How to show $\chi_{{}^{*}P} ={}^{*}\chi_{P}$ by transfer principle?

Question

Let $\mathfrak{R}$ be the real number system, $(\mathbb{R},+,\cdot,<)$ and ${}^{*}\mathfrak{R}$ be the hyperreal number system $({}^{*}\mathbb{R},{}^{*}+,{}^{*}\cdot,{}^{*}<)$. Transfer principle states that , property of $\mathfrak{R}$ that can be expressed in first order language also holds in ${}^{*}\mathfrak{R}$.

But I don't know how to show $\chi_{{}^{*}P} ={}^{*}\chi_{P}$ by transfer principle.

$P$ is a n-ary relation on $\mathbb{R}$, $\chi_{P}$ is its indicator function.

I can show this directly by utilizing the fact the n-tuple $({}^{*}r^0, \ldots {}^{*}r^{n-1}) \in {}^{*}P$ in which ${}^{*}r^k = [(r_0^k,r_1^k,\ldots )] $ ($[ \text{ } ]$ is a equivalence class induced by a free ultrafilter on $\mathbb{N}$,$\mathscr{U}$), if $\{i \in \mathbb{N}: (r_i^0, r_i^1, \ldots, r_{i}^{n-1})\} \in \mathscr{U}$.

Answer 1

You have $\mathfrak R\models (\forall x) ``\chi_P"(x)=1\leftrightarrow ``P"(x)$ where “$\chi_P$” is the $n$-ary function symbol representing $\chi_P$ in $\bf R$, and “$P$” is the $n$-ary predicate symbol representing $P$ in $\bf R$.

$ {}^*\mathfrak R$ is an elementary extension of $\mathfrak R$, so you have ${}^*\mathfrak R\models (\forall x) ``\chi_P"(x)=1\leftrightarrow ``P"(x)$, but this means just the same as ${}^*(\chi_P)(x)=1 \iff {}^*P(x)$, and this is of course the defining characteristic of $\chi_{{}^*P}$.

Edit: For completeness' sake, if you want to be thorough, you should also point out that ${}^*\mathfrak R\models (\forall x) ``\chi_P"(x)=1\lor ``\chi_P"(x)=0$.

Answer 2

The formula expressing $\chi$ in terms of $P$ (and vice versa) is first order. Therefore transfer applies to it. Thus, transfering the relation to the hyperreals gives the same relation between the starred objects.