# Tagged Questions

**1**

vote

**3**answers

291 views

### Can the structure of an ultrafilter determine the structure of its ultrapower?

Usually we work with ultrafilters as pure sets without any structure.
Q1. Is there any important notion of structure on an ultrafilter?
Q2. Is there any non-trivial notion of structure on ...

**5**

votes

**1**answer

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" ...

**1**

vote

**0**answers

104 views

### Compactness-like property for universal generalization?

Hi all! I have the following problem. Suppose I have a sequence of models $M_1,M_2,...$, all of which have the same countable domain (call it $D$). $x_1,x_2,...$ is a well-ordering of $D$. $\phi(x)$ ...

**4**

votes

**2**answers

778 views

### Are all countable, nonstandard models of arithmetic given by ultrapowers?

Countable models of PA fall into two categories: the standard one $(\omega, S)$ and the nonstandard ones (all the rest). The only way I've seen to construct a nonstandard model is through taking an ...

**4**

votes

**1**answer

362 views

### Which properties of ultrafilters on countable sets hold for filters in general?

Background/motivation: I'm investigating the construction of models for a first-order modal system (S5) as products of classical models. Since ultraproducts are all classical models and I need ...

**3**

votes

**2**answers

237 views

### Behavior of externally-infinite elements in ultrapowers of $\langle HF,\epsilon\rangle$

Consider the structure $\langle HF,\epsilon\rangle$ (the hereditarily finite sets with the epsilon-relation). An ultrapower of this structure will have externally-infinite elements -- elements not ...

**4**

votes

**1**answer

650 views

### How can an ultrapower of a model of ZFC be “ill-founded” yet still satisfy ZFC?

My understanding (please correct me if I'm wrong) is that if you have some transitive set M which is an $\epsilon$-model of ZFC, and you take an ultrapower of it using an approprate ultrafilter, you ...