Transfer principle

Remarks

The principle is a special case of Łoś's theorem. That theorem says that a first-order formula holds at a tuple of classes exactly when it holds termwise on a set of indices in the ultrafilter.

Example: the sentence $\forall x \, (x > 0 \to \exists y \, (y \cdot y = x))$ is true in $\R$. By transfer, every positive hyperreal has a square root in ${}^*\R$.

The restriction to first-order sentences, where quantifiers range over elements and not over arbitrary subsets, is necessary. Every nonempty subset of $\R$ that is bounded above has a supremum, but in ${}^*\R$ the set of infinitesimals is bounded above by $1$ and has no supremum. Indeed, a supremum $s$ would be positive; if $s$ were infinitesimal then $2s$ would be a larger infinitesimal, and otherwise $s/2$ would be a smaller upper bound.

Going deeper

For a student

Transfer is what makes nonstandard proofs valid. One proves a statement in ${}^*\R$ using infinitesimals, checks that it is the star version of a first-order statement, and transfers it back to $\R$. For instance, a real function $f$ is continuous at a real $a$ exactly when ${}^*f(x) - f(a)$ is infinitesimal for every $x$ with $x - a$ infinitesimal. Transfer is the key step in proving this equivalence. Statements about all subsets of $\R$ transfer only to statements about internal subsets of ${}^*\R$, such as ${}^*A$ for $A \subseteq \R$.