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

**5**

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**1**answer

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### $\Delta^{1}_{2}$ and degrees of constructibility $\textbf{on sets}$

This is a follow-up to my question on $\Delta^{1}_{2}$ and degrees of constructibility of real numbers that was answered by the user "William", see here: Can $\Delta^{1}_{2}$ separate degrees of ...

**7**

votes

**1**answer

229 views

### Can $\Delta^{1}_{2}$ separate degrees of constructibility?

Suppose that $\phi(x)$ is a $\Delta^{1}_{2}$-formula (without parameters) and let $A:=\{x\subseteq\omega:\phi(x)\}$. It is clear that, e.g. if there are Cohen-generics over $L$, then $A$ cannot be the ...

**2**

votes

**1**answer

119 views

### Encoding sets in locally generic sets

Let $\alpha$ be an ordinal, and let $a\subseteq\alpha$ such that $\alpha$ is countable in $L[a]$. Moreover, let $\beta>\alpha$ be an ordinal such that, in $L[a]$, $\alpha$ and $\beta$ have the same ...

**7**

votes

**1**answer

296 views

### Is the smallest $L_\alpha$ with undefinable ordinals always countable?

Let $\mathfrak{t}$ be the least ordinal such that $L_{\mathfrak{t}}$ has undefinable ordinals; i.e. there is an $\alpha<\mathfrak{t}$ such that $L_{\mathfrak{t}}$ cannot define $\alpha$.
This ...

**18**

votes

**2**answers

774 views

### Does $V = \textit{Ultimate }L$ imply GCH?

In his Midrasha Mathematicae lectures ("In Search of Ultimate $L$", BSL 23 [2017]: 1–109), Woodin notes that $V = \textit{Ultimate }L$ implies $\textrm{CH}$ (Theorem 7.26, p.103). Is it known whether $...

**1**

vote

**1**answer

175 views

### 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 ...

**9**

votes

**1**answer

243 views

### Stability for the Gödel and Jensen hierarchies

Notations: Let $L_\alpha$ stand for the Gödel constructible hierarchy ($L_0=\varnothing$ and $L_{\alpha+1} = \mathrm{def}(L_\alpha)$ is the set of definable subsets of $L_\alpha$ and $L_\delta = \...

**-4**

votes

**2**answers

627 views

### Can only the constructible sets be proven to exist in $ZF$ without benefit of extra assumptions? [closed]

I am interested in asking the following question:
What sets can be proven to exist in $ZF$ without the benefit of extra assumptions? (Thanks to Toshiyasu Arai for inspiring me to ask this variation ...

**5**

votes

**0**answers

253 views

### Inaccessible cardinals and the perfect set property for coanalytic sets

I am wondering who proved the following fact:
($\ast$) If $\omega_1$ is not inaccessible in $L$, then there is an uncountable coanalytic set of reals without a perfect subset.
I have been unable to ...

**9**

votes

**1**answer

313 views

### Minimal cover v.s random reals

The following set theoretical question is inspired by a question from recursion theory:
Question: Is there an $L$-random real $r$ which is a minimal cover over another real $x$?
Where a minimal ...

**6**

votes

**1**answer

251 views

### Problem with definability in the constructible hierarchy

This is a rather technical question. I cannot find my mistake in a proof of the (obviously wrong) following sentence: Every countable ordinal is $\Sigma_2$-definable in $J_{\omega_1 + 1}$ by a formula ...

**14**

votes

**1**answer

445 views

### Category-theoretic characterization of $L$

Does there exist a characterization of Goedel's constructible universe $L$ in purely category-theoretic terms, or is constructibility an 'artifact' of material set theory? If, in fact, ...

**2**

votes

**2**answers

391 views

### What is the largest cardinal consistent with $ZFC$ + $V$=$L$?

What is the largest cardinal consistent with $ZFC$ + $V$=$L$? The reason for the question is this: under the assumption that all of 'ordinary mathematics' (as reverse mathematics understands the ...

**7**

votes

**1**answer

316 views

### How similar are large cardinals, over $L$?

EDIT: Joel's answer shows that no $\Sigma_2$ large cardinal property will do the job - however, $\Pi_2$ properties (such as unfoldability and its relatives) may still be useful.
Throughout this ...

**2**

votes

**1**answer

326 views

### Further research on $\mathrm L_{\infty}$

In the mathoverflow question , "Godel's Constructible Universe in Infinitary Logics (A Possible Solution to $HOD$ Problem), Prof Hamkins answered user46667's question 2
What is $\mathrm L_{\infty}$?...

**2**

votes

**0**answers

133 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

**1**answer

111 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 $L_\mu$}\}...

**6**

votes

**1**answer

229 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

**1**answer

416 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 $\Pi^0_2$.
The ...

**20**

votes

**2**answers

496 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

**0**answers

491 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 ...

**5**

votes

**1**answer

258 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 $...

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**0**answers

299 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

**1**answer

758 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 ...

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votes

**2**answers

589 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

**0**answers

116 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 ...

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votes

**2**answers

379 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

**2**answers

357 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 $\...

**2**

votes

**1**answer

363 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 ...

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votes

**1**answer

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

**3**answers

420 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 ...

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votes

**1**answer

574 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

**1**answer

335 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 ...

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**0**answers

725 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 \...

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votes

**2**answers

542 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

**1**answer

359 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 \...

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votes

**1**answer

279 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 $ \xi&...

**4**

votes

**1**answer

277 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 ...