# Tagged Questions

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### Harrington's unpublished note “The constructible reals can be anything”

Around 1974, Leo Harringto wrote an unpublished note entitled "The constructible reals can be anything", in which he proved that it is consistent that being $\Delta^1_n$ is the same as being ...
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### 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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### 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 ...
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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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### 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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### 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 ...
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 $M ... 2answers 422 views ### Constructible models of New Foundations? Hi all! Is there anything like GĂ¶del's constructible universe for New Foundations? More precisely, I would like a process for taking a model$M$of NF, and using it to build a model$L \subseteq M$... 1answer 238 views ### Acceptability and Soundness of J-structures. I would like an example of a J-structure$(J^A,B)$which is not acceptable and one that is not 1-sound. Edit:Let us recall that a structure$J^A_\alpha$is acceptable if for every limit ordinal$ ...
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 ...