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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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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2
answers
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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 ...
19
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2
answers
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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 $...
15
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1
answer
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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, ...
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When is $A$ "$L$-ish" whenever $B$ is "$L$-ish"?
My question is about a variant of the usual notion of relative constructibility, $\le_c$ (which an earlier version of this question confusingly denoted "$\le_L$"), in set theory.
Fix a ...
14
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2
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Consequences of existence of a certain function from $\omega_1$ to $\omega_1$
In his book [1], 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 ...
14
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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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1
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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 ...
13
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Can $L$ be defined without parameters?
If we omit parameters in the definition of $L$ would the result still be $L$?
That is, we define a successor stage $L_{\alpha+1}$ in the constructible universe $L$, without including parameters; as:
$...
12
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2
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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 $\...
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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 $...
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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 $L \subseteq M$ ...
10
votes
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answer
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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 = \...
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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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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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On the utility of Silver machines
This question arises out of having Devlin's Constructibility [1] 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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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 ...
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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 ...
8
votes
3
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Elementary countable submodels in Gödel's universe
By the downward Lowenheim-Skölem theorem we can find two countable ordinals $\alpha < \beta$ such that $L_\alpha \prec L_{\omega_1}$ and $L_\beta \prec L_{\omega_1}$. That is, $L_\alpha$ and $L_\...
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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 ...
8
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1
answer
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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 ...
8
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Forcing a unique $\Delta_3^1$ generic real
I know Jensen developed a forcing notions in $L$ that adds a unique, minimal and $\Delta_3^1$ $L$-generic real. In his paper Definable sets of minimal degree he says that Solovay had already shown the ...
8
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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 ...
7
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1
answer
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If $L_\alpha \vDash ZFC$, then do we have $L_{\alpha+1} \vDash \alpha\text{ is inaccessible}$?
Here we choose the definition of "is a cardinal" as there is no surjective map from a smaller ordinal to it.
It's easy to prove that, if $L_{\alpha+1} \vDash\ \alpha\text{ is inaccessible}$, ...
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2
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Can countable ordinals start gaps of every order in the constructible universe?
Define "$\alpha$ starts a gap of order $n+1$ and length $\beta$" iff $\mathcal P^n(\omega)\cap (L_{\alpha+\beta}\setminus L_\alpha)=\emptyset\land\forall\gamma\in\alpha: L_\alpha\setminus L_\...
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1
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Why do $\pi$ and $\bigcup$ commute for Gödel-closed extensional classes?
Jech exercise 13.3 states:
If $M$ is closed under Gödel operations and extensional, and $\pi$ is the transitive collapse of $M$, then $\pi(G_i(X,Y))=G_i(\pi X,\pi Y)$ for all $i=1,\ldots,10$ and all $...
7
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1
answer
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Which one of the following two ordinals is larger?
We say that $\alpha$ is $\Sigma_n$-extendable (to $\beta$), if there is $\beta>\alpha$ such that $L_\alpha$ is a $\Sigma_n$ elementary submodel of $L_\beta$.
First ordinal: the least $\alpha_0$ ...
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Capturing the $\omega_1^{\mathrm{CK}}$-th stage of Gödel's constructible hierarchy
For an ordinal $\alpha$, let $L_\alpha$ be the $\alpha^{th}$ set of Gödel's constructible hierarchy and let $\omega_1^{\mathrm{CK}}$ be the first non-recursive ordinal or the first admissible ordinal ...
7
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1
answer
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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 ...
7
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1
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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 ...
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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&...
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1
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For which ordinals do we have $V_\alpha = L_\alpha$?
Some elements of $L$ become constructible only in levels higher than its rank level. So I ask:
Let $V$ be such that $V = L$.
For which ordinals $\alpha$ do we have $V_\alpha = L_\alpha$?
Indeed, we ...
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answer
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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 $$E_\kappa=\{\mu<\kappa: \mbox{there is an elementary substructure of $L_\kappa$ isomorphic to $L_\mu$}\},$$ ...
6
votes
1
answer
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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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1
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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 ...
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0
answers
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Consistency strength of Sy Friedman's result about admissibility spectrum
A result by Sy Friedman in his book "fine structure and class forcing", is that, assume $0^\sharp$ exists, there exists a real number R such that the ordinals admissible in R (called $\...
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Inaccessible cardinals and the perfect set property for coanalytic sets
I am wondering who proved the following fact:
($\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 ...
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3
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Is the power set axiom essential for constructing L?
Take ZFC, remove axiom of Power set, and put instead of it the following axiom:
Axiom of Successor Cardinals: $\forall \kappa\, \exists x \, \forall \alpha \, ( \alpha \leq \kappa \to \alpha \in x)$
...
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2
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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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Is this relation about elementary embedding transitive?
For ordinals $\alpha<\beta$, we say $\alpha<_{el}\beta$, if there is an elementary embedding with domain $L_\beta$ and critical point $\alpha$.
Is $<_{el}$ transitive?
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Sequences of projecta in the constructible hierarchy
For $n$ a natural number, $\alpha$ an ordinal, let $\rho(n,\alpha)$ 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 ...
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answer
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Inner model with a $\mathit{\Delta}^1_3$-good well-ordering of the reals
The constructible universe $L$ has some nice properties:
$L$ has a $\mathit{\Delta}^1_2$-good well-ordering of $\mathbb{R}$. (Gödel, Addison)
For any $\mathit{\Sigma}^1_2$ formula $\varphi(x)$ and a ...
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1
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Height of diamond
Assume $V=L$.
Let $\alpha$ be the least ordinal such that there is a $\Diamond_{\omega_1}$-sequence in $L_\alpha$.
It's obvious that $\omega_1 < \alpha < \omega_2$.
Do we have some better ...
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1
answer
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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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Terminology for ordinals whose constructible level is the least one satisfying some formula
An ordinal $\alpha$ is "meta-definable" by some formula $\varphi$ without free variables if:
$$
\begin{cases}
L_\alpha \models\varphi \\
\forall\beta < \alpha \, L_\beta \not\models \...
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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 ...
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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 \...
4
votes
2
answers
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Can local $0^\#$ exists in L?
Assume $0^\#$ exists and there is an inaccessible cardinal.
Are there two transitive sets $M,N$ s.t. $M\in N,M\vDash ZF+V=L[0^\#],N\vDash ZF+V=L$?
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1
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A doubt about the Gödel condensation lemma
To simplify the notation, assume $V=L$. We have $\lvert V_{\omega_{1}} \rvert=\aleph_{\omega_{1}}$ and $\lvert H(\aleph_{1})\rvert=\aleph_{1}$, so in particular $V_{\omega_{1}} \models \exists x \...
4
votes
1
answer
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Existence of a non-$Q$-set without the perfect set property
We have the following theorem:
Suppose $\omega_1^L=\omega_1$ then there exists a $\Pi_1^1$ subset of reals without the perfect set property
Moreover, under the same hypotheses, we can prove actually ...