# 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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### On a particular proof of “if the sharp of every real exists and every club contains a club constructible from a real, then $\delta^1_2 = \omega_2$”

I am referring to the proof of (4) implies (1) in Theorem 3.16 of Woodin's The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal. His proof leverages on the fact that if the sharp of ...
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### On the utility of Silver machines

This question arises out of having Devlin's Constructibility  in my collection of books at home during the lockdown. Chapter IX of the book deals with Silver machines, which are presented as ...
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### Consequences of existence of a certain function from $\omega_1$ to $\omega_1$

In his book , Paul Larson remarks (Remark 1.1.22) that in L there is a function $h:\omega_1\rightarrow\omega_1$ such that for any countable elementary submodel $X$ of $V_\gamma$ (where $\gamma$ is ...
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### End-extension in Gödel's constructible universe

Given two ordinals $\alpha < \beta$, considering the subsets of Gödel's constructible universe, one say that $L_\beta$ is a $\Sigma_n$ end-extension of $L_\alpha$ (and $L_\alpha$ is an $\Sigma_n$ ...
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### Is there a 'Constructible Universe' that is a submodel of a non-transitive model of $ZF$?

In Barwise's book, Admissible Sets and Structures, the following statement is made (on pg. 8): "...if $ZF$ is consistent, so is $ZF$ + "There is no transitive model of $ZF$" He mentions this fact ...
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### Is there a “hereditary” construction for $L$?

Recall that $L$, Godel's constructible universe is constructed by defining the following hierarchy: $L_0=\varnothing$, for a limit ordinal $\delta$, $L_\delta=\bigcup_{\alpha<\delta}L_\alpha$, and ...
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### Harrington's unpublished note “The constructible reals can be anything”

Around 1974, Leo Harringto wrote an unpublished note entitled "The constructible reals can be anything", in which he proved that it is consistent that being $\Delta^1_n$ is the same as being ...
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### Would a non-constructible set become constructible if we had oracles of arbitrarily high cardinality for the halting problems of ordinal computers?

I still have trouble to grasp the concept of a non-constructible set, my intuition is that we could "avoid" the non-constructibility of many of them if we assume we have "ordinal computers" extended ...
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### Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)

Gödel's constructible universe ($L$) is defined using definable power set operator in first order logic ($\mathcal{L}_{\omega ,\omega}$). One can produce such a universe in infinitary logics in ...
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### In $L$, does there exist a definable non-principal ultrafilter on $\mathbb{N}$

The axiom of constructibility $V=L$ leads to some very interesting consequences, one of which is that it becomes possible to give explicit constructions of some of the "weird" results of AC. For ...
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### V=HOD & The Height of the Large Cardinal Tree

As we know the assumption $V=L$ adds a restriction on the height of the large cardinal tree. Also there is a strict border like $0^{\sharp}$ exists such that all large cardinal axioms which are ...
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### What is the order type of $L$ with Godel's well ordering?

In some sense $Ord$ is a "proper class" ordinal. Unfortunately the notion of a proper class ordinal is not a straight forward generalization of the notion of "set" ordinals because the proper classes ...