notebook/notes/set/natural-numbers.md

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---
title: Natural Numbers
TARGET DECK: Obsidian::STEM
FILE TAGS: set::nat
tags:
- natural-number
- set
---
## Overview
The standard way of representing the natural numbers is as follows:
* $0 = \varnothing$
* $1 = \{0\} = \{\varnothing\}$
* $2 = \{0, 1\} = \{\varnothing, \{\varnothing\}\}$
* $\ldots$
That is, each natural number corresponds to the set of natural numbers smaller than it.
%%ANKI
Basic
How is the number $0$ represented as a set?
Back: $\varnothing$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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Basic
How is the number $1$ represented as a set?
Back: $\{0\} = \{\varnothing\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Basic
How is the number $2$ represented as a set?
Back: $\{0, 1\} = \{\varnothing, \{\varnothing\}\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Basic
Who came up with the standard set representation of natural numbers?
Back: John von Neumann.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Basic
Consider the set representation of $n \in \mathbb{N}$. How many members does $n$ have?
Back: $n$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Basic
Consider the set representation of $n \in \mathbb{N}$. What are the members of $n$?
Back: $0$, $1$, $\ldots$, $n - 1$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Basic
Let $n \in \mathbb{N}$. *Why* is $n \in n + 1$?
Back: $n + 1$ is a set containing all preceding natural numbers.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Basic
Let $n \in \mathbb{N}$. *Why* is $n \subseteq n + 1$?
Back: $n$ and $n + 1$ are sets containing all their preceding natural numbers.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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## Inductive Sets
For any set $a$, its **successor** $a^+$ is defined as $$a^+ = a \cup \{a\}$$
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Basic
How is the successor of a set $a$ denoted?
Back: $a^+$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Basic
How is the successor of a set $a$ defined?
Back: As $a^+ = a \cup \{a\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Basic
Set $\{a, b\}^+$ equals what other set?
Back: $\{a, b, \{a, b\}\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
Set $\{a\}^+$ equals what other set?
Back: $\{a, \{a\}\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Basic
Set $\{a, \{a, b\}, \{a, b, c\}\}$ can be written as the successor of what set?
Back: N/A.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
Set $\{a, b, \{a, b\}\}$ can be written as the successor of what set?
Back: $\{a, b\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
Set $\{a, \{a, b\}\}$ can be written as the successor of what set?
Back: N/A.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
Set $\{a, \{a, b\}, \{a, \{a, b\}\}\}$ can be written as the successor of what set?
Back: $\{a, \{a, b\}\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
If $n \in \mathbb{N}$ then $n \in n + 1$. What analagous statement holds for arbitrary set $a$?
Back: $a \in a^+$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
If $n \in \mathbb{N}$ then $n \subseteq n + 1$. What analagous statement holds for arbitrary set $a$?
Back: $a \subseteq a^+$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
A set $A$ is **inductive** if and only if $\varnothing \in A$ and $\forall a \in A, a^+ \in A$.
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Basic
What does it mean for a set $A$ to be closed under successor?
Back: If $a \in A$, then $a^+ \in A$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Basic
Write "set $B$ is closed under successor" in FOL.
Back: $\forall b \in B, b^+ \in B$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Basic
What does it mean for a set $A$ to be inductive?
Back: $\varnothing \in A$ and $A$ is closed under successor.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Cloze
A set $A$ is inductive iff {$\varnothing \in A$} and {$A$ is closed under successor}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Basic
An inductive set is closed under what operation?
Back: Successor.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Basic
What set is the "seed" of an inductive set?
Back: $\varnothing$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Basic
Let $a \in A$ where $A$ is an inductive set. What other members must belong to $A$?
Back: $a^+$, $a^{++}$, $\ldots$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Basic
What natural number corresponds to $\varnothing^{+++}$?
Back: $3$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Basic
What natural number corresponds to $\varnothing$?
Back: $0$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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A **natural number** is a set that belongs to every inductive set.
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Basic
How is a natural number *defined* in set theory?
Back: As a set belonging to every inductive set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Basic
What greek letter is used to denote the set of natural numbers?
Back: $\omega$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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Basic
What is the smallest inductive set?
Back: $\omega$, i.e. the set of natural numbers.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Basic
How might $\omega$ be defined as an intersection of classes?
Back: $\omega = \bigcap\,\{A \mid A \text{ is inductive}\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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Basic
Suppose $n \in \omega$. What other sets *must* $n$ be a member of?
Back: Every other inductive set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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## Bibliography
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).