# Tagged Questions

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### Is “constructible from” a transitive relation?

In Jech's Set Theory, exercise 13.27, it is hinted that $X \in L[Y]$ and $Y \in L[X]$ together imply $L[X]=L[Y]$. I tried to prove this fact without success, although I suspect the proof is simple. ...
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### Construction of Ultrafilters

I've been doing a lot of work with ultrapowers and saturation recently. In particular, I am reading chapter 6 of Chang and Keisler as well as Keisler's paper on "Ultraproducts which are not ...
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### Why is $\alpha \mapsto L_{\alpha}[A]$ $\Delta_{1}$?

On page 187 of Jech's Set Theory, there is a proof sketch of the fact that $\alpha \mapsto L_{\alpha}$ is $\Delta_{1}$. As far as I can tell, Jech's argument only shows that this operation is ...
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### Sets Constructible Relative To A Unary Predicate

The class $L$ of constructible sets is defined by recursion using the operation def$(M)=\{x \subset M: x$ is definable over $(M, \in) \}$. By adding a unary predicate, $P$, to our language, we can ...
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### Model of complete extension of Zermelo set theory

Chang and Keisler's Model theory gives the following exercise problem: Prove that there is a complete extension $T$ of Zermelo set theory which has arbitrary large natural models. (A model ...
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### Cardinalities of Collections of Models

Let $T$ be a complete theory in a countable language (with only infinite models). Recall the spectrum function: $I(\aleph_\alpha,T)=$ the number of non-isomorphic models of $T$ of cardinality ...
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### On the number of countable models of complete theories of models of ZFC [duplicate]

Fix the language of set theory $\mathcal{L}=\{\in\}$. Let $\langle M,\in\rangle$ be a set or proper class model of ZFC (e.g. $M$ could be $L$, $HOD$, $V_{\kappa}$ for some inaccessible cardinal ...
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### Countable transitive $T \vDash ZFC-P$ with $\approx$ not absolute: why do we need $H(\aleph_3)$?

I am trying to solve Exercise IV.3.31 from Kunen's Foundations of Mathematics. I think I have a solution but I am confused by one of the hints. By request, here is the text of the exercise. ...
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### Absolute confusion! (A question about absolute *sentences*)

I'm seriously confused about absoluteness. A formula in the language of a theory $T$ is absolute for $T$ structures if its truth value is the same in all standard transitive models of $T$ (this may ...
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### How large or small can the gap in Shelah's main gap theorem be, up to consistency?

Shelah's main gap theorem in model theory says: For each first order complete theory $T$ in a countable language if $I(T,\kappa)$ denotes the number of its models of size $\kappa$ then one of ...
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### Name of, and (if I'm lucky) references on, a particular property of an interpretation

So here I am studying the Ackerman interpretation (via Kaye-Wong) to try and suss what the fragment of arithmetic associated with KF (Mac Lane minus Foundation and Infinity, with separation restricted ...
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### Question on existential sentences

A sentence is called existential if it is of the form $\exists x_1 \ldots \exists x_n \ \phi(x_1, \ldots, x_n)$, where $\phi$ is quantifier free. We know that (see Chang-Keisler "Model Theory", ...
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### Defining “structured sets”

In his Notes on Set Theory (p. 44) Moschovakis defines: A structured set is a pair $U = (A,S)$ where $A$ is a set, the space of $U$, and $S$ is an arbitrary object, the frame of $U$. But even ...
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### Showing that a statement is absolute.

After reading about various properties of $V_\alpha$ and how it can be used to model various axioms of Set Theory, Kunen mentions that in $ZFC$, one cannot prove that there is an $\alpha$ such that ...