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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1answer
172 views

On a particular proof of “if the sharp of every real exists and every club contains a club constructible from a real, then $\delta^1_2 = \omega_2$”

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 ...
8
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1answer
248 views

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 ...
14
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2answers
335 views

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 ...
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142 views

End-extension in Gödel's constructible universe

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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2answers
340 views

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: $\forall \kappa \exists x \forall \alpha ( \alpha \leq \kappa \to \alpha \in x)$ where "$...
4
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1answer
419 views

Is ZFC interpretable in a kind of an extended form of second order arithmetic?

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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1answer
211 views

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 ...
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2answers
293 views

Question about Jech's proof of V = L implies GCH

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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0answers
131 views

Can this reflective class theory interpret ZFC?

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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141 views

Is “ZF+ V=L” an upper limit theory for cardinal decidability (per its strength)?

{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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1answer
158 views

What is the strength of this strict constructible iterative hierarchy?

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 ...
5
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1answer
311 views

$\Delta^{1}_{2}$ 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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1answer
245 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 ...
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1answer
124 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
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1answer
408 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 ...
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1k 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 $...
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1answer
192 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 ...
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1answer
290 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 = \...
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2answers
746 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 ...
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304 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
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1answer
327 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 ...
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1answer
290 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
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1answer
457 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, ...
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2answers
447 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 ...
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1answer
338 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 ...
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1answer
336 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 $\mathrm L_{\infty}$?...
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135 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 ...
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1answer
119 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 $L_\mu$}\}...
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1answer
238 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 $$E_\kappa=\{\mu<\kappa: \mbox{there is an elementary substructure of $L_\kappa$ isomorphic to $L_\mu$}\},$$ ...
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1answer
463 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 ...
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2answers
589 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 ...
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561 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
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1answer
342 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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319 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
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1answer
791 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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2answers
704 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 $...
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0answers
120 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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404 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 ...
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2answers
373 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
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1answer
405 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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1answer
2k 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 ...
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3answers
472 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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1answer
656 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 ...
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1answer
389 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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737 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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2answers
591 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
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1answer
370 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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1answer
285 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
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1answer
284 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 ...