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Let $J$ denote Jensen's modification of the constructible hierarchy. For an ordinal $\alpha$ and an $n\in\mathbb N^+$, let $\rho_n^{J_\alpha}$ denote the $\Sigma_n$-projectum of $J_\alpha$, the least $\delta\leq\alpha$ such that there is a $\Sigma_n(J_\alpha)$-definable subset of $\omega\rho$ not in $J_\alpha$. (Note that $x<y$ implies $\rho_x^{J_\alpha}\geq\rho_y^{J_\alpha}$.) For a finite increasing sequence of positive naturals $s=(s_1,\ldots,s_m)$, call $s$ a pattern of projecta if there is some $\alpha$ where the points of decrease in the sequence $(\rho_i^{J_\alpha})_{1\leq i\leq m}$ are just the integers in $s$, i.e. for any $i\in\mathbb N^+$, $\rho_i^{J_\alpha}>\rho_{i+1}^{J_\alpha}$ iff $i$ is an entry of $s$. In this way a pattern of projecta encodes when the sequence of $\Sigma_i$-projecta will "drop" (if $i$ is in the sequence), or will "stay" (if $i$ is not).

What ordering results if we order the patterns of projecta by the size of the least $\alpha$ where $J_\alpha$ has that pattern? We will call this ordering $<_\rho$: set $s<_\rho t$ if the least $\alpha$ where $J_\alpha$ has pattern of projecta $s$ is less than the least $\beta$ where $J_\beta$ has pattern of projecta $t$.

What are some of the basic properties of $<_\rho$, e.g. its order type? If $<_\rho$ is a well-ordering, what ordinal is it isomorphic to?

What I do know is that all finite sequences of naturals are patterns of projecta, a relevant question is MO #67933, and a relevant paper is "Patterns of Projecta" by A. Krawczyk (1981). In the MO question's accepted answer, Philip Welch gives an explicit construction, for arbitrary sequence $s$, of a $J_\alpha$ whose pattern of projecta is $s$. However, for this construction to be used in computing $<_\rho$, we need information about the minimal $\alpha$ in particular. Sufficient for this would be if at each step of the Skolem hulling for $1\leq j\leq m-1$, $\pi(H_m)$ were the minimal $J$-rank with the necessary segment of the pattern so far.

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  • $\begingroup$ I am not sure if I should add the [constructibility] tag or not, since this asks more about the poset $(\{s\in\mathbb N^+\mid s\textrm{ increasing}\},<_\rho)$ than the projecta themselves. $\endgroup$
    – C7X
    Jul 23, 2022 at 23:30

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$<_\rho$ is a wellorder essentially by definition. The ordertype is $\omega^{\omega}$ (ordinal exponentiation of course).

In fact $s<_\rho t$ iff either $\mathrm{lh}(s)<\mathrm{lh}(t)$, or $\mathrm{lh}(s)=\mathrm{lh}(t)$ and $s<_{\mathrm{lex}}t$, i.e. letting $i$ be least such that $s(i)\neq t(i)$, we have $s(i)<t(i)$. This easily yields the ordertype $\omega^{\omega}$.

This is just by a slight variant of Philip Welch's construction. Suppose first $\mathrm{lh}(s)=\mathrm{lh}(t)$. Let $\beta$ be an ordinal which instantiates the pattern of $t$. Let $i$ be least such that $s(i)<t(i)$. So $\beta$ also instantiates the pattern of $s\upharpoonright i$. But $$\rho_{t(k)+1}^{L_\beta}<\rho_{t(k-1)+1}^{L_\beta}<\ldots<\rho_{t(i)+1}^{L_\beta}<\rho_{t(i)}^{L_\beta}=\rho_{s(i)+1}^{L_\beta}=\rho_{s(i)}^{L(\beta)},$$ and the $\rho_{t(k)+1}^{L_\beta},\ldots,\rho_{t(i)+1}^{L_\beta}$ are each cardinals in $L_\beta$. Let $\kappa_j=\rho_{t(j)+1}^{L_\beta}$ for $j\in[i,k]$. Now form the hull $$\mathrm{Hull}_{\Sigma_{s(i)+1}}^{L_\beta}(\kappa_i\cup\{x\}),$$ where $x$ is an appropriate finite set that this is $\Sigma_{s(i)+1}$-elementary and reflects the relevant information (including the relevant standard parameters etc). Let $\beta_i$ be such that $L_{\beta_i}$ is the transitive collapse of the hull. Note that $\beta_i<\beta$, and since $\kappa_i$ is a cardinal in $L_\beta$, therefore so $$\rho_\omega^{L_{\beta_i}}=\kappa_i=\rho_{s(i)+1}^{L_{\beta_i}}<\rho_{s(i)}^{L_{\beta_i}}$$ and $L_{\beta_i}$ instantiates $s\upharpoonright(i+1)$. Since $\kappa_k<\kappa_{k-1}<\ldots<\kappa_i$ and these are cardinals of this model, we can continue forming hulls in this manner at elementarities corresponding to the entries in $s$, and since $\mathrm{lh}(s)=\mathrm{lh}(t)$, there are enough cardinals to do that. This results in an ordinal $\beta_k$ instantiating $s$, and $\beta_k<\beta$, as desired.

Now suppose $\mathrm{lh}(s)<\mathrm{lh}(t)$ and let $\beta$ instantiate $t$. Then we get $\mathrm{lh}(t)$-many $L_\beta$-cardinals $\kappa_k<\ldots<\kappa_0<\beta$ as the resulting projecta. Now $\rho_\omega^{L_{\kappa_0}}=\kappa_0$, so we can proceed like before, taking a $\Sigma_{s(0)+1}$-elementary hull of $L_{\kappa_0}$ in parameters in $\kappa_1\cup\{x\}$ for an appropriate $x$, etc. This shows that $s<_\rho t$.

Note also (by calculation as above) that $\sup_{s}\alpha_s$, where $\alpha_s$ is the least instantiation of $s$, is just the stack of the minimal models of $n$th order arithmetic, for $n<\omega$.

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