All Questions
10 questions
7
votes
0
answers
296
views
A system with distinct infinite cardinalities but no "best" version of $\mathbb{N}$
Let $\mathfrak{S}=(M_z,U_z)_{z\in\mathbb{Z}}$ be a sequence such that for each $z\in\mathbb{Z}$ we have
$M_z\models\mathsf{ZFC}+$ "$U_z$ is a nonprincipal ultrafilter on $\omega$" (so in ...
- 21.2k
7
votes
0
answers
185
views
Interest in the size of ultrapowers in model theory
It seems that in the 60s (at least), there was interest in computing the size of ultrapowers by countably incomplete ultrafilters. For example, given an ultrafilter $U$ on some relatively small set ...
- 18.6k
3
votes
1
answer
271
views
Ultra*powers* in the category of structures and elementary embeddings
This is based on a few previous questions.
Can one characterize ultrapowers in the category of L-structures (modeling a fixed complete theory, say) and elementary embeddings?
Previous posts showed ...
- 151
17
votes
0
answers
1k
views
Non-rigid ultrapowers in $\mathsf{ZFC}$?
Originally asked and bountied at MSE:
Question: Can $\mathsf{ZFC}$ prove that for every countably infinite structure $\mathcal{A}$ in a countable language there is an ultrafilter $\mathcal{U}$ on $\...
- 21.2k
13
votes
1
answer
678
views
When do two ultrafilters yield isomorphic ultrapowers?
Fix a cardinal $\lambda$$\newcommand{\cU}{\mathcal U}\newcommand{\cV}{\mathcal V}$. Consider the equivalence relation on $\beta\lambda$ given by $\cU\sim \cV$ when for all first-order structures $M$ ...
- 8,099
10
votes
3
answers
841
views
Is every field extension of an ultrafield an ultrafield?
Let $K=\lim(K_{i})$ be an ultrafield (over a non-principal ultrafilter), and let $K\hookrightarrow K'$ be a field extension of $K$.
When the field $K'$ is finite over $K$ it is also an ultrafield by ...
- 19.6k
16
votes
1
answer
746
views
Is there a pair of non-isomorphic structures each of which is isomorphic to an ultrapower of the other?
Does there exist a pair of non-isomorphic structures $\mathfrak{A}$ and $\mathfrak{B}$ as well as sets $I$ and $J$ and ultrafilters $\mathcal{U}$ on $I$ and $\mathcal{F}$ on $J$ such that $\mathfrak{A}...
- 8,099
15
votes
4
answers
2k
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 ...
- 32.1k
2
votes
1
answer
292
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 $\mathcal{...
5
votes
1
answer
306
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" ...
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