All Questions
8 questions
10
votes
0
answers
334
views
Definability up to isomorphism versus definability of an isomorphic copy
Question: Is it provable in ZFC that every structure that is ordinal definable up to isomorphism has an ordinal definable isomorphic copy? If not, what are some counterexamples? All structures are ...
- 7,545
8
votes
1
answer
384
views
Iterating definability
An odd -- probably basic -- question about model theory:
For $\mathcal{M}$ a structure in a (first-order) signature $\Sigma$, let $\mathcal{M}'$ be the structure in signature $\Sigma\sqcup\lbrace U\...
- 20.5k
3
votes
2
answers
269
views
Is every countable model of ZFC a subset of some parameter free definable pointwise-definable model of ZFC?
Is it consistent with $\sf ZFC + \text{ countable models of } ZFC \text { exist}$, that every countable model of $\sf ZFC$ is a subset of some parameter free definable pointwise-definable model of $\...
- 11.3k
3
votes
1
answer
303
views
$(\kappa,\lambda)$ - Minimal Models & Stronger Version of Rowbottom's Theorem
Definition 1: Let $M$ be a $\mathcal{L}$ - structure and $A\subseteq Dom(M)$. Define:
$Def_{A}(M):=\{X\subseteq Dom(M)~|~\exists n\in \omega~~\exists \varphi (x,y_1,...,y_n)\in \mathcal{L}-Form~~\...
user42090
3
votes
1
answer
154
views
If a theory speaks of sets that cannot be forced to be parameter free definable, then does this entail a large cardinal property?
If we say that an effectively generated first order theory $\sf T$ extends $\sf ZF$, such that every countable model of $\sf T$ doesn't have a class forcing extension that is pointwise definable. ...
- 11.3k
3
votes
1
answer
169
views
Outer Definability of a Class
Definition: Let $C$ be a class of sets and $\mathcal{L}$ a first order relational language. We say $C$ is "outer definable" by $\mathcal{L}$ if there is a first order theory $T$ and for each $n_{R}$ - ...
user43940
1
vote
1
answer
148
views
Is there a model of each of the following kinds of theories in the first transitive model of ZFC?
The question of whether the minimal transitive model $M$ of $\sf ZFC$ have a model of each consistent extension $\sf T$ of $\sf ZF$, is answered to the negative, basically because $M$ is countable and ...
- 11.3k
1
vote
1
answer
104
views
Are externally pointwise definable models of ZFC subject to the same limitations of the internally pointwise definable ones?
By pointwise definable models, it's meant that every element of those models is definable after a formula in a parameter free manner, but that defining formula is in the language of that model, i.e., ...
- 11.3k