Monotone induction and strong nesting — Problem 10.1

10. Philip Welch

First we recall some basic notions in [1]. Let $\beta$ be an ordinal. Suppose $M$ is a nonstandard model of $\mathsf{KP}$, and $\{\zeta_{i}:i\in\mathbb{N}\}$, $\{s_{i}:i\in\mathbb{N}\}$ are sequences of $M$-ordinals. If $\left(M,\{\zeta_{i}:i\in\mathbb{N}\},\{s_{i}:i\in\mathbb{N}\}\right)$ satisfies the following:

$M$ extends $L_{\beta}$;

for all $i$, $\zeta_{i}<\zeta_{i+1}<\beta$ and $s_{i+1}<^{M}s_{i}$;

for all $i$, $L_{\zeta_{i}}\prec_{\Sigma_{m+1}}L_{s_{i}}^{M}$;

for all $i$, $L_{s_{i+1}}^{M}\prec_{\Sigma_{m-1}}L_{s_{i}}^{M}$,

then we call this triple a strong $\Sigma_{m+1}$-nesting on $\beta$. And we say $\beta$ supports a strong $\Sigma_{m+1}$-nesting if there exist such nonstandard model $M$ of $\mathsf{KP}$ and sequences $\{\zeta_{i}:i\in\mathbb{N}\},\{s_{i}:i\in\mathbb{N}\}$ of $M$-ordinals for $L_{\beta}$.

We call a function $\Phi:\mathbb{R}\to\mathbb{R}$ an operator ($\mathcal{P}(\mathbb{N})$ and $\mathbb{R}$ are identified). An operator $\Phi$ is said to be

$\boldsymbol{\Pi}_{m}^{1}$ if there exists a $\boldsymbol{\Pi}_{m}^{1}$ formula $\phi$ and a parameter $Y$ such that

monotone if $X\subseteq Y$ implies $\Phi(X)\subseteq\Phi(Y)$.

The axiom $\boldsymbol{\Pi}_{m+1}^{1}\mathsf{-MI}$ asserts that for every monotone $\boldsymbol{\Pi}_{m+1}^{1}$ operator $\Phi$, there is some prewellordering $W$ such that for all $a\in\textrm{field}(W)=\textrm{dom}(W)\cup\textrm{ran}(W)$, we have

Here

Note that an operator $\Phi$ can be iterated by setting $\Phi^{0}=\emptyset,\Phi^{<\alpha}=\bigcup_{\gamma<\alpha}\Phi^{\gamma}$ and $\Phi^{\alpha}=\Phi(\Phi^{<\alpha})$. Put $\Phi^{\infty}=\bigcup_{\alpha}\Phi^{\alpha}$. We can see that the field of $W$ is $\Phi^{\infty}$.

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