indiscernible sequence?
Morley sequence?
Ramsey theorem?
Erdos-Rado theorem?
Ehrenfeucht-Fraïssé games (back-and-forth games)
Hrushovski construction?
generic predicate?
An elementary substructure $N \preceq M$ of a model of a first-order theory is an ultraroot of $M$ if $M$ is isomorphic to an ultrapower of $N$.
Formally, recall that any structure $N$ admits a diagonal elementary embedding $N \overset{\Delta}{\to} N^{\mathcal{U}}$ to any ultrapower $N^{\mathcal{U}}$ of $N$.
If $i \colon N \to M$ is an elementary embedding, we say that $N$ is an ultraroot of $M$ if there is some ultrafilter $\mathcal{U}$ on some index set such that there is an isomorphism
in the co-slice category under $N$.
$N$ is an ultraroot of $N^{\mathcal{U}}$.
Assuming the continuum hypothesis, any countable model of a complete-first order theory will be an ultraroot of that theory’s $\aleph_1$-saturated continuum-sized model.
Similarly, taking a countably-indexed ultrapower of a slightly-expanded copy (like a nonstandard model generated by throwing in a single infinitesimal) of the reals still yields the hyperreal numbers.
Closure under taking ultraroots, along with closure under ultraproducts and elementary embeddings, gives a criterion for detecting elementary classes. See also at Birkhoff's HSP theorem.
Last revised on February 24, 2017 at 07:48:15. See the history of this page for a list of all contributions to it.