# Ordering patterns of projecta by least witness

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 $$x implies $$\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.

• 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.
– C7X
Jul 23, 2022 at 23:30

## 1 Answer

$$<_\rho$$ 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 $$s(i). This easily yields the ordertype $$\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 $$s(i). So $$\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 $$n<\omega$$.