The Keisler-Shelah isomorphism theorem characterizes the partially syntactic concept of elementary equivalence in purely semantic form with the help of ultrapowers.

Statement of theorem

In practice, the theorem is usually stated as follows: let $A$ and $B$ be first-order$\mathcal{L}$-structures. Then $A$ and $B$ are elementarily equivalent, written $A \equiv B$, if and only if there is an ultrafilter $\mathcal{U}$ on some index set $I$ such that there is an isomorphism of ultrapowers$A^{\mathcal{U}} \simeq B^{\mathcal{U}}$.

Keisler proved, assuming GCH, that when $A \models T$ and $B \models T$ have cardinality $\leq 2^{|T|}$ then they have isomorphic $|T|$-indexed ultrapowers. Shelah removed the assumption of GCH at the cost of exhibiting the isomorphism for only $2^{|T|}$-indexed ultrapowers instead.

Examples

The Ax-Kochen-Ershov theorem? states that for any non-principal ultrafilter $\mathcal{U}$ on the primes, the valued fields (viewed as structures in ACVF?, where $\mathbb{Q}_p$ is the p-adic field and $\mathbb{F}_p((t))$ is the field of formal Laurent series? over the finite field$\mathbb{F}_p$)

are elementarily equivalent. Assuming the continuum hypothesis (this is an example of where this technical distinction is vital), they are also isomorphic.

Remarks

One could view this theorem as a generalization/variant of the transfer principle from nonstandard analysis: given any two structures with the same theory, there exists a single “nonstandard model” linking the two via their diagonal embeddings into their ultrapowers.