I am reading an article written by W.A.J Luxemburg about nonstandard analysis (https://www.jstor.org/stable/3038221).
My question is: Why does the transfer principle transform sentences and predicates that quantify over subsets of $\mathbb{R}$ to internal subsets of ${^*}\mathbb{R}$ while not adding information to numbers?
I know that a contradiction is reached if one takes a sentence that apllies to subsets of $\mathbb{R}$ and tries to put an external subset of ${^*}\mathbb{R}$. My question is about how to conlude that taking subsets of $\mathbb{R}$ translates to take internal subsets of ${^*}\mathbb{R}$.
Definitions and examples of the article:
In this article, the construction of the hyperreals is based on a $\delta$-incomplete ultrafilter over a set $I$, and the set of functions from $I$ to the superstructure $\widehat{\mathbb{R}}$ of $\mathbb{R}$, ${\widehat{\mathbb{R}}}^I$.
The imbedding from $\widehat{\mathbb{R}}$ to ${\widehat{\mathbb{R}}}^I$ is denoted as $*$, and defined as ${^*}a(i)=a$ for all $a\in\widehat{\mathbb{R}}$. The $*$ transform for any predicate/ sentence is the same but replacing all constants with their $*$ transform. Here are other definitions given:
DEFINITION 3.3. An entity $a$ of the ${}^* \! L$-structure $\hat{R}^I$ is called internal whenever there exists a natural number $n \geqq 0$ such that $a \in {}^* \! R_n$. An internal entity $a$ is called a standard entity whenever there exists an entity $b \in \hat{R}$ such that $a = {}^* \! b$. All entities which are not internal are called external.
The set $\bigcup_{n \geqq 0} {}^* \! R_n$ of all internal entities is called the ultrapower of $\hat{R}$ with respsect to the ultrafilter $\mathscr{U}$ and will be denoted by ${}^* \! (\hat{R}).$
Then, to show examples of how sentences that hold in $\mathbb{R}$ can be interpreted in ${^*}\mathbb{R}$, they present the Archimedian property and the least upper bound property:
The fact that $R$ is Archimedean can be expressed by the following sentence of $K_0$:
$$(\forall x)[x \in R] \implies (\forall n)[n \in N] \implies [[nx \leqq 1] \iff [x \leqq 0]]$$
and so, by the F.T., the following statement holds for ${}^* \! R$.
$$(\forall x)[x \in {}^* \! R] \implies (\forall n)[n \in {}^* \! N] \implies [[nx \leqq 1] \iff [x \leqq 0]],$$
that is, with the proper interpretation of the constants, ${}^* \! R$ is Archimedean with respsect to ${}^* \! N$. It is not Archimedean in the sense of the metalanguage, that is, if …
…
… quantification is over numbers only. Let us now examine a few of the higher order type properties of $R$. One of the important higher order properties which $R$ possesses and which we have already referred to in the beginning of Section 3 is the so-called Dedekind completeness property of $R$ which states that every nonempty subset of $R$ which is bounded above has a least upper bound. This statement about $R$ can easily be expressed by a sentence of $K_0$ which will contain a universal quantifier ranging over subsets of $R$. Then it follows from the F.T. that ${}^* \! R$ satisfies a Dedekind completeness property of the following kind.
(4.1) Every nonempty internal subset of ${}^* \! R$ which is bounded above has a least upper bound.
I am completely new to nonstandard analysis, I tried to put every definition that has to do with the question. If any of the definitions does not provide anything new please let me know. Also, if there is something missing. Thank you!
Best Answer
The transfer principle does not transform sentences that quantify over subsets of $\mathbb R$ to internal sentences that quantify over internal subsets of $\mathbb R^*$. Rather, it transforms sentences that quantify over subsets $S$ of $\mathbb R$ to (the "same") sentences that quantify over subsets $S^*$ of $\mathbb R^*$. It is an additional theorem that the transformed sentence is actually valid over a broader class of sets, namely the internal ones. This is indeed related to the fact that the internal sets are precisely the ones in the star-transform of $\mathcal P(\mathbb R)$.
Such a phenomenon seems a bit miraculous from the model-theoretic viewpoint on NSA, but it becomes more natural when viewed from the viewpoint of the axiomatic approaches to NSA. Here transfer is from sentences ranging over standard entities to sentences ranging over all entities. The "all entities" include standard and nonstandard ones.