543 lines
16 KiB
Markdown
543 lines
16 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 a natural number *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 $n \in \omega$. 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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%%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 $\langle N, S, e \rangle$. With maximum specificity, 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 $\langle N, S, e \rangle$. With maximum specificity, 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 $\langle N, S, e \rangle$. With maximum specificity, 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: 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: 1726364667661-->
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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).
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<!--ID: 1726364667670-->
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END%%
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%%ANKI
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Basic
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Let $\langle N, S, e \rangle$ be a Peano system. *Why* can't there be an $A \subset N$ containing $e$ and closed under $S$?
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Back: The Peano induction postulate states $A$ *must* coincide with $N$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1726364667673-->
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END%%
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%%ANKI
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Basic
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*Why* does Peano system $\langle N, S, e \rangle$ have condition $e \not\in \mathop{\text{ran}}S$?
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Back: To avoid cycles in repeated applications of $S$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1726364667676-->
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END%%
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%%ANKI
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Basic
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Which condition of Peano system $\langle N, S, e \rangle$ does the following depict?
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![[peano-system-i.png]]
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Back: $e \not\in \mathop{\text{ran}}S$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1726364667679-->
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END%%
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%%ANKI
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Basic
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*Why* does Peano system $\langle N, S, e \rangle$ have condition "$S$ is one-to-one"?
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Back: To avoid two members of $N$ mapping to the same element.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1726364667682-->
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END%%
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%%ANKI
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Basic
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Which condition of Peano system $\langle N, S, e \rangle$ does the following depict?
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![[peano-system-ii.png]]
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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: 1726364667685-->
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END%%
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%%ANKI
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Basic
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What is the Peano induction postulate?
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Back: Given Peano system $\langle N, S, e \rangle$, a 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: 1726364667688-->
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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). |