50 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 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. By definition of natural numbers, 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 subsets 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 inductive subsets of \omega?
Back: They 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.
Given \sigma = \{\langle n, n^+ \rangle \mid n \in \omega\}, \langle \omega, \sigma, 0 \rangle is a Peano system.
%%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. 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. 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. 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: The Peano induction postulate.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Given Peano system \langle N, S, e \rangle, what does the Peano induction postulate state?
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?
!
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?
!
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%%
%%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).
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).
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).
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).
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).
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).
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).
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).
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 Aa \in A \Rightarrow a \subseteq AA \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).
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).
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).
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).
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).
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).
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).
END%%
%%ANKI
Basic
Is \varnothing a transitive set?
Back: Yes.
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
END%%
%%ANKI
Basic
Is \omega a transitive set?
Back: Yes.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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.
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).
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).
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).
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).
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).
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).
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).
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.
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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 nm \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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
END%%
%%ANKI
Basic
In what three ways does Enderton denote non-strict ordering of the natural numbers?
Back: \underline{\in}, \subseteq, and \leq.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What does the trichotomy law for \omega state?
Back: For any m, n \in \omega exactly one of m \in n, m = n, or n \in m holds.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m, n \in \omega. If m \in n, why is it that m \subseteq n?
Back: Because n is a transitive set.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m, n \in \omega. If m \in n, why is it that m \subset n?
Back: Because n is a transitive set and no natural number is a member of itself.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic What does Enterton describe is the typical way of using trichotomy in proofs? Back: Showing that two of the three possibilities is false. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Well-Ordering Principle
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.
%%ANKI
Basic
What does the well-ordering principle state?
Back: Every nonempty subset of \omega has a least element.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How is the well-ordering principle stated in FOL?
Back: \forall A \subseteq \omega, A \neq \varnothing \Rightarrow \exists m \in A, \forall n \in A, m \underline{\in} n
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let A be a set of \omega. What condition is necessary for A to have a least element?
Back: A \neq \varnothing
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What principle states every nonempty subset of \omega has a least element?
Back: The well-ordering principle.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What principle states every nonempty subset of \omega has a greatest element?
Back: N/A. This is not true.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose A is a subset of \omega without a least element. What can be said about A?
Back: A = \varnothing
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Why is there no function f \colon \omega \rightarrow \omega such that f(n^+) \in f(n) for all n \in \omega?
Back: \mathop{\text{ran}} f would violate the well-ordering principle.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%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} nBack: The well-ordering principle for \omega.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%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).
END%%
Strong Induction Principle
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.
%%ANKI
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).
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 = \omegaBack: The weak induction principle for \omega.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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 = \omegaBack: The strong induction principle for \omega.
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).
- “Recursion,” in Wikipedia, September 23, 2024, https://en.wikipedia.org/w/index.php?title=Recursion#The_recursion_theorem.