indiscernible sequence?
Morley sequence?
Ramsey theorem?
Erdos-Rado theorem?
Ehrenfeucht-Fraïssé games (back-and-forth games)
The Keisler-Shelah isomorphism theorem characterizes the partially syntactic concept of elementary equivalence in purely semantic form with the help of ultrapowers.
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.
are elementarily equivalent. Assuming the continuum hypothesis (this is an example of where this technical distinction is vital), they are also isomorphic.
Los ultraproduct theorem?
Created on February 26, 2017 at 02:04:52. See the history of this page for a list of all contributions to it.