--- 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). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI 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). END%% ## Inductive Sets For any set $a$, its **successor** $a^+$ is defined as $$a^+ = a \cup \{a\}$$ %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI 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). 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). END%% %%ANKI 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). 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). 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). 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). 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). 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). END%% A set $A$ is **inductive** if and only if $\varnothing \in A$ and $\forall a \in A, a^+ \in A$. %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI Cloze 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). END%% %%ANKI Basic An inductive set is closed under what operation? Back: Successor. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI Basic What natural number corresponds to $\varnothing^{+++}$? Back: $3$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What natural number corresponds to $\varnothing$? Back: $0$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% A **natural number** is a set that belongs to every inductive set. %%ANKI Basic How is the set of natural numbers *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). END%% %%ANKI 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). END%% %%ANKI Basic In set theory, $\omega$ denotes what set? Back: The natural numbers. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI Basic What can be said about a subset of $\omega$? Back: N/A. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What can be said about an inductive subset of $\omega$? Back: It must coincide with $\omega$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Why must every inductive subset of $\omega$ coincide with $\omega$? Back: Because $\omega$ is the smallest inductive set. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What does the induction principle for $\omega$ state? Back: Every inductive subset of $\omega$ coincides with $\omega$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What name is given to the principle, "every inductive subset of $\omega$ coincides with $\omega$?" Back: The induction principle for $\omega$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Inductive sets correspond to what kind of proof method? Back: Proof by induction. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Prove $P(n)$ is true for all $n \in \mathbb{N}$ using induction. What set do we prove is inductive? Back: $\{n \in \mathbb{N} \mid P(n)\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic *How* are inductive sets and proof by induction related? Back: An induction proof corresponds to proving a related set is inductive. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What inductive set do we construct to prove the following by induction? $$\text{Every natural number is nonnegative}$$ Back: $\{n \in \omega \mid 0 \leq n\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What inductive set do we construct to prove the following by induction? $$\text{Every nonzero natural number is the successor of another natural number}$$ Back: $\{n \in \omega \mid n = 0 \lor (\exists m \in \omega, n = m^+)\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Peano System 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: * $e \not\in \mathop{\text{ran}}{S}$; * $S$ is one-to-one; * Any subset $A$ of $N$ that contains $e$ and is closed under $S$ equals $N$ itself. %%ANKI Basic A Peano system is a tuple consisting of how many members? Back: $3$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider Peano system $\langle N, S, e \rangle$. With maximum specificity, what kind of mathematical object is $N$? Back: A set. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider Peano system $\langle N, S, e \rangle$. With maximum specificity, what kind of mathematical object is $S$? Back: A function. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider Peano system $\langle N, S, e \rangle$. What is the domain of $S$? Back: $N$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider Peano system $\langle N, S, e \rangle$. What is the codomain of $S$? Back: $N$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider Peano system $\langle N, S, e \rangle$. With maximum specificity, what kind of mathematical object is $e$? Back: A set or urelement. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic In Peano system $\langle N, S, e \rangle$, $e$ is a member of what set? Back: $N$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic In Peano system $\langle N, S, e \rangle$, $e$ is explicitly *not* a member of what set? Back: $\mathop{\text{ran}}S$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze Consider Peano system $\langle N, S, e \rangle$. Then {1:$e$} $\not\in$ {1:$\mathop{\text{ran} }S$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider Peano system $\langle N, S, e \rangle$. Function $S$ satisfies what additional condition? Back: $S$ is one-to-one. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Consider Peano system $\langle N, S, e \rangle$. What two conditions must be satisfied for $A \subseteq N$ to coincide with $N$? Back: $e \in A$ and $A$ is closed under $S$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What condition of Peano system $\langle N, S, e \rangle$ generalizes the induction principle of $\omega$? Back: Any 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). END%% %%ANKI Basic What name is given to the condition of Peano systems involving closures? Back: The Peano induction postulate. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic The Peano induction postulate of $\langle N, S, e \rangle$ implies $N$ is the smallest set satisfying what? Back: That contains $e$ and is closed under $S$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). 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). 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). 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). 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). 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). 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). END%% ## Bibliography * Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).