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# Differences

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vkuncak
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Let $\mathbb{N} = \{0,​1,​2,​\ldots\}$ denote non-negative integers. ​ Let $D$ be the set of all //finite subsets// of $\mathbb{N}$. ​ We consider the set of interpretations $(D,​\alpha)$ where for each variable $v$ we have $\alpha(v) \in D$, where $\subseteq$ is the subset relation Let $\mathbb{N} = \{0,​1,​2,​\ldots\}$ denote non-negative integers. ​ Let $D$ be the set of all //finite subsets// of $\mathbb{N}$. ​ We consider the set of interpretations $(D,​\alpha)$ where for each variable $v$ we have $\alpha(v) \in D$, where $\subseteq$ is the subset relation
-$+\begin{equation*} ​\alpha({\subseteq}) = \{ (S_1,S_2) \mid S_1 \subseteq S_2 \} ​\alpha({\subseteq}) = \{ (S_1,S_2) \mid S_1 \subseteq S_2 \} -$+\end{equation*}
and the relation $succ(v_1,​v_2)$ is the successor relation on integers lifted to singleton sets: and the relation $succ(v_1,​v_2)$ is the successor relation on integers lifted to singleton sets:
-$+\begin{equation*} ​\alpha(succ) = \{ (\{k\},​\{k+1\}) \mid k \in \mathbb{N} \} ​\alpha(succ) = \{ (\{k\},​\{k+1\}) \mid k \in \mathbb{N} \} -$+\end{equation*}
The meaning of formulas is given by standard [[First-order logic semantics]]. The meaning of formulas is given by standard [[First-order logic semantics]].

Note in particular that quantification is restricted to finite sets (elements of $D$). Note in particular that quantification is restricted to finite sets (elements of $D$).
+

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Then we can define addition $N(Z) = N(X) + N(Y)$ by saying that there exists a set of carry bits $C$ such that the rules for binary addition hold: Then we can define addition $N(Z) = N(X) + N(Y)$ by saying that there exists a set of carry bits $C$ such that the rules for binary addition hold:
\begin{equation*} \begin{equation*}
-  \exists C.\ 0 \notin ​i\ \land \forall i.\ +  \exists C.\ 0 \notin ​C\ \land \forall i.\
​\big(\begin{array}[t]{rcl}    ​\big(\begin{array}[t]{rcl}
((i \in Z) &​\leftrightarrow&​ ((i \in X) \oplus (i \in Y) \oplus (i \in C))\ \land\\     ((i \in Z) &​\leftrightarrow&​ ((i \in X) \oplus (i \in Y) \oplus (i \in C))\ \land\\
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Relations on singleton sets: Relations on singleton sets:
-$+\begin{equation*} r^s_F = \{(p,q) \mid F(\{p\},​\{q\}) \} r^s_F = \{(p,q) \mid F(\{p\},​\{q\}) \} -$+\end{equation*}

Relations on binary representations:​ Relations on binary representations:​
-$+\begin{equation*} r^b_F = \{(p,q) \mid F(N(p),​N(q)) \} r^b_F = \{(p,q) \mid F(N(p),​N(q)) \} -$+\end{equation*}

Addition is not definable as some $r^s_F$, but it is definable as $r^b_F$. Addition is not definable as some $r^s_F$, but it is definable as $r^b_F$.