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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2
votes
0answers
114 views

When do wide initial segments ruin admissibility?

Suppose $\alpha$ is admissible and $\beta<\alpha$. We know that $L_\alpha$ is an admissible set (by definition), but adding subsets of $\beta$ to $L_\alpha$ might break admissibility: while set ...
3
votes
1answer
86 views

Levels of L resembling each other, take 2

(Everything below 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 ...
6
votes
1answer
203 views

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 $$E_\kappa=\{\mu<\kappa: \mbox{there is an elementary substructure of $L_\kappa$ isomorphic to $L_\mu$}\},$$ ...
7
votes
1answer
337 views

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 $\Pi^0_2$. ...
15
votes
2answers
339 views

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 ...
3
votes
0answers
398 views

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 ...
4
votes
1answer
166 views

Simplest non-constructible set of integers compatible with the nonexistence of $0^\sharp$?

What is the simplest non-constructible set of integers (say, in the analytical hierarchy) that is compatible with the nonexistence of $0^\sharp$? In particular, can there still be a non-constructible ...
10
votes
0answers
266 views

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 ...
10
votes
1answer
672 views

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 ...
9
votes
2answers
453 views

How necessary is Godel's Condensation Lemma

It seems that the Godel's Condensation Lemma is typically used to show that certain constructible sets will appear by some stage of the construction of $L$. For example in the proof that CH holds in ...
2
votes
0answers
102 views

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 ...
9
votes
2answers
354 views

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 ...
12
votes
2answers
335 views

Producing no non-constructible reals

The following is stated without proof in Shelah's book "Cardinal arithmetic" (page 276), and is attributed to Uri Abraham: Suppose that $L[A], L[B]$ have no non-constructible reals and that ...
1
vote
1answer
260 views

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 ...
16
votes
1answer
1k views

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 ...
6
votes
3answers
318 views

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 ...
10
votes
1answer
452 views

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 ...
2
votes
1answer
269 views

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 ...
12
votes
0answers
685 views

Can a model of $V\neq L$ contain a class giving the $L$-ordering on all its sets?

This question is inspired by the excellent question by Douglas Ulrich When is $L$-Rank definable in inner models of $V=L$? Suppose $M \in L$ is a countable model of $ZFC$, and furthermore suppose $M ...
10
votes
2answers
473 views

Constructible models of New Foundations?

Hi all! Is there anything like Gödel's constructible universe for New Foundations? More precisely, I would like a process for taking a model $M$ of NF, and using it to build a model $L \subseteq M$ ...
5
votes
1answer
316 views

Higher computability : Constructive ordinal and $\Delta^1_1$ predicates

Everything I know on this subject comes from Sacks book : "Higher recursion theory" Let $\mathcal{O^Y}$ be the set of codes for ordinals constructive in $Y$. We should have the result that $A ...
7
votes
1answer
254 views

Acceptability and Soundness of J-structures.

I would like an example of a J-structure $(J^A,B)$ which is not acceptable and one that is not 1-sound. Edit:Let us recall that a structure $J^A_\alpha$ is acceptable if for every limit ordinal $ ...
4
votes
1answer
251 views

Sequences of projecta in the constructible hierarchy.

For $n$ a natural number, $\alpha$ an ordinal, let $p(n,\alpha)$ be the $n$-th projectum of $J_\alpha$, where $J$ is the Jensen hierarchy for $L$. Call a finite sequence $s:=(x_1,\dots,x_m)$ of ...