nLab
ultraroot

Contents

Idea

An elementary substructure NMN \preceq M of a model of a first-order theory is an ultraroot of MM if MM is isomorphic to an ultrapower of NN.

Definition

Formally, recall that any structure NN admits a diagonal elementary embedding NΔN 𝒰N \overset{\Delta}{\to} N^{\mathcal{U}} to any ultrapower N 𝒰N^{\mathcal{U}} of NN.

If i:NMi \colon N \to M is an elementary embedding, we say that NN is an ultraroot of MM if there is some ultrafilter 𝒰\mathcal{U} on some index set such that there is an isomorphism

(NiM)(NΔN 𝒰) \left(N \overset{i}{\to} M\right) \simeq \left(N \overset{\Delta}{\to} N^{\mathcal{U}}\right)

in the co-slice category under NN.

Examples

  • NN is an ultraroot of N 𝒰N^{\mathcal{U}}.

  • Assuming the continuum hypothesis, any countable model of a complete-first order theory will be an ultraroot of that theory’s 1\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.

Remarks

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.

References

  • See e.g. page 7 of these notes by Ward Henson on continuous model theory,

Last revised on February 24, 2017 at 02:48:15. See the history of this page for a list of all contributions to it.