All Questions
62 questions
5
votes
1
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345
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A question on the size of an admissible ordinal
Let $\mathbf{L}_{\varsigma}$ be the level of ordinal $\varsigma$ of Gödel's constructible universe $\mathbf{L}$. Let $\Sigma_{3}$-KP be Kripke-Platek set theory with infinity and $\Sigma_{3}$-$...
- 3,574
9
votes
0
answers
358
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 ...
- 39.8k
9
votes
1
answer
862
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Harrington's unpublished note "The constructible reals can be anything"
Around 1974, Leo Harringto wrote an unpublished note entitled "The constructible reals can be anything", in which he proved that it is consistent that being $\Delta^1_n$ is the same as being ...
- 32.1k
14
votes
3
answers
777
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 ...
- 20.5k
11
votes
2
answers
437
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 $\...
- 32.1k
23
votes
1
answer
3k
views
Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)
Gödel's constructible universe ($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 ...
user46667
13
votes
1
answer
971
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 ...
user43940
2
votes
1
answer
665
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 ...
user36136
14
votes
1
answer
966
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 \...
- 1,146
10
votes
2
answers
737
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 $L \subseteq M$ ...
- 413
6
votes
1
answer
315
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&...
- 163
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 ...
- 521