5
votes
1answer
268 views

Ultraproduct of Forcing Extensions & Forcing Extension of Ultraproduct

Notation: $M[{\mathbb{P}}:G]$ denotes the forcing extension of $M$ by $\mathbb{P}$-generic filter $G$. $\prod_{\mathcal{F}}\langle M_i~|~i\in I\rangle$ denotes the ultraproduct of models using the ...
2
votes
1answer
120 views

How many elementary equivalent models are unifiable by ultrapower?

Definition. A class $\mathcal{C}$ of pairwise elementary equivalent $\mathcal{L}$-structures is unifiable by ultrapower if there is an index set $I$ and an ultrafilter $F$ on it such that $\forall ...
10
votes
3answers
851 views

Is it ever a good idea to use Keisler-Shelah to show elementary equivalence?

The most useful way I know to show that two structures are elementarily equivalent is Ehrenfeucht-Fraisse games. These are quite nice and intuitive, and even when I can't use them to solve my problem ...
1
vote
0answers
174 views

Ultrapowers of ultrapowers

Suppose that you have some structure $S$, and you want to construct an ultrapower of cardinality $\kappa$ to obtain $S^*_\kappa$. Then, say you want to construct a new ultrapower from $S^*_\kappa$, ...
2
votes
1answer
194 views

Can we flex the rigid models by enough power?

Definition (1): ‎An ‎‎$‎‎‎\mathcal{L}‎$ -‎ ‎structure ‎‎$‎‎‎\mathcal{M}‎$ ‎called ‎"‎‎rigid" ‎iff ‎‎there ‎is ‎no ‎non-trivial automorphism on ‎$‎\mathcal{M}‎$.‎ ‎‎ Definition (2): ‎An ...
5
votes
1answer
140 views

Is there a truth approximation on a‎ cumulative hierarchy‎‎‎‎?

‎‎‎Note ‎to ‎the ‎following ‎well known theorem:‎ Theorem (1): ‎If ‎‎$‎‎\kappa‎‎$ ‎be a ‎‎"measurable" ‎cardinal ‎and ‎‎$‎‎‎\mathcal{F}‎$ be a‎ ‎"non-principal ‎$‎‎‎\kappa‎$-complete normal‎" ...
5
votes
2answers
308 views

Set forcing and ultrapowers

The following is a result of Woodin (the proof is found after Theorem 5 of "Generalizations of the Kunen Inconsistency" by J.D.Hamkins, G.Kirmayer and N.L.Perlmutter): (Woodin) Let $V[G]$ be a ...
5
votes
1answer
415 views

An Extender is a Generalization of an Ultrafilter?

I'm not sure whether this belongs here or on math stackexchange, but I'll give a try here. I've heard that an extender is a generalization of an ultrafilter. This is not to say that an ultrafilter ...
3
votes
1answer
153 views

cardinality of local bases in the non-standard reals

Given a index set $S$ together with a ultrafilter $\mu$ on $S$ (such that no set of cardinality $< |S|$ has measure $1$). Let the ordered field $\mathbb{R}(S,\mu)$ denote the ultrapower of ...
5
votes
1answer
294 views

The closure of a generic ultrapower

Let $I$ be a normal ideal on $P_{\kappa} (\lambda)$. Let $V$ denote our ground model. Now we force with the $I$-positive sets, and if $G$ is the resulting generic filter, it can be shown that $G$ is a ...