From 1e7eeb5604bbe6d5046688b86472b2c72775da65 Mon Sep 17 00:00:00 2001
From: Joshua Potter <jrpotter2112@gmail.com>
Date: Sun, 15 Sep 2024 16:23:10 -0600
Subject: [PATCH] Peano systems.

---
 .../plugins/obsidian-to-anki-plugin/data.json |  24 ++-
 notes/_journal/2024-09-15.md                  |   9 +
 notes/_journal/{ => 2024-09}/2024-09-12.md    |   2 +-
 notes/_journal/2024-09/2024-09-13.md          |  11 ++
 notes/_journal/2024-09/2024-09-14.md          |  11 ++
 notes/algebra/sequences/triangular-numbers.md |   2 +-
 notes/set/functions.md                        |  67 +++++++
 notes/set/images/peano-system-i.png           | Bin 0 -> 17003 bytes
 notes/set/images/peano-system-ii.png          | Bin 0 -> 18837 bytes
 notes/set/natural-numbers.md                  | 169 ++++++++++++++++++
 notes/set/order.md                            |   2 +-
 11 files changed, 288 insertions(+), 9 deletions(-)
 create mode 100644 notes/_journal/2024-09-15.md
 rename notes/_journal/{ => 2024-09}/2024-09-12.md (61%)
 create mode 100644 notes/_journal/2024-09/2024-09-13.md
 create mode 100644 notes/_journal/2024-09/2024-09-14.md
 create mode 100644 notes/set/images/peano-system-i.png
 create mode 100644 notes/set/images/peano-system-ii.png

diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
index b03f2fa..fbd6e51 100644
--- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
+++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
@@ -186,7 +186,14 @@
     "infinity.png",
     "nan.png",
     "triangular-gnomon.png",
-    "pascals-triangle.png"
+    "pascals-triangle.png",
+    "function-bijective.png",
+    "function-injective.png",
+    "function-surjective.png",
+    "function-general.png",
+    "function-kernel.png",
+    "peano-system-i.png",
+    "peano-system-ii.png"
   ],
   "File Hashes": {
     "algorithms/index.md": "3ac071354e55242919cc574eb43de6f8",
@@ -326,7 +333,7 @@
     "algebra/sequences/index.md": "208174a5a078b120fa11e296ad1d09c1",
     "_journal/2024-03-02.md": "08c3cae1df0079293b47e1e9556f1ce1",
     "_journal/2024-03/2024-03-01.md": "70da812300f284df72718dd32fc39322",
-    "algebra/sequences/triangular-numbers.md": "bf08ea7759b24defb7d5d3912cf04503",
+    "algebra/sequences/triangular-numbers.md": "aafaf54e5aff9ca3c7354591fce9f833",
     "algebra/sequences/square-numbers.md": "171f7c5a8dac088afba40923ab86c68e",
     "_journal/2024-03-03.md": "c4977a3778ed227b768c3f9ad5512670",
     "_journal/2024-03/2024-03-02.md": "8136792b0ee6e08232e4f60c88d461d2",
@@ -578,7 +585,7 @@
     "_journal/2024-06/2024-06-12.md": "f82dfa74d0def8c3179d3d076f94558e",
     "_journal/2024-06-14.md": "5d12bc272238ac985a1d35d3d63ea307",
     "_journal/2024-06/2024-06-13.md": "e2722a00585d94794a089e8035e05728",
-    "set/functions.md": "6716f8a32af73e5a4d1b2cbf6987b60f",
+    "set/functions.md": "59e449d6756b57c846cdf07b0a1b4330",
     "_journal/2024-06-15.md": "92cb8dc5c98e10832fb70c0e3ab3cec4",
     "_journal/2024-06/2024-06-14.md": "5d12bc272238ac985a1d35d3d63ea307",
     "lambda-calculus/beta-reduction.md": "a8e2825c84e842ceef7aa638a493b91a",
@@ -738,7 +745,7 @@
     "_journal/2024-08/2024-08-15.md": "7c3a96a25643b62b0064bf32cb17d92f",
     "_journal/2024-08-17.md": "b06a551560c377f61a1b39286cd43cee",
     "_journal/2024-08/2024-08-16.md": "da1127a1985074a3930b4c3512344025",
-    "set/order.md": "3bf63dd9c8ce6d2b4c6905dab0bd4aad",
+    "set/order.md": "b69f922200514975b7a7028eef030b59",
     "_journal/2024-08-18.md": "6f8aec69e00401b611db2a377a3aace5",
     "ontology/philosophy/properties.md": "41b32249d3e4c23d73ddb3a417d65a4c",
     "_journal/2024-08-19.md": "94836e52ec04a72d3e1dbf3854208f65",
@@ -755,7 +762,7 @@
     "_journal/2024-08/2024-08-21.md": "1637b8ec8475cf3eb4f41d1d86cbf5df",
     "_journal/2024-08/2024-08-20.md": "e8bec308d1b29e411c6799ace7ef6571",
     "_journal/2024-08-23.md": "3b2feab2cc927e267263cb1e9c173d50",
-    "set/natural-numbers.md": "f37647a51f457cb7d335e4e4fff227de",
+    "set/natural-numbers.md": "bf73972ec1ca619ba0124169c25b1c39",
     "_journal/2024-08-24.md": "563fad24740e44734a87d7c3ec46cec4",
     "_journal/2024-08/2024-08-23.md": "7b5a40e83d8f07ff54cd9708017d029c",
     "_journal/2024-08/2024-08-22.md": "050235d5dc772b542773743b57ce3afe",
@@ -797,7 +804,12 @@
     "_journal/2024-09/2024-09-10.md": "71a766783213f58552990b3ab1baeb50",
     "_journal/2024-09/2024-09-08.md": "0949eaf8df8d7e35cc0734d3a823921a",
     "_journal/2024-09/2024-09-07.md": "807e46a75e8b4b414141fb0c7d3f03e4",
-    "_journal/2024-09/2024-09-06.md": "7ea6a87f77cf49943eb76dd1052bd736"
+    "_journal/2024-09/2024-09-06.md": "7ea6a87f77cf49943eb76dd1052bd736",
+    "_journal/2024-09-14.md": "774019f651e728faa288041ce4b265d3",
+    "_journal/2024-09/2024-09-13.md": "8c8f33fdd8242e5ab9adaa797dea7995",
+    "_journal/2024-09/2024-09-12.md": "30968fa3d73c005bdb4acc2025b34e11",
+    "_journal/2024-09-15.md": "a203f489d0205246b9b625354123046c",
+    "_journal/2024-09/2024-09-14.md": "1050e9ae0dfe4196a419105c43c2162f"
   },
   "fields_dict": {
     "Basic": [
diff --git a/notes/_journal/2024-09-15.md b/notes/_journal/2024-09-15.md
new file mode 100644
index 0000000..32f6825
--- /dev/null
+++ b/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)
\ No newline at end of file
diff --git a/notes/_journal/2024-09-12.md b/notes/_journal/2024-09/2024-09-12.md
similarity index 61%
rename from notes/_journal/2024-09-12.md
rename to notes/_journal/2024-09/2024-09-12.md
index 1be126e..2112cbc 100644
--- a/notes/_journal/2024-09-12.md
+++ b/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.)
diff --git a/notes/_journal/2024-09/2024-09-13.md b/notes/_journal/2024-09/2024-09-13.md
new file mode 100644
index 0000000..c1c470c
--- /dev/null
+++ b/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).
\ No newline at end of file
diff --git a/notes/_journal/2024-09/2024-09-14.md b/notes/_journal/2024-09/2024-09-14.md
new file mode 100644
index 0000000..b6d1622
--- /dev/null
+++ b/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]].
\ No newline at end of file
diff --git a/notes/algebra/sequences/triangular-numbers.md b/notes/algebra/sequences/triangular-numbers.md
index 5c6ef91..b65cbcc 100644
--- a/notes/algebra/sequences/triangular-numbers.md
+++ b/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%%
diff --git a/notes/set/functions.md b/notes/set/functions.md
index c4bff1f..5bcdd04 100644
--- a/notes/set/functions.md
+++ b/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 $$x \sim y \Leftrightarrow f(x) = f(y)$$
diff --git a/notes/set/images/peano-system-i.png b/notes/set/images/peano-system-i.png
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HcmV?d00001

diff --git a/notes/set/natural-numbers.md b/notes/set/natural-numbers.md
index 61ee1fe..3111ec9 100644
--- a/notes/set/natural-numbers.md
+++ b/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).
\ No newline at end of file
diff --git a/notes/set/order.md b/notes/set/order.md
index 2e7478b..40720c1 100644
--- a/notes/set/order.md
+++ b/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-->