Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1724606314394-->
END%%
%%ANKI
Basic
What can be said about an inductive subset of $\omega$?
Back: It must coincide with $\omega$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1724606314396-->
END%%
%%ANKI
Basic
Why must every inductive subset of $\omega$ coincide with $\omega$?
Back: Because $\omega$ is the smallest inductive set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1724606314397-->
END%%
%%ANKI
Basic
What does the induction principle for $\omega$ state?
Back: Every inductive subset of $\omega$ coincides with $\omega$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1724606314399-->
END%%
%%ANKI
Basic
What name is given to the principle, "every inductive subset of $\omega$ coincides with $\omega$?"
Back: The induction principle for $\omega$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1724606314400-->
END%%
%%ANKI
Basic
Inductive sets correspond to what kind of proof method?
Back: Proof by induction.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1724606314401-->
END%%
%%ANKI
Basic
Prove $P(n)$ is true for all $n \in \mathbb{N}$ using induction. What set do we prove is inductive?
Back: $\{n \in \mathbb{N} \mid P(n)\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1724606314403-->
END%%
%%ANKI
Basic
*How* are inductive sets and proof by induction related?
Back: An induction proof corresponds to proving a related set is inductive.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1724606314404-->
END%%
%%ANKI
Basic
What inductive set do we construct to prove the following by induction? $$\text{Every natural number is nonnegative}$$
Back: $\{n \in \omega \mid 0 \leq n\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1724606314405-->
END%%
%%ANKI
Basic
What inductive set do we construct to prove the following by induction? $$\text{Every nonzero natural number is the successor of another natural number}$$
Back: $\{n \in \omega \mid n = 0 \lor (\exists m \in \omega, n = m^+)\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
A **Peano system** is a triple $\langle N, S, e \rangle$ consisting of a set $N$, a function $S \colon N \rightarrow N$, and a member $e \in N$ such that the following three conditions are met:
* $e \not\in \mathop{\text{ran}}{S}$;
* $S$ is one-to-one;
* Any subset $A$ of $N$ that contains $e$ and is closed under $S$ equals $N$ itself.
A set $A$ is said to be **transitive** iff every member of a member of $A$ is itself a member of $A$. We can equivalently express this using any of the following formulations:
* $x \in a \in A \Rightarrow x \in A$
* $\bigcup A \subseteq A$
* $a \in A \Rightarrow a \subseteq A$
* $A \subseteq \mathscr{P}A$
%%ANKI
Basic
What does it mean for $A$ to be a transitive set?
Back: Every member of a member of $A$ is itself a member of $A$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209150-->
END%%
%%ANKI
Basic
In what way is the term "transitive set" ambiguous?
Back: This term can also be used to describe a transitive relation.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209152-->
END%%
%%ANKI
Cloze
A transitive {1:set} is to {2:membership} whereas a transitive {2:relation} is to {1:related}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209154-->
END%%
%%ANKI
Cloze
$A$ is a transitive set iff {$x \in a \in A$} $\Rightarrow$ {$x \in A$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209155-->
END%%
%%ANKI
Cloze
$A$ is a transitive set iff {$\bigcup A$} $\subseteq$ {$A$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209157-->
END%%
%%ANKI
Cloze
$A$ is a transitive set iff {$a \in A$} $\Rightarrow$ {$a \subseteq A$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209158-->
END%%
%%ANKI
Cloze
$A$ is a transitive set iff {$A$} $\subseteq$ {$\mathscr{P} A$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209159-->
END%%
%%ANKI
Basic
Is $\varnothing$ a transitive set?
Back: Yes.
<!--ID: 1726797209160-->
END%%
%%ANKI
Basic
*Why* isn't $\{0, 1\}$ a transitive set?
Back: N/A. It is.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209161-->
END%%
%%ANKI
Basic
*Why* isn't $\{1\}$ a transitive set?
Back: Because $0 \in 1$ but $0 \not\in \{1\}$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209163-->
END%%
%%ANKI
Basic
*Why* isn't $\{\varnothing\}$ a transitive set?
Back: N/A. It is.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209164-->
END%%
%%ANKI
Basic
*Why* isn't $\{\{\varnothing\}\}$ a transitive set?
Back: Because $\varnothing \in \{\varnothing\}$ but $\varnothing \not\in \{\{\varnothing\}\}$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209165-->
END%%
%%ANKI
Basic
Suppose $a$ is a transitive set. *Why* does $\bigcup a \cup a = a$?
Back: Because transitivity holds if and only if $\bigcup a \subseteq a$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209166-->
END%%
%%ANKI
Basic
Suppose $A \cup B = A$. What relation immediately follows?
Back: $B \subseteq A$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209167-->
END%%
%%ANKI
Basic
Suppose $A \cap B = A$. What relation immediately follows?
The recursion theorem guarantees recursively defined functions exist. More formally, let $A$ be a set, $a \in A$, and $F \colon A \rightarrow A$. Then there exists a unique function $h \colon \omega \rightarrow A$ such that, for every $n \in \omega$, $$\begin{align*} h(0) & = a \\ h(n^+) & = F(h(n)) \end{align*}$$
%%ANKI
Basic
*Why* is the recursion theorem important?
Back: It guarantees recursively defined functions exist.
Reference: “Recursion,” in _Wikipedia_, September 23, 2024, [https://en.wikipedia.org/w/index.php?title=Recursion#The_recursion_theorem](https://en.wikipedia.org/w/index.php?title=Recursion&oldid=1247328220#The_recursion_theorem).
What entities does the recursion theorem presume the existence of?
Back: A set $A$, an element $a \in A$, and a function $F \colon A \rightarrow A$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1727492422632-->
END%%
%%ANKI
Basic
Let $a \in A$ and $F \colon A \rightarrow A$. The recursion theorem implies existence of what?
Back: A unique function $h \colon \omega \rightarrow A$ such that $h(0) = a$ and $h(n^+) = F(h(n))$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1727492422636-->
END%%
%%ANKI
Basic
What function "signature" is considered in the consequent of the recursion theorem?
Back: $h \colon \omega \rightarrow A$ for some set $A$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1727492422666-->
END%%
%%ANKI
Basic
What function "signature" is considered in the antecedent of the recursion theorem?
Back: $F \colon A \rightarrow A$ for some set $A$ and function $F$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1727492422673-->
END%%
%%ANKI
Basic
Suppose the recursion theorem proves $h \colon \omega \rightarrow A$ exists. What does $h(0)$ equal?
Back: A fixed member of $A$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1727492422679-->
END%%
%%ANKI
Basic
The recursion theorem proves $h \colon \omega \rightarrow A$ exists. What does $h(n^+)$ equal?
Back: $F(h(n))$ for a fixed $F \colon A \rightarrow A$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1727492422685-->
END%%
%%ANKI
Basic
*Why* is the recursion theorem named the way it is?
Back: It guarantees recursively defined functions exist.
Reference: “Recursion,” in _Wikipedia_, September 23, 2024, [https://en.wikipedia.org/w/index.php?title=Recursion#The_recursion_theorem](https://en.wikipedia.org/w/index.php?title=Recursion&oldid=1247328220#The_recursion_theorem).
<!--ID: 1727492422693-->
END%%
%%ANKI
Basic
The recursion theorem proves function $h$ exists. What is the domain of $h$?
Back: $\omega$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1727492422707-->
END%%
%%ANKI
Basic
The recursion theorem proves function $h$ exists. What is the codomain of $h$?
Back: A fixed set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1727492422711-->
END%%
%%ANKI
Basic
The recursion theorem proves $h \colon \omega \rightarrow A$ exists. How do we compute $h(n)$?
Back: By applying $F$ to a fixed initial element $n$ times.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
In Enderton's recursion theorem proof, desired $h \colon \omega \rightarrow A$ is {a function} because {$\{ n \in \omega \mid \text{at most one } y \text{ such that } \langle n, y \rangle \in h \}$} is {an inductive set}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1727627702468-->
END%%
%%ANKI
Basic
In Enderton's recursion theorem proof, how is it shown the domain of desired $h \colon \omega \rightarrow A$ equals $\omega$?
Back: By proving $\mathop{\text{dom}} h$ is an inductive set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1727627702469-->
END%%
%%ANKI
Cloze
In Enderton's recursion theorem proof,desired $h \colon \omega \rightarrow A$ is {unique} because {$\{ n \in \omega \mid h_1(n) = h_2(n) \}$} is {an inductive set}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1727627702470-->
END%%
%%ANKI
Basic
*Why* is there no function $h \colon \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all $n \in \mathbb{Z}$, $$\begin{align*} h(0) & = 0 \\ h(n + 1) & = h(n) + 1 \end{align*}$$
Back: Because $\mathbb{Z}$ has no "starting point" to ground the recursive definition.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
*Why* is there no function $h \colon \mathbb{N} \rightarrow \mathbb{N}$ such that for all $n \in \mathbb{N}$, $$\begin{align*} h(0) & = 0 \\ h(n + 1) & = h(n) + 1 \end{align*}$$
For each $m \in \omega$, there exists (by the recursion theorem) a unique function $A_m \colon \omega \rightarrow \omega$ such that for all $n \in \omega$, $$\begin{align*} A_m(0) & = m, \\ A_m(n^+) & = A_m(n)^+ \end{align*}$$
**Addition** ($+$) is the binary operation on $\omega$ such that for any $m, n \in \omega$, $$m + n = A_m(n).$$
%%ANKI
Basic
Let $A_m \colon \omega \rightarrow \omega$ denote recursively defined addition. How is $A_m(n)$ more traditionally denoted?
Back: As $m + n$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914175-->
END%%
%%ANKI
Basic
Let $A_m \colon \omega \rightarrow \omega$ denote recursively defined addition. How is $m + n$ defined in terms of $A_m$?
Back: As $A_m(n)$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914179-->
END%%
%%ANKI
Basic
Let $A_m \colon \omega \rightarrow \omega$ denote recursively defined addition. What does $A_m(0)$ evalute to?
Back: $m$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914180-->
END%%
%%ANKI
Basic
Let $A_m \colon \omega \rightarrow \omega$ denote recursively defined addition. What does $A_m(n^+)$ evalute to?
Back: $A_m(n)^+$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914181-->
END%%
%%ANKI
Basic
Let $A_m \colon \omega \rightarrow \omega$ denote recursively defined addition. How do we know $A_m$ exists?
Back: The recursion theorem states it does.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914182-->
END%%
%%ANKI
Basic
How is $m + n$ defined as a relation?
Back: $\{ \langle m, n, A_m(n) \rangle \mid m, n \in \omega \}$ where $A_m \colon \omega \rightarrow \omega$ is recursively defined addition.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914183-->
END%%
%%ANKI
Basic
Is $+$ (addition) a function, operation, both, or neither?
Back: Both.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914184-->
END%%
%%ANKI
Basic
Without introducing new notation, what does $m + 0$ evaluate to?
Back: $m$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914185-->
END%%
%%ANKI
Basic
Without introducing new notation, what does $m + n^+$ evaluate to?
Back: $(m + n)^+$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914186-->
END%%
### Multiplication
For each $m \in \omega$, there exists (by the recursion theorem) a unique function $M_m \colon \omega \rightarrow \omega$ such that for all $n \in \omega$, $$\begin{align*} M_m(0) & = 0, \\ M_m(n^+) & = M_m(n) + m \end{align*}$$
**Multiplication** ($\cdot$) is the binary operation on $\omega$ such that for any $m, n \in \omega$, $$m \cdot n = M_m(n).$$
%%ANKI
Basic
Let $M_m \colon \omega \rightarrow \omega$ denote recursively defined multiplication. How is $M_m(n)$ more traditionally denoted?
Back: As $m \cdot n$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914187-->
END%%
%%ANKI
Basic
Let $M_m \colon \omega \rightarrow \omega$ denote recursively defined multiplication. How is $m \cdot n$ defined in terms of $M_m$?
Back: As $M_m(n)$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914188-->
END%%
%%ANKI
Basic
Let $M_m \colon \omega \rightarrow \omega$ denote recursively defined multiplication. What does $M_m(0)$ evalute to?
Back: $0$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914189-->
END%%
%%ANKI
Basic
Let $M_m \colon \omega \rightarrow \omega$ denote recursively defined multiplication. What does $M_m(n^+)$ evalute to?
Back: $M_m(n) + m$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914190-->
END%%
%%ANKI
Basic
Let $M_m \colon \omega \rightarrow \omega$ denote recursively defined multiplication. How do we know $M_m$ exists?
Back: The recursion theorem states it does.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914191-->
END%%
%%ANKI
Basic
How is $m \cdot n$ defined as a relation?
Back: $\{ \langle m, n, M_m(n) \rangle \mid m, n \in \omega \}$ where $M_m \colon \omega \rightarrow \omega$ is recursively defined multiplication.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914192-->
END%%
%%ANKI
Basic
Is $\cdot$ (multiplication) a function, operation, both, or neither?
Back: Both.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914193-->
END%%
%%ANKI
Basic
Without introducing new notation, what does $m \cdot 0$ evaluate to?
Back: $0$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914194-->
END%%
%%ANKI
Basic
Without introducing new notation, what does $m \cdot n^+$ evaluate to?
Back: $m \cdot n + m$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914195-->
END%%
### Exponentiation
For each $m \in \omega$, there exists (by the recursion theorem) a unique function $E_m \colon \omega \rightarrow \omega$ such that for all $n \in \omega$, $$\begin{align*} E_m(0) & = 1, \\ E_m(n^+) & = E_m(n) \cdot m \end{align*}$$
**Exponentiation** is the binary operation on $\omega$ such that for any $m, n \in \omega$, $$m^n = E_m(n).$$
%%ANKI
Basic
Let $E_m \colon \omega \rightarrow \omega$ denote recursively defined exponentiation. How is $E_m(n)$ more traditionally denoted?
Back: As $m^n$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914196-->
END%%
%%ANKI
Basic
Let $E_m \colon \omega \rightarrow \omega$ denote recursively defined exponentiation. How is $m^n$ defined in terms of $E_m$?
Back: As $E_m(n)$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914197-->
END%%
%%ANKI
Basic
Let $E_m \colon \omega \rightarrow \omega$ denote recursively defined exponentiation. What does $E_m(0)$ evalute to?
Back: $1$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914198-->
END%%
%%ANKI
Basic
Let $E_m \colon \omega \rightarrow \omega$ denote recursively defined exponentiation. What does $E_m(n^+)$ evalute to?
Back: $E_m(n) \cdot m$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914199-->
END%%
%%ANKI
Basic
Let $E_m \colon \omega \rightarrow \omega$ denote recursively defined exponentiation. How do we know $E_m$ exists?
Back: The recursion theorem states it does.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914200-->
END%%
%%ANKI
Basic
How is $m^n$ defined as a relation?
Back: $\{ \langle m, n, E_m(n) \rangle \mid m, n \in \omega \}$ where $E_m \colon \omega \rightarrow \omega$ is recursively defined exponentiation.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914201-->
END%%
%%ANKI
Basic
Is exponentiation a function, operation, both, or neither?
Back: Both.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914202-->
END%%
%%ANKI
Basic
Without introducing new notation, what does $m^0$ evaluate to?
Back: $1$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1729804914203-->
END%%
%%ANKI
Basic
Without introducing new notation, what does $m^{(n^+)}$ evaluate to?
Back: $m^n \cdot m$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
* “Recursion,” in _Wikipedia_, September 23, 2024, [https://en.wikipedia.org/w/index.php?title=Recursion#The_recursion_theorem](https://en.wikipedia.org/w/index.php?title=Recursion&oldid=1247328220#The_recursion_theorem).
