16 KiB
| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Natural Numbers | Obsidian::STEM | set::nat |
|
Overview
The standard way of representing the natural numbers is as follows:
0 = \varnothing1 = \{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 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).
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};Sis one-to-one;- Any subset
AofNthat containseand is closed underSequalsNitself.
%%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 depict?
!
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 depict?
!
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).