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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3 votes
1 answer
494 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 ...
Zuhair Al-Johar's user avatar
4 votes
1 answer
563 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 ...
Zuhair Al-Johar's user avatar
19 votes
2 answers
2k 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 $...
Beau Madison Mount's user avatar
14 votes
3 answers
764 views

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 ...
Noah Schweber's user avatar
10 votes
1 answer
846 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Harrington's unpublished note "The constructible reals can be anything"

Around 1974, Leo Harringto wrote an unpublished note entitled "The constructible reals can be anything", in which he proved that it is consistent that being $\Delta^1_n$ is the same as being ...
Mohammad Golshani's user avatar
7 votes
1 answer
280 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 ...
M Carl's user avatar
  • 437
6 votes
0 answers
427 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 ...
Trevor Wilson's user avatar
6 votes
1 answer
254 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 $$E_\kappa=\{\mu<\kappa: \mbox{there is an elementary substructure of $L_\kappa$ isomorphic to $L_\mu$}\},$$ ...
Noah Schweber's user avatar
6 votes
1 answer
501 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 ...
Reflecting_Ordinal's user avatar
5 votes
1 answer
362 views

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 ...
M Carl's user avatar
  • 521
5 votes
3 answers
772 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: $\forall \kappa\, \exists x \, \forall \alpha \, ( \alpha \leq \kappa \to \alpha \in x)$ ...
Zuhair Al-Johar's user avatar
5 votes
1 answer
470 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 $...
Jesse Elliott's user avatar
3 votes
0 answers
120 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 ...
Zuhair Al-Johar's user avatar
3 votes
1 answer
140 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 $L_\mu$}\}...
Noah Schweber's user avatar
2 votes
1 answer
146 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 ...
Zuhair Al-Johar's user avatar
1 vote
1 answer
198 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 ...
Zuhair Al-Johar's user avatar