# Questions tagged [constructibility]

This tag is for questions about Gödel's constructible universe $L$, and related constructions such as $L[X]$ and $L(X)$.

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### When is a $\Sigma_n$ Skolem hull a proper submodel?

For $M$ an amenable structure and $X \subset M$, the $\Sigma_n$ Skolem hull of $X$ is a $\Sigma_n$-elementary submodel of $M$. That is, as presentend in Devlin, Constructibility, pp. 85-88, for $h_n$ ...
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### What does the Concordant constructible universe model?

Define a ranking function $\cal R$ as: $\mathcal{R}: V \to ON; \,\mathcal {R}(x)= \min \alpha \, \forall y \in x: \alpha > \mathcal {R}(y)$ Now the constructible rank $\mathcal R^c$ of a set $X$ ...
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### Can all unions of sets above some level be constructible before the sets in some relative constructible universe?

Can we have some relative constructible universe $L(A) \ (or \ L[A])$ such that for some infinite ordinal $\gamma \leq |A|$ we have: for every subset $u$ of $A$, if $u$ is the union of a set $\sf U$ ...
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### Is $\mathsf{ZFC+V=L}$ consistently $\omega$-complete?

This was previously asked and bountied on MSE: For brevity, let $T$ be $\mathsf{ZFC+V=L}$. Say that an extension of $\mathsf{ZFC}$ is $\omega$-complete iff it has exactly one $\omega$-model up to ...
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### Fine structure question: when do levels of $L$ look "a lot" like each other?

(Everything is assuming $V=L$.) Fix an uncountable regular cardinal $\kappa$, and let $$E_\kappa=\{\mu<\kappa: \mbox{there is an elementary substructure of L_\kappa isomorphic to L_\mu}\},$$ ...
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### Is every ordinal potentially definable?

It is easy to see that, if $V\models\alpha>\omega_1^{CK}$, then $\alpha$ is not recursive in any forcing extension of $V$. The argument goes as follows: The relation "$\Phi_e=r$" is $\Pi^0_2$. The ...
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### Can $L$ be thin?

I have recently been wondering if the following is consistent with ZFC: For every infinite ordinal $\alpha$: $|V_\alpha\cap L|=|\alpha|$. Intuitively, this states that for $L$ is very "thin", in ...
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### Can there be ordinals larger than those contained in Ord, and if so, can they be used to extend the constructible universe $L$?

Can there be ordinals larger than those contained in Ord, the class of all ordinals,and if so, can these ordinals be used to extend the constructible universe $L$? In a simplified form, my question ...
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### Does $\forall X \in L_{\kappa}: P(X) \subset L_{\kappa}$ hold in $L$

While working on an exercise in Jech's Set Theory, I tried to prove that if $V=L$, then $\forall X \in L_{\kappa}: P(X) \subset L_{\kappa}$, where $k$ is any cardinal. I was hoping someone could ...
### When is $A$ "$L$-ish" whenever $B$ is "$L$-ish"?
My question is about a kind of relative constructibility in set theory. Fix a countable transitive model $W\models ZFC$ which is much bigger than $L^W$. There is a natural way within $W$ to compare ...