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It is well known that Gödel proved the following theorem:

  • $\mathsf{ZFC + V=L}$ has a $\mathit{\Delta}^1_2$-good well-ordering of $\mathbb{R}$. (Gödel, Addison)

So:

  • Is there an inner model for KP/Z/... has a $\mathit{\Delta}^1_1$-good well-ordering of $\mathbb{R}$?
  • Is it possible to reach lower hierarchy?
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    $\begingroup$ What does "good" mean here? $\endgroup$
    – Noah Schweber
    Commented Oct 6, 2023 at 16:21
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    $\begingroup$ The definition of goodness doesn't appear there either ... $\endgroup$
    – Noah Schweber
    Commented Oct 6, 2023 at 16:49
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    $\begingroup$ The word "good" doesn't appear anywhere in that linked post! Do you just mean "lightface"? I'm familiar with lightface/boldface, but I really don't know what you mean by "good" here. You don't have to give a complete definition, but a link to a place where some definition is given would be great (neither of the links provided does this). $\endgroup$
    – Noah Schweber
    Commented Oct 6, 2023 at 17:10
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    $\begingroup$ @NoahSchweber I believe "good $\Delta^1_2$ well ordering" means that not only is the ordering relation $\Delta^1_2$ but there is a $\Delta^1_2$ function mapping each real to a code for its (countable) set of predecessors in the ordering. I think this is in Moschovakis's "Descriptive Set Theory". Goodness is used to produce $\Delta^1_2$ things whose production in ZFC would involve choice and thus would be undefinable (e.g., ultrafilters on $\omega$, Hamel bases for $\mathbb R$, Vitali sets). $\endgroup$
    – Andreas Blass
    Commented Oct 6, 2023 at 21:10
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    $\begingroup$ @AndreasBlass Thanks! $\endgroup$
    – Noah Schweber
    Commented Oct 6, 2023 at 23:30

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Re the first question with KP, yes, $L_{\omega_1^{\mathrm{ck}}}$. (Use the order of constructibility and Ville's Lemma.)

The second question is a bit vague. But for each ordinal $\alpha<\omega_1^{\mathrm{ck}}$, the constructibility order on the reals in $J_\alpha$ is a $\Delta_1^{J_\alpha}$-good wellorder, and also a $(\Delta_1^1)^{J_\alpha}$-good wellorder, and one can convert $(\Delta_1^1)^{J_\alpha}$ into a natural bound on complexity strictly within $\Delta^1_1$.

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    $\begingroup$ What is Ville's lemma and why does it produce a $\Delta^1_1$ well-ordering of the reals in $L_{\omega_1^{ck}}$? $\endgroup$
    – ikrto
    Commented Oct 31, 2023 at 4:09
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    $\begingroup$ Something along the lines of: if $M$ is a model of enough (but a small fragment of) set theory which has wellfounded $\omega$, then its wellfounded part models KP, and hence $L_{\omega_1^{\mathrm{ck}}}\subseteq M$. Therefore there is no such $M$ in $L_{\omega_1^{\mathrm{ck}}}$. So working in $L_{\omega_1^{\mathrm{ck}}}$, the set of codes $x\in\mathbb{R}$ for levels $L_\alpha$ of $L$ (necessarily with $\alpha<\omega_1^{\mathrm{ck}}$) is $\Delta^1_1$ (i.e. they are the same as the codes for models $M$ with wellfounded $\omega$ which satisfy "$V=L$"). $\endgroup$
    – Farmer S
    Commented Nov 1, 2023 at 10:33
  • $\begingroup$ Thank you! Just for the benefit of my own understanding: in $L_{\omega_1^{ck}}$, a well-ordering of the reals can be defined as "$x\prec y$ iff there is a code of an $\omega$-model of $V=L$ that thinks $x<_L y$" (and $x\not\prec y$ iff $y\prec x$). Ville's lemma is used to ensure that, every $\omega$-model of $V=L$ in $L_{\omega_1^{ck}}$ is actually well-founded, because otherwise their standard part would contain $L_{\omega_1^{ck}}$. Is that the reasoning? $\endgroup$
    – ikrto
    Commented Nov 1, 2023 at 18:37
  • $\begingroup$ What I'm slightly unsure about is that the lemma by Ville typically is stated as "if $M$ is an $\omega$-model of $KP$, then so is its standard part." But there are no $\omega$-models of $KP$ in $L_{\omega_1^{ck}}$. Does this mean that, in the assumption of Ville's lemma, $M$ can be assumed to satisfy something weaker than $KP$? $\endgroup$
    – ikrto
    Commented Nov 1, 2023 at 18:41
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    $\begingroup$ Right, we can instead require that $M$ models "$V=L$" + some very basic set theory + $\Sigma_0$-separation + "for every $x$ and every $\Sigma_1$ formula $\varphi$, if $\varphi(x)$ is true then there is a least ordinal $\alpha$ such that $L_\alpha$ models $\varphi(x)$". Any model $M$ of this theory with wellfounded $\omega$ has wellfounded part isomorphic to $L_\beta$ for some ordinal $\beta$ such that $L_\beta$ models KP. $\endgroup$
    – Farmer S
    Commented Nov 2, 2023 at 1:48

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