# 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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### How does the cardinality of a set and its powerset compare in the hereditarily rank-concordant constructible world?

Working in the constructible universe "$L$", if we define two kinds of ranks for any constructible set $x$, one being the ordinal index of the first $L_\alpha$ where $x$ appears as a subset ...
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### Models of second-order arithmetic closed under relative constructibility

I know little to nothing about second-order arithmetic and its subsystems. However, I would like to understand when a model of (a subsystem of) second-order arithmetic ($\mathsf{Z}_2$) is downward ...
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### Inner model for KP and a Well-Ordering of the Reals

It is well known that Gödel proved the following theorem: $\mathsf{ZFC + V=L}$ has a $\mathit{\Delta}^1_2$-good well-ordering of $\mathbb{R}$. (Gödel, Addison) So: Is there an inner model for KP/Z/....
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### Can the Constructible Universe be built in absence of Unions and Power?

Can $L$ be built in $\sf ZF$ $\sf-Regularity-Union-Power+ Boolean \ Union$? We know that $L$ can be built in $\sf KP$, but here we don't have Set Union. If the answer is to the negative, then would ...
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### Parameter-free effective cardinals

In the paper "Effective cardinals and determinacy in third order arithmetic" by Juan Aguilera, effective cardinals is defined. I'm curious about its little variation, parameter-free ...
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### Is stable ordinals in non-well-founded model the same as well founded models?

Let $BST$ be the axiom system $KP$ - $\Delta_0$ collection. For an ordinal $\alpha$, we say that $\alpha$ is $\varphi$-$\Sigma_n$-stable, if there is a $\beta>\alpha$ satisfies the formula $φ$ such ...
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### Higher-order oracle computation of reals and axiom of constructibility

Certain real numbers can be approximated arbitrarily well by computable functions. If we introduce halting oracles, then more real numbers can be "computed", like Chaitin's constant or the ...
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### Are all constructible from below sets parameter free definable?

Lets take the intersection of the theory of $L_{\omega_1^{CK}}$ and $\sf ZF + [V=L]$, this is equivalent to the theory of constructability from below + limit stages. Can this theory prove the ...
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