Peano systems.

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Joshua Potter 2024-09-15 16:23:10 -06:00
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"Basic": [

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* Watched [Query Execution 1](https://www.youtube.com/watch?v=I0QdCSu06_o&list=PLSE8ODhjZXjaKScG3l0nuOiDTTqpfnWFf&index=13) and [Parallel Query Execution](https://www.youtube.com/watch?v=FG_wr-0QYg4&list=PLSE8ODhjZXjaKScG3l0nuOiDTTqpfnWFf&index=14).

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* Notes on [[natural-numbers#Peano System|Peano systems]].

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@ -14,7 +14,7 @@ The $n$th term of the **triangular numbers** $(T_n)_{n \geq 0}$ is the sum of wh
%%ANKI
Basic
What is a polygonal number?
Back: A number of pebbles that can be arranged into the shape of a regular polygon.
Back: A number of pebbles that can be arranged into the shape of a regular filled polygon.
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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END%%

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@ -1641,6 +1641,73 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1720819771087-->
END%%
### Closures
If $S$ is a function and $A$ is a subset of $\mathop{\text{dom}}S$, then $A$ is said to be **closed** under $S$ if and only if whenever $x \in A$, then $S(x) \in A$. This is equivalently expressed as $S[\![A]\!] \subseteq A$.
%%ANKI
Basic
Let $A$ be closed under $S$. Then $A$ is a subset of what other set?
Back: $\mathop{\text{dom}}S$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726363069981-->
END%%
%%ANKI
Basic
Let $A$ be closed under $S$. With maximum specificity, what kind of mathematical object is $A$?
Back: A set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726363069988-->
END%%
%%ANKI
Basic
Let $A$ be closed under $S$. 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).
<!--ID: 1726363069991-->
END%%
%%ANKI
Basic
In FOL, what does it mean for set $A$ to be closed under function $S$?
Back: $\forall x \in A, S(x) \in A$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726363069994-->
END%%
%%ANKI
Basic
What concept is being expressed in "$\forall x \in A, S(x) \in A$"?
Back: Set $A$ is closed under $S$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726363069997-->
END%%
%%ANKI
Basic
How can we more compactly express "$\forall x \in A, S(x) \in A$"?
Back: $S[\![A]\!] \subseteq A$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726363070000-->
END%%
%%ANKI
Cloze
If $S[\![A]\!] \subseteq A$, then {1:$A$} is closed {2:under} {1:$S$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726363070010-->
END%%
%%ANKI
Basic
Suppose $A$ is closed under function $S$. What imagery does the term "closed" invoke?
Back: Applying a member of $A$ to $S$ always yields an element in $A$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726363070015-->
END%%
## Kernels
Let $F \colon A \rightarrow B$. Define [[relations#Equivalence Relations|equivalence relation]] $\sim$ as $$x \sim y \Leftrightarrow f(x) = f(y)$$

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@ -369,6 +369,175 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
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END%%
## Peano System
A **Peano system** is a triple $\langle N, S, e \rangle$ consisting of a set $N$, a function $S \colon N \rightarrow N$, and a member $e \in N$ such that the following three conditions are met:
* $e \not\in \mathop{\text{ran}}{S}$;
* $S$ is one-to-one;
* Any subset $A$ of $N$ that contains $e$ and is closed under $S$ equals $N$ itself.
%%ANKI
Basic
A Peano system is a tuple consisting of how many members?
Back: $3$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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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).
<!--ID: 1726364667620-->
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).
<!--ID: 1726364667623-->
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).
<!--ID: 1726364667626-->
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).
<!--ID: 1726364667629-->
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).
<!--ID: 1726364667632-->
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).
<!--ID: 1726364667635-->
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).
<!--ID: 1726364667639-->
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).
<!--ID: 1726364667643-->
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).
<!--ID: 1726364667648-->
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).
<!--ID: 1726364667655-->
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).
<!--ID: 1726364667661-->
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).
<!--ID: 1726364667666-->
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).
<!--ID: 1726364667670-->
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).
<!--ID: 1726364667673-->
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).
<!--ID: 1726364667676-->
END%%
%%ANKI
Basic
Which condition of Peano system $\langle N, S, e \rangle$ does the following depict?
![[peano-system-i.png]]
Back: $e \not\in \mathop{\text{ran}}S$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726364667679-->
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).
<!--ID: 1726364667682-->
END%%
%%ANKI
Basic
Which condition of Peano system $\langle N, S, e \rangle$ does the following depict?
![[peano-system-ii.png]]
Back: $S$ is one-to-one.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726364667685-->
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).
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END%%
## Bibliography
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).

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%%ANKI
Basic
Consider an equivalence class of $x$ (modulo $R$). What kind of mathematical object is $R$?
Consider an equivalence class of $x$ (modulo $R$). With maximum specificity, what kind of mathematical object is $R$?
Back: A relation.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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