Peano systems.

notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json

@@ -186,7 +186,14 @@
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notes/_journal/2024-09-15.md

@@ -0,0 +1,9 @@
+---
+title: "2024-09-15"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)

notes/_journal/2024-09/2024-09-12.md

@@ -2,7 +2,7 @@
 title: "2024-09-12"
 ---
 
-- [ ] Anki Flashcards
+- [x] Anki Flashcards
 - [x] KoL
 - [x] OGS
 - [ ] Sheet Music (10 min.)

notes/_journal/2024-09/2024-09-13.md

@@ -0,0 +1,11 @@
+---
+title: "2024-09-13"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Watched [Query Execution 1](https://www.youtube.com/watch?v=I0QdCSu06_o&list=PLSE8ODhjZXjaKScG3l0nuOiDTTqpfnWFf&index=13) and [Parallel Query Execution](https://www.youtube.com/watch?v=FG_wr-0QYg4&list=PLSE8ODhjZXjaKScG3l0nuOiDTTqpfnWFf&index=14).

notes/_journal/2024-09/2024-09-14.md

@@ -0,0 +1,11 @@
+---
+title: "2024-09-14"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Notes on [[natural-numbers#Peano System|Peano systems]].

notes/algebra/sequences/triangular-numbers.md

@@ -14,7 +14,7 @@ The $n$th term of the **triangular numbers** $(T_n)_{n \geq 0}$ is the sum of wh
 %%ANKI
 Basic
 What is a polygonal number?
-Back: A number of pebbles that can be arranged into the shape of a regular polygon.
+Back: A number of pebbles that can be arranged into the shape of a regular filled polygon.
 Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
 <!--ID: 1709419325851-->
 END%%

notes/set/functions.md

@@ -1641,6 +1641,73 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1720819771087-->
 END%%
 
+### Closures
+
+If $S$ is a function and $A$ is a subset of $\mathop{\text{dom}}S$, then $A$ is said to be **closed** under $S$ if and only if whenever $x \in A$, then $S(x) \in A$. This is equivalently expressed as $S[\![A]\!] \subseteq A$.
+
+%%ANKI
+Basic
+Let $A$ be closed under $S$. Then $A$ is a subset of what other set?
+Back: $\mathop{\text{dom}}S$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726363069981-->
+END%%
+
+%%ANKI
+Basic
+Let $A$ be closed under $S$. With maximum specificity, what kind of mathematical object is $A$?
+Back: A set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726363069988-->
+END%%
+
+%%ANKI
+Basic
+Let $A$ be closed under $S$. With maximum specificity, what kind of mathematical object is $S$?
+Back: A function.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726363069991-->
+END%%
+
+%%ANKI
+Basic
+In FOL, what does it mean for set $A$ to be closed under function $S$?
+Back: $\forall x \in A, S(x) \in A$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726363069994-->
+END%%
+
+%%ANKI
+Basic
+What concept is being expressed in "$\forall x \in A, S(x) \in A$"?
+Back: Set $A$ is closed under $S$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726363069997-->
+END%%
+
+%%ANKI
+Basic
+How can we more compactly express "$\forall x \in A, S(x) \in A$"?
+Back: $S[\![A]\!] \subseteq A$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726363070000-->
+END%%
+
+%%ANKI
+Cloze
+If $S[\![A]\!] \subseteq A$, then {1:$A$} is closed {2:under} {1:$S$}.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726363070010-->
+END%%
+
+%%ANKI
+Basic
+Suppose $A$ is closed under function $S$. What imagery does the term "closed" invoke?
+Back: Applying a member of $A$ to $S$ always yields an element in $A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726363070015-->
+END%%
+
 ## Kernels
 
 Let $F \colon A \rightarrow B$. Define [[relations#Equivalence Relations|equivalence relation]] $\sim$ as $$x \sim y \Leftrightarrow f(x) = f(y)$$

notes/set/natural-numbers.md

@@ -369,6 +369,175 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1724606314406-->
 END%%
 
+## Peano System
+
+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.
+
+%%ANKI
+Basic
+A Peano system is a tuple consisting of how many members?
+Back: $3$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667616-->
+END%%
+
+%%ANKI
+Basic
+Consider Peano system $\langle N, S, e \rangle$. With maximum specificity, what kind of mathematical object is $N$?
+Back: A set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667620-->
+END%%
+
+%%ANKI
+Basic
+Consider Peano system $\langle N, S, e \rangle$. With maximum specificity, what kind of mathematical object is $S$?
+Back: A function.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667623-->
+END%%
+
+%%ANKI
+Basic
+Consider Peano system $\langle N, S, e \rangle$. What is the domain of $S$?
+Back: $N$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667626-->
+END%%
+
+%%ANKI
+Basic
+Consider Peano system $\langle N, S, e \rangle$. What is the codomain of $S$?
+Back: $N$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667629-->
+END%%
+
+%%ANKI
+Basic
+Consider Peano system $\langle N, S, e \rangle$. With maximum specificity, what kind of mathematical object is $e$?
+Back: A set or urelement.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667632-->
+END%%
+
+%%ANKI
+Basic
+In Peano system $\langle N, S, e \rangle$, $e$ is a member of what set?
+Back: $N$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667635-->
+END%%
+
+%%ANKI
+Basic
+In Peano system $\langle N, S, e \rangle$, $e$ is explicitly *not* a member of what set?
+Back: $\mathop{\text{ran}}S$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667639-->
+END%%
+
+%%ANKI
+Cloze
+Consider Peano system $\langle N, S, e \rangle$. Then {1:$e$} $\not\in$ {1:$\mathop{\text{ran} }S$}.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667643-->
+END%%
+
+%%ANKI
+Basic
+Consider Peano system $\langle N, S, e \rangle$. Function $S$ satisfies what additional condition?
+Back: $S$ is one-to-one.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667648-->
+END%%
+
+%%ANKI
+Basic
+Consider Peano system $\langle N, S, e \rangle$. What two conditions must be satisfied for $A \subseteq N$ to coincide with $N$?
+Back: $e \in A$ and $A$ is closed under $S$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667655-->
+END%%
+
+%%ANKI
+Basic
+What condition of Peano system $\langle N, S, e \rangle$ generalizes the induction principle of $\omega$?
+Back: Any set $A \subseteq N$ containing $e$ and closed under $S$ coincides with $N$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667661-->
+END%%
+
+%%ANKI
+Basic
+What name is given to the condition of Peano systems involving closures?
+Back: The Peano induction postulate.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667666-->
+END%%
+
+%%ANKI
+Basic
+The Peano induction postulate of $\langle N, S, e \rangle$ implies $N$ is the smallest set satisfying what?
+Back: That contains $e$ and is closed under $S$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667670-->
+END%%
+
+%%ANKI
+Basic
+Let $\langle N, S, e \rangle$ be a Peano system. *Why* can't there be an $A \subset N$ containing $e$ and closed under $S$?
+Back: The Peano induction postulate states $A$ *must* coincide with $N$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667673-->
+END%%
+
+%%ANKI
+Basic
+*Why* does Peano system $\langle N, S, e \rangle$ have condition $e \not\in \mathop{\text{ran}}S$?
+Back: To avoid cycles in repeated applications of $S$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667676-->
+END%%
+
+%%ANKI
+Basic
+Which condition of Peano system $\langle N, S, e \rangle$ does the following depict?
+![[peano-system-i.png]]
+Back: $e \not\in \mathop{\text{ran}}S$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667679-->
+END%%
+
+%%ANKI
+Basic
+*Why* does Peano system $\langle N, S, e \rangle$ have condition "$S$ is one-to-one"?
+Back: To avoid two members of $N$ mapping to the same element.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667682-->
+END%%
+
+%%ANKI
+Basic
+Which condition of Peano system $\langle N, S, e \rangle$ does the following depict?
+![[peano-system-ii.png]]
+Back: $S$ is one-to-one.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667685-->
+END%%
+
+%%ANKI
+Basic
+What is the Peano induction postulate?
+Back: Given Peano system $\langle N, S, e \rangle$, a set $A \subseteq N$ containing $e$ and closed under $S$ coincides with $N$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1726364667688-->
+END%%
+
 ## Bibliography
 
 * Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).

notes/set/order.md

@@ -487,7 +487,7 @@ END%%
 
 %%ANKI
 Basic
-Consider an equivalence class of $x$ (modulo $R$). What kind of mathematical object is $R$?
+Consider an equivalence class of $x$ (modulo $R$). With maximum specificity, what kind of mathematical object is $R$?
 Back: A relation.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1721098094144-->