Questions tagged [constructibility]
This tag is for questions about Gödel's constructible universe $L$, and related constructions such as $L[X]$ and $L(X)$.
109 questions
9 votes
1 answer
634 views
Automorphisms over models of L?
let $M$ be a model of $\mathrm {ZF + V=L}$ , and let $j$ be an external automorphism on $M$. Is it possible to have an infinite cardinal $\kappa \in L$ such that $j(\kappa)=\kappa^+$? Note that $\...
- 13k
6 votes
1 answer
202 views
If there's an $α$-Erdős cardinal for every countable $α$, is there an $L_λ$ with a set of order-indiscernibles of order-type $λ<ω_1$?
Let $\alpha$ be a countable limit ordinal. An $\alpha$-Erdős cardinal exists iff there is some cardinal $\kappa$ satisfying the partition relation $\kappa\rightarrow(\alpha)^{<\omega}_2$. Given $\...
- 83
1 vote
2 answers
320 views
Can this semi-constructible structure satisfy existence of a measurable cardinal?
If we add a primitive unary function symbol $\mathfrak L$ to the first order language of set theory. Axiom of semi-constructibility: if $\phi^\alpha (y,x_1,\ldots,x_n)$ is a formula in which all and ...
- 13k
7 votes
1 answer
626 views
Does this ZFC+V=L like theory, have a limit on large cardinal properties?
Let $\sf T$ be a theory that has as axioms every axiom of $\sf ZFC$, and every theorem of $\sf ZFC + [V=L]$ that is neither provable nor disprovable by $\sf ZFC$, whose addition or addition of its ...
- 13k
7 votes
2 answers
436 views
At what ordinal $\chi$ does $\mathrm{L}_\chi$ contain a surjection from $\omega$ to $\mathrm{L}_{\beta_0}$?
Let $\mathrm{ZF^-}$ be $\mathrm{ZF}$ minus power set, and let $\beta_0$ be the ordinal for ramified analysis so that $\mathrm{L}_{\beta_0}$ is the least $\mathrm{L}$-model of $\mathrm{ZF}^-$. Clearly, ...
- 4,122
9 votes
1 answer
387 views
Complexity of definable global choice functions
It is well-known that $L$ has a $\Sigma_{1}$-definable global choice function; it is also known that there are other transitive class models of ZFC with this property. I wonder about the complexity ...
- 93
1 vote
1 answer
159 views
Constructible cardinality downslides and their consistency strengths?
Posting "Large cardinals and constructible universe" mentions that $\omega_1^L < \omega_1$ if we assume Ramsey cardinal. My question can we have more downslides like for example $\omega_2^...
- 13k
15 votes
2 answers
895 views
Ordinal realizability vs the constructible universe
Koepke's paper Turing Computations on Ordinals defines a notion of "ordinal computability" using Turing machines with a tape the length of Ord and that can run for Ord-many steps, and shows ...
- 68.7k
4 votes
1 answer
220 views
Impact of coining $L$ in $\mathcal L_{\omega_1, \omega}$ on which large cardinals it can satisfy?
What happens to the constructible universe $L$ if we build it in the first of infinitary languages $\mathcal L_{\omega_1, \omega}$? Would the usual limitation of $L$ not satisfying existence of a ...
- 13k
13 votes
1 answer
2k views
Are some interesting mathematical statements minimal?
Gödel's set $\mathrm{L}$, of constructible sets, decides many interesting mathematical statements, as the Continuum hypothesis and the Axiom of Choice. Are some interesting mathematical questions, ...
- 4,122
1 vote
0 answers
209 views
How does the cardinality of a set and its powerset compare in the hereditarily rank-concordant constructible world?
Working in the constructible universe "$L$", if we define two kinds of ranks for any constructible set $x$, one being the ordinal index of the first $L_\alpha$ where $x$ appears as a subset ...
- 13k
5 votes
2 answers
499 views
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 ...
- 2,308
2 votes
1 answer
361 views
Inner model for KP and a Well-Ordering of the Reals
It is well known that Gödel proved the following theorem: $\mathsf{ZFC + V=L}$ has a $\mathit{\Delta}^1_2$-good well-ordering of $\mathbb{R}$. (Gödel, Addison) So: Is there an inner model for KP/Z/....
- 1,391
1 vote
0 answers
182 views
Can the Constructible Universe be built in absence of Unions and Power?
Can $L$ be built in $\sf ZF$ $\sf-Regularity-Union-Power+ Boolean \ Union$? We know that $L$ can be built in $\sf KP$, but here we don't have Set Union. If the answer is to the negative, then would ...
- 13k
1 vote
0 answers
180 views
Must models of the following theory satisfying opposing infinitary sentences, satisfy opposing finitary sentences?
This is a follow-up to posting titled "Is this theory finitary first order complete?" Recall the theory presented at that posting. Replace the size axiom by the following: $\textbf{...
- 13k
2 votes
1 answer
227 views
Is this theory finitary first order complete?
If we coin a theory in $\mathcal L_{\omega_1, \omega}$ that begins with constructing pure true well founded finite sets, then the set of all true well founded hereditarily finite sets, then builds up ...
- 13k
14 votes
1 answer
823 views
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: $...
- 13k
6 votes
1 answer
659 views
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 ...
- 1,652
2 votes
1 answer
341 views
Is stable ordinals in non-well-founded model the same as well founded models?
Let $BST$ be the axiom system $KP$ - $\Delta_0$ collection. For an ordinal $\alpha$, we say that $\alpha$ is $\varphi$-$\Sigma_n$-stable, if there is a $\beta>\alpha$ satisfies the formula $φ$ such ...
- 1,652
2 votes
0 answers
189 views
Higher-order oracle computation of reals and axiom of constructibility
Certain real numbers can be approximated arbitrarily well by computable functions. If we introduce halting oracles, then more real numbers can be "computed", like Chaitin's constant or the ...
- 481
3 votes
1 answer
617 views
Are all constructible from below sets parameter free definable?
Lets take the intersection of the theory of $L_{\omega_1^{CK}}$ and $\sf ZF + [V=L]$, this is equivalent to the theory of constructability from below + limit stages. Can this theory prove the ...
- 13k
5 votes
0 answers
209 views
Some questions on a paper of Gerald Sacks
I've been reading Sacks' Countable admissible ordinals and hyperdegrees as I'm interested in Theorem 5.3 of the paper: Let $M$ be a countable standard model of $\mathsf{ZF}$ and $V=L$. Suppose $\...
- 2,308
7 votes
1 answer
405 views
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 $\alpha_0$ ...
- 1,652
2 votes
1 answer
286 views
End elementary extension in infinitary logic of some $L_\alpha$ producing a $L_\beta$
Let $L_\alpha$ be some admissible level of the constructible hierarchy and $M \supseteq L_\alpha$ an extension of $L_\alpha$. I am looking for conditions under which $M \simeq L_\beta$. It is not ...
- 531
5 votes
1 answer
335 views
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 ...
- 2,308
5 votes
2 answers
445 views
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 \...
- 531
8 votes
1 answer
379 views
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 ...
- 2,308
1 vote
0 answers
301 views
Is Jensen's covering lemma meaningful in a platonist's view?
The typical applications of fine structure theory are finding out the lower bounds of consistency strength of axiom systems. In such a proccess, we also constructs many combinatorial objects in core ...
- 1,652
1 vote
1 answer
345 views
Recursively inaccessible ordinals and non locally countable ordinals
This answer seems to imply that: for an ordinal $\alpha$, to be recursively inaccessible (i.e. $\alpha$ is admissible and limit of admissible) implies to be not locally countable (i.e. $L_\alpha \...
- 531
8 votes
3 answers
637 views
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_\...
- 531
4 votes
1 answer
584 views
What's the order type of the following set?
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 $...
- 1,652
4 votes
0 answers
209 views
Slicing large countable ordinal properties, from $\Pi_3$-reflection to $\Sigma_2$-admissibility
Edit 2024: This post was based on an incorrect premise, as can be seen by my conversation with Farmer S in the comments. However the mistake I made and the conversation in comments may be instructive (...
- 2,848
7 votes
1 answer
133 views
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 $...
- 1,219
3 votes
1 answer
201 views
Weak form of $\text{CH}$ in $L(\mathbb{R})$
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{ ...
- 2,308
2 votes
1 answer
158 views
A continuous map relating co-constructible reals
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) = ...
- 2,308
4 votes
0 answers
292 views
Is this recursion theoretic analogue of a criterion of weakly compact cardinal accurate?
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 $\kappa$, ...
- 1,652
3 votes
0 answers
161 views
At which large cardinal property this second order ordinal arithmetic stops?
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 ...
- 13k
1 vote
1 answer
248 views
At which large cardinal, the theory of the minimal transitive model of ZFC starts proving its absence?
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 ...
- 13k
3 votes
1 answer
332 views
Do all limit $\alpha \in \omega_1^L$ satisfy $L_\alpha \models V=HC$?
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}(\...
- 431
2 votes
1 answer
342 views
Why can't $L_\beta$ contain a real coding a well-ordering of order-type $\beta$, when $\beta$ is a gap ordinal?
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 ...
- 431
4 votes
1 answer
297 views
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 ...
- 2,308
3 votes
1 answer
362 views
When does $\Pi_2$-reflection on $X$ fail to imply iterated $\Pi_1$-reflection on $X$?
Let lowercase Greek letters denote ordinals. Recall from Richter and Aczel's "Inductive definitions and reflecting properties of admissible ordinals", for a set of formulae $\Gamma$ and a ...
- 2,848
1 vote
0 answers
209 views
Can Jensen's covering lemma be proven easier in generic extensions of L?
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 ...
- 1,652
7 votes
2 answers
653 views
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_\...
- 534
9 votes
2 answers
509 views
Can local $0^\#$ exists in L?
Assume $0^\#$ exists and there is an inaccessible cardinal. Are there two transitive sets $M,N$ s.t. $M\in N,M\vDash ZF+V=L[0^\#],N\vDash ZF+V=L$?
- 1,652
0 votes
0 answers
207 views
How to define BHO alternatives below admissible ordinals?
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 ...
- 1,652
7 votes
1 answer
469 views
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 $L_{\alpha+1} \vDash\ \alpha\text{ is inaccessible}$, ...
- 1,652
4 votes
1 answer
1k views
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 \...
- 391
5 votes
1 answer
456 views
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?
- 1,652
7 votes
0 answers
236 views
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 $\...
- 1,652