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compactness example (#346)
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* Removed \div as function on rationals.

* A tiny bit more explanation in the changes for the infinitesimal example

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Co-authored-by: Richard Zach <[email protected]>
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greleigh and rzach authored Dec 5, 2023
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\end{ex}

\begin{ex}
Consider a language $\Lang{L}$ containing the !!{predicate}~$<$,
!!{constant}s $\Obj{0}$, $\Obj{1}$, and !!{function}s $+$, $\times$,
$-$, $\div$. Let $\Gamma$ be the set of all !!{sentence}s in this
language true in $\Struct{Q}$ with domain $\Rat$ and the obvious
interpretations. $\Gamma$ is the set of all !!{sentence}s
of~$\Lang{L}$ true about the rational numbers. Of course, in $\Rat$
(and even in $\Real$), there are no numbers which are greater than~$0$
Consider !!a{language} $\Lang{L}$ containing the !!{predicate}~$<$,
!!{constant}s $\Obj{0}$, $\Obj{1}$, and !!{function}s $+$, $\times$, and
$-$. Let $\Gamma$ be the set of all !!{sentence}s in this
!!{language} true in the !!{structure}~$\Struct{Q}$ with domain~$\Rat$ and the obvious
interpretations. $\Gamma$~is the set of all !!{sentence}s
of~$\Lang{L}$ true about the rational numbers. Of course, in~$\Rat$
(and even in~$\Real$), there are no numbers~$r$ which are greater than~$0$
but less than $1/k$ for all $k \in \PosInt$. Such a number, if it
existed, would be an \emph{infinitesimal:} non-zero, but infinitely
small. The compactness theorem shows that there are models
of~$\Gamma$ in which infinitesimals exist: Let $\Delta$ be $\{0<c\}
\cup \Setabs{c < (\Obj{1} \div \num{k})}{k \in \PosInt}$ (where
small. The compactness theorem can be used to show that there are
models of~$\Gamma$ in which infinitesimals exist. We do not have
!!a{function} for division in our language (division by zero is
undefined, and !!{function}s have to be interpreted by total functions).
However, we can still express that $r < 1/k$, since this is the case iff
$r \cdot k < 1$. Now let $c$ be a new !!{constant} and let $\Delta$ be
\[
\{0<c\}
\cup \Setabs{ c \times \num k < \Obj{1} }{k \in \PosInt}
\]
(where
$\num{k} = (\Obj{1} + (\Obj{1} + \dots + (\Obj{1} + \Obj{1})\dots))$
with $k$ $\Obj{1}$'s). For any finite subset~$\Delta_0$ of~$\Delta$
there is a $K$ such that all the !!{sentence}s $c < (\Obj{1} \div \num{k})$ in
$\Delta_0$ have $k < K$. If we expand $\Struct{Q}$ to $\Struct{Q'}$
with $k$~$\Obj{1}$'s). For any finite subset~$\Delta_0$ of~$\Delta$
there is a~$K$ such that for all the !!{sentence}s $c \times \num{k} < \Obj{1}$
in~$\Delta_0$ have $k < K$. If we expand $\Struct{Q}$ to~$\Struct{Q'}$
with $\Assign{c}{Q'} = 1/K$ we have that $\Sat{Q'}{\Gamma \cup
\Delta_0}$, and so $\Gamma \cup \Delta$ is finitely satisfiable
(Exercise: prove this in detail). By compactness, $\Gamma \cup \Delta$
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