# 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 integers projectum-representable iff there is an ordinal $$\alpha$$ such that $$\rho(1,\alpha)=\rho(2,\alpha)=\dots=\rho(x_1,\alpha)\gt$$ $$\rho(x_1+1,\alpha)=\dots=\rho(x_2,\alpha)\gt$$ $$\rho(x_2+1,\alpha)=\dots=\rho(x_3,\alpha)\gt$$ $$\dots=\rho(x_m,\alpha),$$ i.e. if the sequence of projecta of $$J_\alpha$$ consists of a sequence of $$x_1$$ identical terms, then drops, then has $$x_2$$ identical terms, then drops again etc.

Which finite sequences are projectum-representable?

By condensation arguments, it can be seen that every representable sequence is already representable with a countable ordinal $$\alpha$$.

References would also be welcome.

The short answer is that all finite sequences are representable.

Suppose you want the general constellation you gave. We work inside $$L$$. For $$X=\langle X,\in \rangle$$ a model of V=L (not necessarily transitive) define $$S^X_n(Y)$$ to be the $$\Sigma_n$$-Skolem Hull in $$X$$ of the set $$Y$$. Let $$\tau$$ be the least ordinal $$> \omega_{m-1}$$ with $$L_\tau$$ a model of $$\Sigma_{x_1}$$-Separation scheme. (This ensures that $$\rho(x_1,\tau)=\tau$$.) Set $$H_0= L_\tau$$. Now define by recursion on $$j$$ for $$1\leq j\leq m-1$$: $$H_j = S^{H_{j-1}}_{x_j +1}(\omega_{m-1-j}).$$

Let $$L_\alpha \simeq H_m$$.

This is admittedly easier to see with an example: say $$(x_1,x_2,x_3) = (2,4,6)$$ with $$m=3$$ "drops". Let $$\tau$$ be the least above $$\omega_{m-1} =\omega_2$$ with $$L_\tau$$ a model of $$\Sigma_2$$-Separation. Then $$\tau = \rho(\tau,2)> \omega_2 =\rho(\tau,3) \ldots$$. Then $$H_1 = S^{L_\tau}_5(\omega_1).$$ (If we were to take the Mostowski-Shepherdson collapse of this hull it would be an $$L_{\alpha'}$$ with $$\rho(\alpha',k)=\omega_1$$ for $$k\geq 5$$. One should check, e.g., that $$\rho(\alpha',4)\neq \omega_1$$: but if this failed then there would be a $$\Sigma_4(L_{\alpha'})$$ map of $$\omega_1$$ cofinally into $$\alpha'$$; but this is a $$\Pi_5$$ statement, and would then be true in $$L_\tau$$ which is absurd.) Then take $$H_2 =S^{H_1}_7(\omega).$$

Let $$L_\alpha \simeq H_2$$. Similar checks reveal that $$\rho(\alpha,6)=\omega_1> \rho(\alpha,7) \ldots$$ etc. and we have the correct pattern.

I do not know of any references for this.

• Thanks for this thorough answer! The construction is really elegant. Nov 19, 2011 at 15:16
• It might be known before but "Patterns of Projecta" By Adam Krawczyk could be one of the references Aug 25, 2017 at 10:43