# nLab Keisler-Shelah isomorphism theorem

model theory

## Dimension, ranks, forking

• forking and dividing?

• Morley rank?

• Shelah 2-rank?

• Lascar U-rank?

• Vapnik–Chervonenkis dimension?

# Contents

## Idea

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 $\mathbb{F}_p$)
$\displaystyle \prod_{p} \mathbb{Q}_p/\mathcal{U} \equiv \displaystyle \prod_{p} \mathbb{F}_p((t)) / \mathcal{U}$

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.

ultraroot

ultrapower

ultraproduct

Los ultraproduct theorem?