# Questions tagged [constructibility]

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

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

77
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Fix a positive integer n. Assume $Lan=\{R_0,R_1,...,R_n\}$ be a language of first order logic, where every $R_i$ is a 2-ary relation symbol.
Assume $M$ is an Lan-model, where the underlying set is $...

7
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1
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96
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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 $...

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Working within $\sf ZF-Reg.$, is there anything that forbids having a set $A$ and a transitive model $M$ of $\sf ZF+ V=L(A)$, that admits an external bijection $j$ between two limit stages $L_\alpha(A)...

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I was wandering whether this weak form of $\text{CH}$ holds in $L(\mathbb{R})$ provably in $\text{ZF}+\text{DC}$
$(\text{ZF}+\text{DC}) \ L(\mathbb{R})\vDash \forall X\subseteq\mathbb{R} ( X \text{ ...

2
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105
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My question is the following:
Given $x,y \in \omega^\omega$ such that $x\equiv_c y$ is there an $L$-definable continuous map $\varphi: \omega^\omega\rightarrow \omega^\omega$ such that $\varphi(x) = ...

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Jensen proved that, if V=L, and $\kappa$ is a regular cardinal, then if for any stationary $A\subseteq \kappa$, the set $\{\alpha\mid A \text{ is stationary below }\alpha\}$ is stationary in $\kappa$, ...

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110
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Language: Second order logic, with as usual predicates written in upper case, and objects in lower case. Let $<$ be a primitive constant binary relation symbol.
Equality between objects is ...

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123
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Let's take the minimal transitive model of $\sf ZFC$ which, I came to know, is some minimal $L_\kappa$ for a countable $\kappa$, that models $\sf ZFC$, and since its minimal so no subset of it can be ...

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147
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In Gaps in the constructible universe, Marek and Srebrny, 1973 a gap ordinal and the start of a gap are defined as follows
$\alpha$ is a gap ordinal iff $(L_{\alpha+1}-L_\alpha)\bigcap \mathcal{P}(\...

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In Gaps in the constructible universe, Marek and Srebrny, 1973 a gap ordinal is defined as follows
$\alpha$ is a gap ordinal iff $(L_{\alpha+1}-L_\alpha)\cap \mathcal{P}(\omega) = \emptyset$
Their ...

4
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1
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192
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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 ...

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Jensen's covering lemma, stating that if there is no $0^\#$ in V, then some covering property holds true, has a very complex proof.
In any generic extension L[G] of L, $0^\#$ don't exist, so the ...

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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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250
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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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91
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Bachmann-Howard ordinal is a recursive ordinal. It's not that large compared to those proof-theoretic ordinals of stronger theories, but the definition of BHO is sufficient to illustrate how ...

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272
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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 $L_{\alpha+1} \vDash\ \alpha\text{ is inaccessible}$, ...

4
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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 \...

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359
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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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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 $\...

4
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1
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258
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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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0
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Define stratified $L$, denoted by $^S L$, as:
Let $S$ be the set of all stratified formulas in first order language of set theory.
Define:
${ }^S Def (X) = \{\{y \in X \mid (X, \in) \models \phi(y,z_1,...

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Hartley Rogers Jr., on pg. 120 of his text, Theory of Recursive Functions and Effective Computability, presents and discusses the following characterization of the sets in $\mathscr P(\omega)$:
$\...

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For $M$ an amenable structure and $X \subset M$, the $\Sigma_n$ Skolem hull of $X$ is a $\Sigma_n$-elementary submodel of $M$. That is, as presentend in Devlin, Constructibility, pp. 85-88, for $h_n$ ...

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282
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Define a ranking function $\cal R$ as:
$\mathcal{R}: V \to ON; \,\mathcal {R}(x)= \min \alpha \, \forall y \in x: \alpha > \mathcal {R}(y) $
Now the constructible rank $\mathcal R^c$ of a set $X$ ...

7
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423
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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 ...

3
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292
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The definition of $L$ only permits bounded quantifiers. If we allow a certain number of unbounded quantifiers, does this result in a strict superset of $L$? For example:
$$
\operatorname{Def}^{\...

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210
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Let $V$ denote the von Neumann universe and $L$ Gödel's constructible universe. For any set $X$, let $P(X)$ denote the power set of $X$.
Assume that $0^\sharp$ exists (and ZFC).
What is the smallest ...

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294
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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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1
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248
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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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542
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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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388
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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 ...

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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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Take ZFC, remove axiom of Power set, and put instead of it the following axiom:
Axiom of Successor Cardinals: $\forall \kappa \exists x \forall \alpha ( \alpha \leq \kappa \to \alpha \in x)$
where "$...

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512
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Informally the following theory is a kind of extension of second order arithmetic, where numbers are not limited to naturals, instead here we have formation of further numbers by setting limits on ...

5
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343
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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 ...

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On pg 190 of Jech's Set Theory, he proves V = L implies GCH. I understand it all except the following:
Thus let X ⊂ $ω_α$. There exists a limit ordinal δ>$ω_α$ such that X ∈ $L_δ$.
Let M be an ...

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Reflective Set Theory $\mathsf{RfST}$ is formulated in first order predicate logic with extra-logical primitives of equality $``="$, membership $``\in"$, and a single primitive constant symbol $V$ ...

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{EDIT: this posting has been edited, the additional text is in italics}
If $\varphi, \psi$ are two parameter free formulas in the language of set theory $T$ such that there is a theorem of $T$ that ...

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Begin with the empty set then construct the set of the empty set, then construct the set of all subsets of the latter set, then at each level of construction construct the next level as the set all ...

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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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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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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 ...

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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 ...

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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 $...

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1
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208
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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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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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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 ...

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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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335
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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
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386
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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 ...