1561 lines
50 KiB
Markdown
1561 lines
50 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: $A \subseteq B$
|
|
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).
|
|
<!--ID: 1727492422724-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
In Enderton's recursion theorem proof, function $h \colon \omega \rightarrow A$ is defined as the union of what?
|
|
Back: The set of "acceptable" functions.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727627702457-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
In Enderton's recursion theorem proof, what is the domain of an acceptable function?
|
|
Back: A subset of $\omega$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727627702459-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
In Enderton's recursion theorem proof, what is the codomain of an acceptable function?
|
|
Back: A subset of some fixed set.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727627702461-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
In Enderton's recursion theorem proof, what follows if $0 \in \mathop{\text{dom}} v$ for acceptable function $v$?
|
|
Back: $v(0) = a$ for some fixed $a \in A$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727627702462-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
In Enderton's recursion theorem proof, what follows if $n^+ \in \mathop{\text{dom}} v$ for acceptable function $v$?
|
|
Back: $n \in \mathop{\text{dom}} v$ and $v(n^+) = F(v(n))$ for some fixed $F \colon A \rightarrow A$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727627702464-->
|
|
END%%
|
|
\
|
|
%%ANKI
|
|
Basic
|
|
In Enderton's recursion theorem proof, what term is used to refer to the "approximating" functions?
|
|
Back: They are called "acceptable".
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727627702465-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
In Enderton's recursion theorem proof, desired $h \colon \omega \rightarrow A$ is defined as $\bigcup$ {$\{ v \mid v \text{ is acceptable} \}$}.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727627702466-->
|
|
END%%
|
|
|
|
%%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}.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727627702468-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
In Enderton's recursion theorem proof, how is it shown the domain of desired $h \colon \omega \rightarrow A$ equals $\omega$?
|
|
Back: By proving $\mathop{\text{dom}} h$ is an inductive set.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727627702469-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
In Enderton's recursion theorem proof,desired $h \colon \omega \rightarrow A$ is {unique} because {$\{ n \in \omega \mid h_1(n) = h_2(n) \}$} is {an inductive set}.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727627702470-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* is there no function $h \colon \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all $n \in \mathbb{Z}$, $$\begin{align*} h(0) & = 0 \\ h(n + 1) & = h(n) + 1 \end{align*}$$
|
|
Back: Because $\mathbb{Z}$ has no "starting point" to ground the recursive definition.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727629020369-->
|
|
END%%
|
|
|
|
%%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.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1727629020375-->
|
|
END%%
|
|
|
|
%%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).
|
|
<!--ID: 1727629232445-->
|
|
END%%
|
|
|
|
### Addition
|
|
|
|
For each $m \in \omega$, there exists (by the recursion theorem) a unique function $A_m \colon \omega \rightarrow \omega$ such that for all $n \in \omega$, $$\begin{align*} A_m(0) & = m, \\ A_m(n^+) & = A_m(n)^+ \end{align*}$$
|
|
|
|
**Addition** ($+$) is the binary operation on $\omega$ such that for any $m, n \in \omega$, $$m + n = A_m(n).$$
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $A_m \colon \omega \rightarrow \omega$ denote recursively defined addition. How is $A_m(n)$ more traditionally denoted?
|
|
Back: As $m + n$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914175-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $A_m \colon \omega \rightarrow \omega$ denote recursively defined addition. How is $m + n$ defined in terms of $A_m$?
|
|
Back: As $A_m(n)$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914179-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $A_m \colon \omega \rightarrow \omega$ denote recursively defined addition. What does $A_m(0)$ evalute to?
|
|
Back: $m$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914180-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $A_m \colon \omega \rightarrow \omega$ denote recursively defined addition. What does $A_m(n^+)$ evalute to?
|
|
Back: $A_m(n)^+$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914181-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $A_m \colon \omega \rightarrow \omega$ denote recursively defined addition. How do we know $A_m$ exists?
|
|
Back: The recursion theorem states it does.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914182-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is $m + n$ defined as a relation?
|
|
Back: $\{ \langle m, n, A_m(n) \rangle \mid m, n \in \omega \}$ where $A_m \colon \omega \rightarrow \omega$ is recursively defined addition.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914183-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Is $+$ (addition) a function, operation, both, or neither?
|
|
Back: Both.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914184-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Without introducing new notation, what does $m + 0$ evaluate to?
|
|
Back: $m$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914185-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Without introducing new notation, what does $m + n^+$ evaluate to?
|
|
Back: $(m + n)^+$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914186-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the order-preserving property of addition on $\omega$ in FOL?
|
|
Back: $\forall m, n, p \in \omega, m \in n \Leftrightarrow m + p \in n + p$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1731170040087-->
|
|
END%%
|
|
|
|
### Multiplication
|
|
|
|
For each $m \in \omega$, there exists (by the recursion theorem) a unique function $M_m \colon \omega \rightarrow \omega$ such that for all $n \in \omega$, $$\begin{align*} M_m(0) & = 0, \\ M_m(n^+) & = M_m(n) + m \end{align*}$$
|
|
|
|
**Multiplication** ($\cdot$) is the binary operation on $\omega$ such that for any $m, n \in \omega$, $$m \cdot n = M_m(n).$$
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $M_m \colon \omega \rightarrow \omega$ denote recursively defined multiplication. How is $M_m(n)$ more traditionally denoted?
|
|
Back: As $m \cdot n$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914187-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $M_m \colon \omega \rightarrow \omega$ denote recursively defined multiplication. How is $m \cdot n$ defined in terms of $M_m$?
|
|
Back: As $M_m(n)$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914188-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $M_m \colon \omega \rightarrow \omega$ denote recursively defined multiplication. What does $M_m(0)$ evalute to?
|
|
Back: $0$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914189-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $M_m \colon \omega \rightarrow \omega$ denote recursively defined multiplication. What does $M_m(n^+)$ evalute to?
|
|
Back: $M_m(n) + m$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914190-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $M_m \colon \omega \rightarrow \omega$ denote recursively defined multiplication. How do we know $M_m$ exists?
|
|
Back: The recursion theorem states it does.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914191-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is $m \cdot n$ defined as a relation?
|
|
Back: $\{ \langle m, n, M_m(n) \rangle \mid m, n \in \omega \}$ where $M_m \colon \omega \rightarrow \omega$ is recursively defined multiplication.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914192-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Is $\cdot$ (multiplication) a function, operation, both, or neither?
|
|
Back: Both.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914193-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Without introducing new notation, what does $m \cdot 0$ evaluate to?
|
|
Back: $0$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914194-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Without introducing new notation, what does $m \cdot n^+$ evaluate to?
|
|
Back: $m \cdot n + m$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914195-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the order-preserving property of multiplication on $\omega$ in FOL?
|
|
Back: $\forall m, n, p \in \omega, m \in n \Leftrightarrow m \cdot p^+ \in n \cdot p^+$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1731170040116-->
|
|
END%%
|
|
|
|
### Exponentiation
|
|
|
|
For each $m \in \omega$, there exists (by the recursion theorem) a unique function $E_m \colon \omega \rightarrow \omega$ such that for all $n \in \omega$, $$\begin{align*} E_m(0) & = 1, \\ E_m(n^+) & = E_m(n) \cdot m \end{align*}$$
|
|
|
|
**Exponentiation** is the binary operation on $\omega$ such that for any $m, n \in \omega$, $$m^n = E_m(n).$$
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $E_m \colon \omega \rightarrow \omega$ denote recursively defined exponentiation. How is $E_m(n)$ more traditionally denoted?
|
|
Back: As $m^n$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914196-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $E_m \colon \omega \rightarrow \omega$ denote recursively defined exponentiation. How is $m^n$ defined in terms of $E_m$?
|
|
Back: As $E_m(n)$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914197-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $E_m \colon \omega \rightarrow \omega$ denote recursively defined exponentiation. What does $E_m(0)$ evalute to?
|
|
Back: $1$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914198-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $E_m \colon \omega \rightarrow \omega$ denote recursively defined exponentiation. What does $E_m(n^+)$ evalute to?
|
|
Back: $E_m(n) \cdot m$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914199-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $E_m \colon \omega \rightarrow \omega$ denote recursively defined exponentiation. How do we know $E_m$ exists?
|
|
Back: The recursion theorem states it does.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914200-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is $m^n$ defined as a relation?
|
|
Back: $\{ \langle m, n, E_m(n) \rangle \mid m, n \in \omega \}$ where $E_m \colon \omega \rightarrow \omega$ is recursively defined exponentiation.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914201-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Is exponentiation a function, operation, both, or neither?
|
|
Back: Both.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914202-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Without introducing new notation, what does $m^0$ evaluate to?
|
|
Back: $1$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914203-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Without introducing new notation, what does $m^{(n^+)}$ evaluate to?
|
|
Back: $m^n \cdot m$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1729804914204-->
|
|
END%%
|
|
|
|
## Ordering
|
|
|
|
For natural numbers $m$ and $n$, define $m$ to be **less than $n$** if and only if $m \in n$. The following biconditionals hold true:
|
|
|
|
* $m \in n \Leftrightarrow m^+ \in n^+$
|
|
* $m \in n \Leftrightarrow m \subset n$
|
|
* $m \underline{\in} n \Leftrightarrow m \subseteq n$
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $m, n \in \omega$. How does Enderton prefer denoting $m$ is less than $n$?
|
|
Back: As $m \in n$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1730118488824-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $m, n \in \omega$. What dual meaning does Enderton give $m \in n$?
|
|
Back: Set membership and ordering.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1730118488827-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $m, n \in \omega$. How does Enderton prefer denoting $m$ is less than or equal to $n$?
|
|
Back: As $m \underline\in n$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1730118488830-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $m, n \in \omega$. How is $m \underline\in n$ defined?
|
|
Back: As $m \in n \lor m = n$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1730118488833-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $m, n \in \omega$. How is $m = n \lor m \in n$ more compactly denoted?
|
|
Back: As $m \underline\in n$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1730118488837-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
Let $m, n \in \omega$. $m$ {$\in$} $n^+ \Leftrightarrow m$ {$\underline\in$} $n$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1730118488842-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
Let $m, n \in \omega$. $m$ {$\in$} $n \Leftrightarrow m^+$ {$\in$} $n^+$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1730118488846-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $m, n \in \omega$. What is the strict analog of $m \underline{\in} n \Leftrightarrow m \subseteq n$?
|
|
Back: $m \in n \Leftrightarrow m \subset n$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1731168085673-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $m, n \in \omega$. What is the non-strict analog of $m \in n \Leftrightarrow m \subset n$?
|
|
Back: $m \underline{\in} n \Leftrightarrow m \subseteq n$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1731168085679-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
In what three ways does Enderton denote strict ordering of the natural numbers?
|
|
Back: $\in$, $\subset$, and $<$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1731170040122-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
In what three ways does Enderton denote non-strict ordering of the natural numbers?
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Back: $\underline{\in}$, $\subseteq$, and $\leq$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1731170040128-->
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END%%
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%%ANKI
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Basic
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What does the trichotomy law for $\omega$ state?
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Back: For any $m, n \in \omega$ exactly one of $m \in n$, $m = n$, or $n \in m$ holds.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1730118488850-->
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END%%
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%%ANKI
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Basic
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Let $m, n \in \omega$. If $m \in n$, why is it that $m \subseteq n$?
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Back: Because $n$ is a transitive set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1731168085682-->
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END%%
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%%ANKI
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Basic
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Let $m, n \in \omega$. If $m \in n$, why is it that $m \subset n$?
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Back: Because $n$ is a transitive set and no natural number is a member of itself.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1731168085685-->
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END%%
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%%ANKI
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Basic
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What does Enterton describe is the typical way of using trichotomy in proofs?
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Back: Showing that two of the three possibilities is false.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1731168085688-->
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END%%
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### Well-Ordering Principle
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Let $A$ be a nonempty subset of $\omega$. Then there is some $m \in A$ such that $m \underline{\in} n$ for all $n \in A$.
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%%ANKI
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Basic
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What does the well-ordering principle state?
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Back: Every nonempty subset of $\omega$ has a least element.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1731200524848-->
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END%%
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%%ANKI
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Basic
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How is the well-ordering principle stated in FOL?
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Back: $\forall A \subseteq \omega, A \neq \varnothing \Rightarrow \exists m \in A, \forall n \in A, m \underline{\in} n$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1731200524851-->
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END%%
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%%ANKI
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Basic
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Let $A$ be a set of $\omega$. What condition is necessary for $A$ to have a least element?
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Back: $A \neq \varnothing$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1731200524854-->
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END%%
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%%ANKI
|
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Basic
|
|
What principle states every nonempty subset of $\omega$ has a least element?
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Back: The well-ordering principle.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1731200524857-->
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END%%
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%%ANKI
|
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Basic
|
|
What principle states every nonempty subset of $\omega$ has a greatest element?
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Back: N/A. This is not true.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1731200524861-->
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END%%
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%%ANKI
|
|
Basic
|
|
Suppose $A$ is a subset of $\omega$ without a least element. What can be said about $A$?
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Back: $A = \varnothing$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1731200524864-->
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END%%
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%%ANKI
|
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Basic
|
|
*Why* is there no function $f \colon \omega \rightarrow \omega$ such that $f(n^+) \in f(n)$ for all $n \in \omega$?
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|
Back: $\mathop{\text{ran}} f$ would violate the well-ordering principle.
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|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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|
<!--ID: 1731200524868-->
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END%%
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%%ANKI
|
|
Basic
|
|
The following is a FOL representation of what principle?
|
|
$$\forall A \subseteq \omega, A \neq \varnothing \Rightarrow \exists m \in A, \forall n \in A, m \underline{\in} n$$
|
|
Back: The well-ordering 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: 1731203636938-->
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END%%
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%%ANKI
|
|
Basic
|
|
How can we show set $S$ coincides with $\omega$ using the well-ordering principle?
|
|
Back: By showing $\omega - S$ has no least element.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1731204485586-->
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END%%
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|
|
### Strong Induction Principle
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|
|
Let $A$ be a subset of $\omega$ and assume that for every $n \in \omega$, $$\text{if every number less than } n \text{ is in } A, \text{then } n \in A.$$
|
|
Then $A = \omega$.
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%%ANKI
|
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Basic
|
|
Let $A \subseteq \omega$. The strong induction principle for $\omega$ assumes what about every $n \in \omega$?
|
|
Back: If every number less than $n$ is in $A$, then $n \in A$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1731203636943-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The following is a FOL representation of what principle?
|
|
$$[\forall A \subseteq \omega, 0 \in A \land (\forall n \in \omega, n^+ \in \omega)] \Rightarrow A = \omega$$
|
|
Back: The weak induction principle for $\omega$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1731203636947-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The following is a FOL representation of what principle?
|
|
$$[\forall A \subseteq \omega, \forall n \in \omega, (\forall m \in n, m \in A) \Rightarrow n \in A] \Rightarrow A = \omega$$
|
|
Back: The strong induction principle for $\omega$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1731203636951-->
|
|
END%%
|
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|
|
## Bibliography
|
|
|
|
* 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). |