1077 lines
34 KiB
Markdown
1077 lines
34 KiB
Markdown
---
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title: Natural Numbers
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TARGET DECK: Obsidian::STEM
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FILE TAGS: set::nat
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tags:
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- natural-number
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- set
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---
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## Overview
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The standard way of representing the natural numbers is as follows:
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* $0 = \varnothing$
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* $1 = \{0\} = \{\varnothing\}$
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* $2 = \{0, 1\} = \{\varnothing, \{\varnothing\}\}$
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* $\ldots$
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That is, each natural number corresponds to the set of natural numbers smaller than it.
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%%ANKI
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Basic
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How is the number $0$ represented as a set?
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Back: $\varnothing$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724485233219-->
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END%%
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%%ANKI
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Basic
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How is the number $1$ represented as a set?
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Back: $\{0\} = \{\varnothing\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724485233247-->
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END%%
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%%ANKI
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Basic
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How is the number $2$ represented as a set?
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Back: $\{0, 1\} = \{\varnothing, \{\varnothing\}\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724485233252-->
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END%%
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%%ANKI
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Basic
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Who came up with the standard set representation of natural numbers?
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Back: John von Neumann.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724485233257-->
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END%%
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%%ANKI
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Basic
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Consider the set representation of $n \in \mathbb{N}$. How many members does $n$ have?
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Back: $n$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724485233263-->
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END%%
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%%ANKI
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Basic
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Consider the set representation of $n \in \mathbb{N}$. What are the members of $n$?
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Back: $0$, $1$, $\ldots$, $n - 1$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724485233269-->
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END%%
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%%ANKI
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Basic
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Let $n \in \mathbb{N}$. *Why* is $n \in n + 1$?
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Back: $n + 1$ is a set containing all preceding natural numbers.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724485233274-->
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END%%
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%%ANKI
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Basic
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Let $n \in \mathbb{N}$. *Why* is $n \subseteq n + 1$?
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Back: $n$ and $n + 1$ are sets containing all their preceding natural numbers.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724485233279-->
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END%%
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## Inductive Sets
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For any set $a$, its **successor** $a^+$ is defined as $$a^+ = a \cup \{a\}$$
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%%ANKI
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Basic
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How is the successor of a set $a$ denoted?
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Back: $a^+$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724485233287-->
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END%%
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%%ANKI
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Basic
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How is the successor of a set $a$ defined?
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Back: As $a^+ = a \cup \{a\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724485233291-->
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END%%
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%%ANKI
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Basic
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Set $\{a, b\}^+$ equals what other set?
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Back: $\{a, b, \{a, b\}\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724485233295-->
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END%%
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%%ANKI
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Basic
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Set $\{a\}^+$ equals what other set?
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Back: $\{a, \{a\}\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724485233299-->
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END%%
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%%ANKI
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Basic
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Set $\{a, \{a, b\}, \{a, b, c\}\}$ can be written as the successor of what set?
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Back: N/A.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724485516768-->
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END%%
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%%ANKI
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Basic
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Set $\{a, b, \{a, b\}\}$ can be written as the successor of what set?
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Back: $\{a, b\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724485516774-->
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END%%
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%%ANKI
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Basic
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Set $\{a, \{a, b\}\}$ can be written as the successor of what set?
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Back: N/A.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724485516777-->
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END%%
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%%ANKI
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Basic
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Set $\{a, \{a, b\}, \{a, \{a, b\}\}\}$ can be written as the successor of what set?
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Back: $\{a, \{a, b\}\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724485516780-->
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END%%
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%%ANKI
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Basic
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If $n \in \mathbb{N}$ then $n \in n + 1$. What analagous statement holds for arbitrary set $a$?
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Back: $a \in a^+$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724485233303-->
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END%%
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%%ANKI
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Basic
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If $n \in \mathbb{N}$ then $n \subseteq n + 1$. What analagous statement holds for arbitrary set $a$?
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Back: $a \subseteq a^+$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724485233283-->
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END%%
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A set $A$ is **inductive** if and only if $\varnothing \in A$ and $\forall a \in A, a^+ \in A$.
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%%ANKI
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Basic
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What does it mean for a set $A$ to be closed under successor?
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Back: If $a \in A$, then $a^+ \in A$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724486269548-->
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END%%
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%%ANKI
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Basic
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Write "set $B$ is closed under successor" in FOL.
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Back: $\forall b \in B, b^+ \in B$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724486269552-->
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END%%
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%%ANKI
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Basic
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What does it mean for a set $A$ to be inductive?
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Back: $\varnothing \in A$ and $A$ is closed under successor.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724486269555-->
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END%%
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%%ANKI
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Cloze
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Set $A$ is inductive iff {$\varnothing \in A$} and {$A$ is closed under successor}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724486269558-->
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END%%
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%%ANKI
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Basic
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An inductive set is closed under what operation?
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Back: Successor.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724486269562-->
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END%%
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%%ANKI
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Basic
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What set is the "seed" of an inductive set?
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Back: $\varnothing$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724486269565-->
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END%%
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%%ANKI
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Basic
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Let $a \in A$ where $A$ is an inductive set. What other members must belong to $A$?
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Back: $a^+$, $a^{++}$, $\ldots$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724486269568-->
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END%%
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%%ANKI
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Basic
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What natural number corresponds to $\varnothing^{+++}$?
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Back: $3$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724486269571-->
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END%%
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%%ANKI
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Basic
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What natural number corresponds to $\varnothing$?
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Back: $0$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724486269575-->
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END%%
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A **natural number** is a set that belongs to every inductive set.
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%%ANKI
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Basic
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How is the set of natural numbers *defined* in set theory?
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Back: As a set belonging to every inductive set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724486756997-->
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END%%
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%%ANKI
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Basic
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What greek letter is used to denote the set of natural numbers?
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Back: $\omega$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724486757001-->
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END%%
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%%ANKI
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Basic
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In set theory, $\omega$ denotes what set?
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Back: The natural numbers.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724606314391-->
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END%%
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%%ANKI
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Basic
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What is the smallest inductive set?
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Back: $\omega$, i.e. the set of natural numbers.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724486757004-->
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END%%
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%%ANKI
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Basic
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How might $\omega$ be defined as an intersection of classes?
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Back: $\omega = \bigcap\,\{A \mid A \text{ is inductive}\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724486757007-->
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END%%
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%%ANKI
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Basic
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Suppose $n \in \omega$. By definition of natural numbers, what other sets must $n$ be a member of?
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Back: Every other inductive set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724486757010-->
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END%%
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%%ANKI
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Basic
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What can be said about a subset of $\omega$?
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Back: N/A.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724606314394-->
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END%%
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%%ANKI
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Basic
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What can be said about an inductive subset of $\omega$?
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Back: It must coincide with $\omega$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724606314396-->
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END%%
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%%ANKI
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Basic
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Why must every inductive subset of $\omega$ coincide with $\omega$?
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Back: Because $\omega$ is the smallest inductive set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724606314397-->
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END%%
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%%ANKI
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Basic
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What does the induction principle for $\omega$ state?
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Back: Every inductive subset of $\omega$ coincides with $\omega$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724606314399-->
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END%%
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%%ANKI
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Basic
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What name is given to the principle, "every inductive subset of $\omega$ coincides with $\omega$?"
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Back: The induction principle for $\omega$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724606314400-->
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END%%
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%%ANKI
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Basic
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Inductive sets correspond to what kind of proof method?
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Back: Proof by induction.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724606314401-->
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END%%
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%%ANKI
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Basic
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Prove $P(n)$ is true for all $n \in \mathbb{N}$ using induction. What set do we prove is inductive?
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Back: $\{n \in \mathbb{N} \mid P(n)\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724606314403-->
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END%%
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%%ANKI
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Basic
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*How* are inductive sets and proof by induction related?
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Back: An induction proof corresponds to proving a related set is inductive.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724606314404-->
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END%%
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%%ANKI
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Basic
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What inductive set do we construct to prove the following by induction? $$\text{Every natural number is nonnegative}$$
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Back: $\{n \in \omega \mid 0 \leq n\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724606314405-->
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END%%
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%%ANKI
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Basic
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What inductive set do we construct to prove the following by induction? $$\text{Every nonzero natural number is the successor of another natural number}$$
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Back: $\{n \in \omega \mid n = 0 \lor (\exists m \in \omega, n = m^+)\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1724606314406-->
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END%%
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## Peano System
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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:
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* $e \not\in \mathop{\text{ran}}{S}$;
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* $S$ is one-to-one;
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* Any subset $A$ of $N$ that contains $e$ and is closed under $S$ equals $N$ itself.
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Given $\sigma = \{\langle n, n^+ \rangle \mid n \in \omega\}$, $\langle \omega, \sigma, 0 \rangle$ is a Peano system.
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%%ANKI
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Basic
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A Peano system is a tuple consisting of how many members?
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Back: $3$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1726364667616-->
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END%%
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%%ANKI
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Basic
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Consider Peano system $\langle N, S, e \rangle$. What kind of mathematical object is $N$?
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Back: A set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1726364667620-->
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END%%
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%%ANKI
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Basic
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Consider Peano system $\langle N, S, e \rangle$. What kind of mathematical object is $S$?
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Back: A function.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1726364667623-->
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END%%
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%%ANKI
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Basic
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Consider Peano system $\langle N, S, e \rangle$. What is the domain of $S$?
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Back: $N$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1726364667626-->
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END%%
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%%ANKI
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Basic
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Consider Peano system $\langle N, S, e \rangle$. What is the codomain of $S$?
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Back: $N$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1726364667629-->
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END%%
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%%ANKI
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Basic
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Consider Peano system $\langle N, S, e \rangle$. What kind of mathematical object is $e$?
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Back: A set or urelement.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1726364667632-->
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END%%
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%%ANKI
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Basic
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In Peano system $\langle N, S, e \rangle$, $e$ is a member of what set?
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Back: $N$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1726364667635-->
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END%%
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%%ANKI
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Basic
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In Peano system $\langle N, S, e \rangle$, $e$ is explicitly *not* a member of what set?
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Back: $\mathop{\text{ran}}S$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1726364667639-->
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END%%
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%%ANKI
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Cloze
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Consider Peano system $\langle N, S, e \rangle$. Then {1:$e$} $\not\in$ {1:$\mathop{\text{ran} }S$}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1726364667643-->
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END%%
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%%ANKI
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Basic
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Consider Peano system $\langle N, S, e \rangle$. Function $S$ satisfies what additional condition?
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Back: $S$ is one-to-one.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1726364667648-->
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END%%
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%%ANKI
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Basic
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Consider Peano system $\langle N, S, e \rangle$. What two conditions must be satisfied for $A \subseteq N$ to coincide with $N$?
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Back: $e \in A$ and $A$ is closed under $S$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1726364667655-->
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END%%
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%%ANKI
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Basic
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What condition of Peano system $\langle N, S, e \rangle$ generalizes the induction principle of $\omega$?
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Back: The Peano induction postulate.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1726364667661-->
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END%%
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%%ANKI
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Basic
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Given Peano system $\langle N, S, e \rangle$, what does the Peano induction postulate state?
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Back: Any set $A \subseteq N$ containing $e$ and closed under $S$ coincides with $N$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1727895401785-->
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END%%
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%%ANKI
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Basic
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What name is given to the condition of Peano systems involving closures?
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Back: The Peano induction postulate.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1726364667666-->
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END%%
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%%ANKI
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Basic
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The Peano induction postulate of $\langle N, S, e \rangle$ implies $N$ is the smallest set satisfying what?
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Back: That contains $e$ and is closed under $S$.
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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 violate?
|
|
![[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 violate?
|
|
![[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%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which Peano system serves as the prototypical example?
|
|
Back: $\langle \omega, \sigma, 0 \rangle$ where $\sigma$ denotes the successor restricted to the natural numbers.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1726928580006-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $\langle \omega, \sigma, 0 \rangle$ be a Peano system. How is $\omega$ defined?
|
|
Back: As the set of natural numbers.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1726928580037-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $\langle \omega, \sigma, 0 \rangle$ be a Peano system. How is $\sigma$ defined?
|
|
Back: $\{\langle n, n^+ \rangle \mid n \in \omega\}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1726928580064-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $\langle \omega, \sigma, 0 \rangle$ be a Peano system. What kind of mathematical object is $\sigma$?
|
|
Back: A function.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1726928580069-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $\langle \omega, \sigma, 0 \rangle$ be a Peano system. What is the domain of $\sigma$?
|
|
Back: $\omega$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1726928580075-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $\langle \omega, \sigma, 0 \rangle$ be a Peano system. What is the codomain of $\sigma$?
|
|
Back: $\omega$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1726928580081-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $\langle \omega, \sigma, 0 \rangle$ be a Peano system. Its Peano induction postulate goes by what other name?
|
|
Back: The induction principle for $\omega$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1726928580087-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $\langle \omega, \sigma, 0 \rangle$ be a Peano system. The induction principle for $\omega$ satisfies what postulate of the system?
|
|
Back: The Peano induction postulate.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1726928580092-->
|
|
END%%
|
|
|
|
## Transitivity
|
|
|
|
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?
|
|
Back: $B = A$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1726797814900-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
$A$ is a transitive set iff {$\bigcup$}$A^+ =$ {$A$}.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1726797209168-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which sets serve as the prototypical example of transitive sets?
|
|
Back: The natural numbers.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1726857149204-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Is $n \in \omega$ a transitive set?
|
|
Back: Yes.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1726857149214-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Is $\omega$ a transitive set?
|
|
Back: Yes.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1726857149225-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How can we alternatively state "$\omega$ is a transitive set"?
|
|
Back: Every natural number is a set of natural numbers.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1726976055230-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How can we more concisely state "every natural number is a set of natural numbers"?
|
|
Back: $\omega$ is a transitive set.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1726976055239-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What does $\mathscr{P}\,0$ evaluate to?
|
|
Back: $1$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727019806525-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What does $\mathscr{P}\,1$ evaluate to?
|
|
Back: $2$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727019806532-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What does $\mathscr{P}\,2$ evaluate to?
|
|
Back: $\{0, 1, 2, \{1\}\}$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727019806534-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $X$ is transitive. Is $\bigcup X$ transitive?
|
|
Back: Yes.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727019806538-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose for all $x \in X$, $x$ is transitive. Is $X$ transitive?
|
|
Back: Not necessarily.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727019806541-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose for all $x \in X$, $x$ is transitive. Is $\bigcup X$ transitive?
|
|
Back: Yes.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727019806545-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose for all $x \in X$, $x$ is transitive. Is $\bigcap X$ transitive?
|
|
Back: N/A. If $X = \varnothing$, $\bigcap X$ is undefined.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727019806550-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $X \neq \varnothing$ and for all $x \in X$, $x$ is transitive. Is $\bigcap X$ transitive?
|
|
Back: Yes.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727019806554-->
|
|
END%%
|
|
|
|
## Recursion Theorem
|
|
|
|
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).
|
|
<!--ID: 1727492422625-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The recursion theorem on $\omega$ assumes existence of what Peano system?
|
|
Back: $\langle \omega, \sigma, 0 \rangle$ where $\sigma$ is the successor operation restricted to the natural numbers.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727629020357-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
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).
|
|
<!--ID: 1727492422716-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The recursion theorem shows existence of $h \colon \omega \rightarrow A$. What is $A$?
|
|
Back: A set fixed before application of the recursion theorem.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727629020364-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $a \in A$ and $F \colon A \rightarrow A$. Using the recursion theorem, how else is $F(F(F(F(a))))$ expressed?
|
|
Back: The recursion theorem implies existence of $h \colon \omega \rightarrow A$ satisfying $h(4) = F(F(F(F(a))))$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727492422721-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which theorem in set theory implies existence of recursively defined functions?
|
|
Back: The recursion theorem (on $\omega$).
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1727492422724-->
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END%%
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%%ANKI
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Basic
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In Enderton's recursion theorem proof, function $h \colon \omega \rightarrow A$ is defined as the union of what?
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Back: The set of "acceptable" functions.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1727627702457-->
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END%%
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%%ANKI
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Basic
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In Enderton's recursion theorem proof, what is the domain of an acceptable function?
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Back: A subset of $\omega$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1727627702459-->
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END%%
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%%ANKI
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Basic
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In Enderton's recursion theorem proof, what is the codomain of an acceptable function?
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Back: A subset of some fixed set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1727627702461-->
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END%%
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%%ANKI
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Basic
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In Enderton's recursion theorem proof, what follows if $0 \in \mathop{\text{dom}} v$ for acceptable function $v$?
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Back: $v(0) = a$ for some fixed $a \in A$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1727627702462-->
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END%%
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%%ANKI
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Basic
|
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In Enderton's recursion theorem proof, what follows if $n^+ \in \mathop{\text{dom}} v$ for acceptable function $v$?
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Back: $n \in \mathop{\text{dom}} v$ and $v(n^+) = F(v(n))$ for some fixed $F \colon A \rightarrow A$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1727627702464-->
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END%%
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\
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%%ANKI
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Basic
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|
In Enderton's recursion theorem proof, what term is used to refer to the "approximating" functions?
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Back: They are called "acceptable".
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1727627702465-->
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END%%
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%%ANKI
|
|
Cloze
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In Enderton's recursion theorem proof, desired $h \colon \omega \rightarrow A$ is defined as $\bigcup$ {$\{ v \mid v \text{ is acceptable} \}$}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1727627702466-->
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END%%
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%%ANKI
|
|
Cloze
|
|
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}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1727627702468-->
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END%%
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%%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.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
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<!--ID: 1727627702469-->
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END%%
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%%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}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
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<!--ID: 1727627702470-->
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END%%
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%%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).
|
|
<!--ID: 1727629020369-->
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END%%
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%%ANKI
|
|
Basic
|
|
*Why* is there no function $h \colon \mathbb{N} \rightarrow \mathbb{N}$ such that for all $n \in \mathbb{N}$, $$\begin{align*} h(0) & = 0 \\ h(n + 1) & = h(n) + 1 \end{align*}$$
|
|
Back: N/A. The resursive theorem of $\omega$ states such an $h$ exists.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727629020375-->
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END%%
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%%ANKI
|
|
Basic
|
|
In what natural way could we generalize the recursion theorem on $\omega$?
|
|
Back: By stating the theorem in terms of arbitrary Peano systems.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
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<!--ID: 1727629232445-->
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END%%
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## Bibliography
|
|
|
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* 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). |