ and $\lambda$, respectively. %%ANKI Basic Let $\kappa$ and $\lambda$ be any cardinal numbers. How is $\kappa \cdot \lambda$ defined? Back: As $\mathop{\text{card}}(K \times L)$ where $K$ and $L$ are sets with cardinality $\kappa$ and $\lambda$, respectively. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $K$ and $L$ be sets. What does $\mathop{\text{card}}(K \times L)$ evaluate to? Back: As $\kappa \cdot \lambda$ where $\kappa = \mathop{\text{card}} K$ and $\lambda = \mathop{\text{card}} L$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $K \approx \kappa$ and $L \approx \lambda$. What is necessary for $\mathop{\text{card}}(K \times L) \approx \kappa \cdot \lambda$? Back: N/A. This is true by definition. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze {Multiplication} of cardinal numbers is defined in terms of the {Cartesian product} of sets. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How do we prove $2 \cdot 2 = 4$ using the recursion theorem? Back: By proving $M_2(2) = 2 + 2 = 4$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How do we prove $2 \cdot 2 = 4$ using cardinal numbers? Back: By proving for sets $K \approx 2$ and $L \approx 2$, that $K \times L \approx 4$ holds. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $m, n \in \omega$. What does $m \cdot n$ evaluate to in terms of cardinal numbers? Back: $\mathop{\text{card}}(m \times n)$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ### Exponentiation Let $\kappa$ and $\lambda$ be any cardinal numbers. Then $\kappa^\lambda = \mathop{\text{card}}(^LK)$, where $K$ and $L$ are any sets of cardinality $\kappa$ and $\lambda$, respectively. %%ANKI Basic Let $\kappa$ and $\lambda$ be any cardinal numbers. How is $\kappa^\lambda$ defined? Back: As $\mathop{\text{card}}(^LK)$ where $K$ and $L$ are sets with cardinality $\kappa$ and $\lambda$, respectively. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $K$ and $L$ be sets. What does $\mathop{\text{card}}(^LK)$ evaluate to? Back: As $\kappa^\lambda$ where $\kappa = \mathop{\text{card}} K$ and $\lambda = \mathop{\text{card}} L$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $K$ and $L$ be sets. How is $\mathop{\text{card}}(^KL)$ expressed in terms of cardinal numbers? Back: As $\lambda^\kappa$ where $\kappa = \mathop{\text{card}} K$ and $\lambda = \mathop{\text{card}} L$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $K \approx \kappa$ and $L \approx \lambda$. What is necessary for $\mathop{\text{card}}(^LK) \approx \kappa^\lambda$? Back: N/A. This is true by definition. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze {Exponentiation} of cardinal numbers is defined in terms of the {set of functions} between sets. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How do we prove $2^2 = 4$ using the recursion theorem? Back: By proving $E_2(2) = 2 \cdot 2 = 4$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How do we prove $2^2 = 4$ using cardinal numbers? Back: By proving for sets $K \approx 2$ and $L \approx 2$, that $^LK \approx 4$ holds. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $m, n \in \omega$. What does $m^n$ evaluate to in terms of cardinal numbers? Back: $\mathop{\text{card}}(^nm)$. 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).