From d10dc3d24bfb24201a5775b50c29125eb6f76d59 Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Mon, 16 Dec 2024 11:37:31 -0700 Subject: [PATCH] Additional notes on cardinal numbers. --- .../plugins/obsidian-to-anki-plugin/data.json | 4 +- notes/_journal/2024-12-16.md | 3 +- notes/set/cardinality.md | 82 +++++++++++++++++-- 3 files changed, 81 insertions(+), 8 deletions(-) diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 72b2964..a64a3c1 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -966,7 +966,7 @@ "_journal/2024-11/2024-11-21.md": "951b6034d60a40dbd8201c50abf0dbb9", "_journal/2024-11/2024-11-20.md": "951b6034d60a40dbd8201c50abf0dbb9", "_journal/2024-11/2024-11-19.md": "d879f57154cb27cb168eb1f1f430e312", - "set/cardinality.md": "1a9a31dcc1d1bddce9a316f9a3062324", + "set/cardinality.md": "9609eca0fe79de40e43d56cd0ef45985", "geometry/area.md": "7f947bb5ac782495a1fb4a63bb2463e7", "_journal/2024-11-23.md": "911f82ab8aede5ecdb96493aef64b0b9", "_journal/2024-11/2024-11-22.md": "51117030e2364dbce3a8d507dead86ae", @@ -1023,7 +1023,7 @@ "_journal/2024-12/2024-12-13.md": "d2223f90fd1ce3d82a4fbb6828a1ec56", "_journal/2024-12/2024-12-12.md": "59e71caa4e9ebdb11a7c7549c33bed20", "linkers/object-files.md": "77767f310330b8650a5023dd0522226c", - "_journal/2024-12-16.md": "77c27920d5b44b6315c2ea22b990eefc", + "_journal/2024-12-16.md": "6c03d49c05eb47486aae48aa40dcef9f", "_journal/2024-12/2024-12-15.md": "be66c8808d8bb66d4e7b91db7c93c94a", "linkers/elf.md": "83a27a7d71ca26f8f034770a31d58fa0", "c17/strings/printf.md": "8b67cfbccaf35dd9488b73e7e5555405", diff --git a/notes/_journal/2024-12-16.md b/notes/_journal/2024-12-16.md index 898a5e7..313979d 100644 --- a/notes/_journal/2024-12-16.md +++ b/notes/_journal/2024-12-16.md @@ -8,4 +8,5 @@ title: "2024-12-16" - [ ] Sheet Music (10 min.) - [ ] Korean (Read 1 Story) -* Notes on sections found within an [[elf|ELF]] file. \ No newline at end of file +* Notes on sections found within an [[elf|ELF]] file. +* Additional examples on cardinal numbers and their relation to the recursion theorem. \ No newline at end of file diff --git a/notes/set/cardinality.md b/notes/set/cardinality.md index 8d533c6..03b1dab 100644 --- a/notes/set/cardinality.md +++ b/notes/set/cardinality.md @@ -643,15 +643,15 @@ END%% %%ANKI Basic -Let $K$ and $L$ be sets. How is $\mathop{\text{card}}(K \cup L)$ expressed in terms of cardinal numbers? -Back: N/A. $K$ and $L$ must be *disjoint* sets for this to make sense. +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. How is $\mathop{\text{card}}(K \cup L)$ expressed in terms of cardinal numbers? +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). @@ -672,6 +672,30 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre 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. @@ -686,7 +710,7 @@ END%% %%ANKI Basic -Let $K$ and $L$ be sets. How is $\mathop{\text{card}}(K \times L)$ expressed in terms of cardinal numbers? +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). @@ -707,6 +731,30 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre 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. @@ -721,7 +769,7 @@ END%% %%ANKI Basic -Let $K$ and $L$ be sets. How is $\mathop{\text{card}}(^LK)$ expressed in terms of cardinal numbers? +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). @@ -750,6 +798,30 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre 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). \ No newline at end of file