42 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 = \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
. 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 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
ofN
that containse
and is closed underS
equalsN
itself.
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 A
a \in A \Rightarrow a \subseteq A
A \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%%
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%%
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%%
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.