**10**

votes

**0**answers

222 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**

votes

**1**answer

604 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### Harrington's unpublished note “The constructible reals can be anything”

Around 1974, Leo Harringto wrote an unpublished note entitled "The constructible reals can be anything", in which he proved that it is consistent that being $\Delta^1_n$ is the same as being ...

**9**

votes

**2**answers

353 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 ...

**2**

votes

**0**answers

92 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 ...

**8**

votes

**2**answers

313 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 ...

**12**

votes

**2**answers

314 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 ...

**1**

vote

**1**answer

194 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 ...

**15**

votes

**1**answer

900 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 ...

**5**

votes

**3**answers

233 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 ...

**10**

votes

**1**answer

359 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 ...

**2**

votes

**1**answer

211 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 ...

**11**

votes

**0**answers

636 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 ...

**10**

votes

**2**answers

426 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**

votes

**1**answer

275 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 ...

**7**

votes

**1**answer

238 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 $ ...

**4**

votes

**1**answer

238 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 ...