The Order of Reflection

TL;DR

This paper characterizes the linear order of reflection patterns involving $oldsymbol{orall ext{-}}oldsymbol{ ext{and}}oldsymbol{ ext{-}}oldsymbol{ ext{exists}}$-reflection in descriptive set theory, establishing it as a well-ordered structure of length $oldsymbol{oldsymbol{ ext{omega}}^oldsymbol{ ext{omega}}}$.

Contribution

It extends classical results by determining the order between all patterns of iterated $oldsymbol{ ext{Sigma}}^1_1$ and $oldsymbol{ ext{Pi}}^1_1$ reflection, revealing a complex prewellordering structure.

Findings

The linear reflection order is a prewellordering of length ω^ω.

Established relationships between linear and non-linear reflection patterns.

Identified the structure of reflection patterns involving simultaneous $oldsymbol{ ext{Sigma}}^1_1$ and $oldsymbol{ ext{Pi}}^1_1$ reflection.

Abstract

Extending Aanderaa's classical result that $\pi^1_1<\sigma^1_1$, we determine the order between any two patterns of iterated $\Sigma^1_1$- and $\Pi^1_1$-reflection. We show that this \emph{linear reflection order} is a prewellordering of length $\omega^\omega$. This requires considering the relationship between linear and some \emph{non-linear} reflection patterns, such as $\sigma^1_1\wedge\pi^1_1$, the pattern of simultaneous $\Sigma^1_1$- and $\Pi^1_1$-reflection.

Equations

$$L_{\alpha}\models\phi(a_{1},\ldots,a_{n})\text{ implies }\exists\beta<\alpha\,\Big{(}a_{1},\ldots,a_{n}\in L_{\beta}\wedge L_{\beta}\models\phi(a_{1},\ldots,a_{n})\Big{)}.$$

$$L_{\alpha }{\prec }_1{L}_{\beta }.$$

$${\sigma }_1^1({\sigma }_1^1)<{\sigma }_1^1({\pi }_1^1({\sigma }_1^1\wedge {\pi }_1^1))$$

$${\pi }_1^1({\sigma }_1^1)<{\pi }_1^1({\sigma }_1^1)\wedge {\pi }_1^1({\pi }_1^1).$$

$$\delta_{\alpha}=\sup\big{\{}\delta:\delta\text{ is the length of an $\alpha$-recursive wellordering of a subset of $\alpha$}\big{\}},$$

$$L_{\sigma }{\prec }_1{L}_{\gamma }.$$

$$xRy\leftrightarrow L_{\sigma }\models \psi (x,y).$$

$$x{R}_{{\sigma }^{\prime }}y\leftrightarrow L_{{\sigma }^{\prime }}\models \psi (x,y).$$

$$M\models L_{\sigma }{\prec }_1{L}_{{\delta }_{\sigma }}.$$

$$M\models L_{\sigma }{\prec }_1{L}_{{\delta }_{\sigma }},$$

$$M\models ‘‘L_{{\delta }_{\sigma }+1}\models \varphi (a).\text{”}$$

$$M\models ‘‘L_{{\delta }_{\tau }+1}\models \varphi (a).\text{”}$$

$${\pi }_1^1({\sigma }_1^1)<{\sigma }_1^1({\sigma }_1^1).$$

$$\text{if }\sigma \text{ is s-reflecting,}\text{ then it is t-reflecting},$$

$${\sigma }_1^1({\pi }_1^1({\sigma }_1^1))<{\sigma }_1^1({\sigma }_1^1).$$

$${\sigma }_1^1\wedge {\pi }_1^1={\sigma }_1^1\wedge {\pi }_1^1({\sigma }_1^1)={\sigma }_1^1({\pi }_1^1({\sigma }_1^1))\wedge {\pi }_1^1={\sigma }_1^1\wedge {\pi }_1^1({\sigma }_1^1({\pi }_1^1({\sigma }_1^1)))$$

$${\sigma }_1^1({\pi }_1^1({\sigma }_1^1))<{\sigma }_1^1\wedge {\pi }_1^1.$$

$${\sigma }_1^1({\sigma }_1^1)={\sigma }_1^1({\sigma }_1^1\wedge {\pi }_1^1),$$

$${\sigma }_1^1({\pi }_1^1({\sigma }_1^1))<{\sigma }_1^1\wedge {\pi }_1^1<{\sigma }_1^1({\sigma }_1^1),$$

$${\sigma }_1^1={\sigma }_1^1({\pi }_1^1)={\sigma }_1^1({\pi }_1^1({\pi }_1^1))={\sigma }_1^1({\pi }_1^1({\pi }_1^1({\pi }_1^1)))=\dots .$$

$${\pi }_1^1<{\pi }_1^1({\pi }_1^1)<{\pi }_1^1({\pi }_1^1({\pi }_1^1))<\cdots <{\sigma }_1^1,$$

$$M\models$$

$$\gamma <{\delta }_{\sigma }^M.$$

$$M\models$$

$$L_{{\tau }^{+}}\models {\varphi }^{\ast }(a,L_{\tau }),$$

$${\pi }_1^1({\sigma }_1^1)<{\sigma }_1^1({\pi }_1^1({\sigma }_1^1))<{\sigma }_1^1\wedge {\pi }_1^1.$$

$${\pi }_1^1({\sigma }_1^1)={\pi }_1^1({\sigma }_1^1)\wedge {\pi }_1^1({\pi }_1^1)={\pi }_1^1({\sigma }_1^1)\wedge {\pi }_1^1({\pi }_1^1({\pi }_1^1))=\dots$$

$$\pi^{1}_{1}\Big{(}\sigma^{1}_{1}\wedge\pi^{1}_{1}(\sigma^{1}_{1}\wedge\pi^{1}_{1})\Big{)}<\sigma^{1}_{1}(\sigma^{1}_{1}).$$

Full text

The proofs involve linking the lengths of $\alpha$-recursive wellorderings to various forms of stability and reflection properties satisfied by ordinals $\alpha$ within standard and non-standard models of set theory.

1. 1 Introduction
2. 2 Stability and Gandy Ordinals
3. 3 Reflection Transfer Theorems
4. 4 Patterns Below $\sigma^{1}_{1}(\sigma^{1}_{1})$
5. 5 Linear Patterns
6. 6 Concluding Remarks

1. Introduction

Let $L_{\alpha}$ denote the $\alpha$th level of Gödel’s constructible hierarchy, given by $L_{0}=\varnothing$, $L_{\alpha+1}=$ all sets definable over $L_{\alpha}$ with parameters, and $L_{\eta}=\bigcup_{\alpha<\eta}L_{\alpha}$ at limit stages. In $\alpha$-recursion theory, one lifts the usual notion of “computation” over the natural numbers (or, equivalently, over $L_{\omega}$) to $L_{\alpha}$, for sufficiently closed $\alpha$. As became evident from early work by Kreisel, Kripke, Platek, Sacks, Takeuti, and others (see e.g., Simpson [11]), facts about recursion on $L_{\alpha}$ can be translated into facts about recursion on $L_{\omega}$ in various ways. In particular, the termination of simple inductive definitions of sets of natural numbers is deeply connected with the reflecting structure of $L$ (see e.g., Cenzer [7] or Aczel and Richter [3]). The purpose of this article is to study the order in which various reflecting properties given in terms of iterated $\Sigma^{1}_{1}$- and $\Pi^{1}_{1}$-reflection first occur in the constructible hierarchy.

A formula in the language of set theory is $\Sigma^{1}_{1}$ if it contains only existential second-order quantifiers (i.e., ranging over classes) followed by arbitrary first-order quantifiers. An ordinal $\alpha$ is said to be $\Sigma^{1}_{1}$-reflecting if whenever $\phi$ is a $\Sigma^{1}_{1}$ formula in the language of set theory and $a_{1},\ldots,a_{n}$ are finitely many elements of $L_{\alpha}$, then

[TABLE]

Given a class of ordinals $X$, an ordinal $\alpha$ is said to be $\Sigma^{1}_{1}$-reflecting on $X$ if one can additionally demand that the ordinal $\beta$ above belong to $X$. The least $\Sigma^{1}_{1}$-reflecting ordinal is denoted by $\sigma^{1}_{1}$, and $\pi^{1}_{1}$ is defined dually.

An ordinal $\alpha$ is said to be $\beta$-stable if $L_{\alpha}$ is a $\Sigma_{1}$-elementary substructure of $L_{\beta}$; in symbols:

[TABLE]

Given an ordinal $\alpha$, write $\alpha^{+}$ for the smallest admissible ordinal greater than $\alpha$. Aczel and Richter [3] showed that $\pi^{1}_{1}\neq\sigma^{1}_{1}$ and that a countable ordinal $\alpha$ is $\Pi^{1}_{1}$-reflecting if, and only if, it is $\alpha^{+}$-stable. Afterwards, Aanderaa [1] showed that $\pi^{1}_{1}<\sigma^{1}_{1}$. Gostanian [8] showed that $\sigma^{1}_{1}$ is smaller than the least $\alpha$ which is $(\alpha^{+}+1)$-reflecting; in fact, he showed that any $\alpha$ which is both $(\alpha^{+}+1)$-stable and locally countable is also $\Sigma^{1}_{1}$-reflecting. Later Gostanian and Hrbacek [9] employed Gostanian’s method to give a new proof of Aanderaa’s theorem. A third, apparently folklore proof appears in Simpson [11]. Aanderaa’s theorem is also an immediate consequence of Proposition 10 below, although the proof of Proposition 10 has a similar flavor to the argument in Simpson [11].

Let us now generalize the definitions of $\sigma^{1}_{1}$ and $\pi^{1}_{1}$ as follows:

Definition 1**.**

The notion of a reflection pattern is given inductively: the empty set is a reflection pattern; if $s$ and $t$ are reflection patterns, then so too are $s\wedge t$, $\sigma^{1}_{1}(s)$, and $\pi^{1}_{1}(s)$.

We write $\sigma^{1}_{1}$ for $\sigma^{1}_{1}(\varnothing)$ and $\pi^{1}_{1}$ for $\pi^{1}_{1}(\varnothing)$.

Definition 2**.**

A reflection pattern is linear if it contains no conjunctions, and non-linear otherwise.

Definition 3**.**

An ordinal is said to be $\sigma^{1}_{1}(\varnothing)$-reflecting if it is $\Sigma^{1}_{1}$-reflecting; it is said to be $\pi^{1}_{1}(\varnothing)$-reflecting if it is $\Pi^{1}_{1}$-reflecting. Let $s$ and $t$ be reflection patterns. Inductively, an ordinal $\alpha$ is said to be $\sigma^{1}_{1}(s)$-reflecting if it reflects $\Sigma^{1}_{1}$ statements onto $s$-reflecting ordinals; it is said to be $\pi^{1}_{1}(s)$-reflecting if it reflects $\Pi^{1}_{1}$ statements onto $s$-reflecting ordinals; it is said to be $s\wedge t$-reflecting if it is both $s$-reflecting and $t$-reflecting.

We may alternate between uppercase $\Sigma$ and $\Pi$ and lowercase $\sigma$ and $\pi$ in speaking about patterns of reflection.

The ordering problem is: given two reflection patterns $s$ and $t$, determine whether the least $s$-reflecting ordinal is smaller than the least $t$-reflecting ordinal. We will identify a pattern $s$ with the least $s$-reflecting ordinal. Thus, instances of the ordering problem are e.g., determining whether

[TABLE]

or whether

[TABLE]

Other related problems emerge. For instance, one may ask whether $\sigma^{1}_{1}(\sigma^{1}_{1})$ is the least $\Sigma^{1}_{1}$-reflecting ordinal which is also a limit of $\Sigma^{1}_{1}$-reflecting ordinals. (Incidentally, the answer to all three questions is “no.”)

In this article, we solve the ordering problem for linear patterns of reflection: we exhibit a way of assigning ordinals to linear patterns in a way that respects their ordering; in particular, we show:

Theorem 4**.**

The linear order of reflection is a prewellordering of length $\omega^{\omega}$.

The proof requires analyzing the structure of the non-linear, or full, reflection order, to a certain extent. We shall see that all reflection patterns are witnessed for the first time by ordinals between the least $\alpha$ which is $\alpha^{+}$-stable and the least $\alpha$ which is $(\alpha^{+}+1)$-stable. In addition, we show:

Theorem 5**.**

The linear reflection order is cofinal in the full reflection order.

This raises the question of whether the full reflection order also has length $\omega^{\omega}$. This turns out to be false:

Theorem 6**.**

The patterns $\pi^{1}_{1}$, $\sigma^{1}_{1}$, $\sigma^{1}_{1}\wedge\pi^{1}_{1}$, and $\sigma^{1}_{1}(\sigma^{1}_{1})$ have ranks $1$, $\omega$, $\omega^{2}$, and $\omega^{\omega}$ in the reflection order, respectively.

In the course of proving these theorems, we find various easier results which we believe to be of independent interest; these are labelled “propositions.”

Convention

Even if not mentioned explicitly, every ordinal in this article is assumed to be both countable and locally countable (i.e., for all $\beta<\alpha$, there is a surjection from $\omega$ to $\beta$ in $L_{\alpha}$). These are the hypotheses for the theorems of Gostanian and Aczel-Richter mentioned above, respectively.

2. Stability and Gandy Ordinals

For an admissible ordinal $\alpha$, write

[TABLE]

where a subset of $\alpha$ is said to be $\alpha$-recursive if it is $\Delta_{1}$-definable over $L_{\alpha}$ with parameters. The value of $\delta_{\alpha}$ remains unchanged if one replaces “$\alpha$-recursive” by “$\alpha$-r.e.” in the definition. For every admissible $\alpha$, $\delta_{\alpha}$ is easily seen to be a limit and e.g., additively indecomposable. We always have $\delta_{\alpha}\leq\alpha^{+}$; an ordinal $\alpha$ is Gandy if $\delta_{\alpha}=\alpha^{+}$. Gostanian [8] showed that $\sigma^{1}_{1}$ is the smallest ordinal which is not Gandy. In fact, he showed that a locally countable ordinal is not Gandy if, and only if, it is $\Sigma^{1}_{1}$-reflecting. Abramson and Sacks [2] showed that $(\aleph_{\omega}^{L})^{+}$ is Gandy, so not every Gandy ordinal is locally countable.

Since we know what the degree of stability of $\pi^{1}_{1}$ is (viz. $(\pi^{1}_{1})^{+}$), a possible first question is that of the degree of stability of $\sigma^{1}_{1}$.

Proposition 7**.**

$\sigma^{1}_{1}$* is not $(\delta_{\sigma^{1}_{1}}+1)$-stable.*

Proof.

Let $\delta=\delta_{\sigma^{1}_{1}}$. Since $\delta<(\sigma^{1}_{1})^{+}$, it is not admissible. As we observed before, $\delta$ is a limit ordinal; thus, the failure of admissibility must be due to an instance of collection. Choose some $\Delta_{0}$ formula $\psi$ such that for some $\vec{a}\in L_{\delta}$, $L_{\delta}\not\models\psi(\vec{a})\text{-collection}.$ To see that $\sigma^{1}_{1}$ is not $(\delta+1)$-stable, consider the formula $\phi$ in the language of set theory asserting that there are sets $A$, $B$ such that:

1. (1)

$A$ and $B$ are transitive sets satisfying V=L, $A$ is admissible, $A\in B$, and there is $\vec{a}\in B$ such that $B$ does not satisfy $\psi(\vec{a})\text{-collection}$; 2. (2)

for each $({\mathsf{Ord}}\cap A)$-recursive linear ordering $R\in B$, either there is an infinite descending sequence $b$ through $R$ with $b\in A$, or there is an ordinal $\beta\in B$ and an isomorphism $f\in B$ from $R$ to $\beta$; 3. (3)

for each $\beta\in B$, there is an $({\mathsf{Ord}}\cap A)$-recursive linear ordering $R\in B$ and an isomorphism $f\in B$ from $R$ to $\beta$.

Notice that $\phi$ is a $\Sigma_{1}$ formula, since the only unbounded quantifier is the one on $B$. Moreover, it does not hold in $L_{\sigma^{1}_{1}}$, for the sets $A$ and $B$ would need to be of the form $L_{\alpha}$ and $L_{\beta}$, with $\alpha<\beta<\sigma^{1}_{1}$. Conditions (2) and (3) together imply that $\beta=\delta_{\alpha}$, but Gostanian’s characterization of $\sigma^{1}_{1}$ then implies $\beta=\alpha^{+}$, contradicting condition (1). Finally, it does hold in $L_{\delta+1}$, as witnessed by $A=L_{\sigma^{1}_{1}}$ and $B=L_{\delta}$. To see that (2) holds, recall a theorem of Gostanian [8, Theorem 3.2] by which if $\alpha$ is $\Sigma^{1}_{1}$-reflecting, then every $\alpha$-recursive linear ordering which is not a wellordering has an infinite descending sequence in $L_{\alpha}$. Thus, every $\sigma^{1}_{1}$-recursive linear ordering $R$ either has an infinite descending sequence in $L_{\sigma^{1}_{1}}$, or else is isomorphic to some ordinal $\beta<\delta$. One can construct an isomorphism witnessing this by transfinite recursion: at stage $\gamma<\beta$, one has defined $f\upharpoonright\gamma$ and sets $f(\gamma)$ equal to the $R$-least element not in the range of $f\upharpoonright\gamma$. Since this process takes $\beta$-many stages and $R\in L_{\sigma^{1}_{1}+1}$, such an isomorphism belongs to $L_{\sigma^{1}_{1}+\delta}$. Since $\delta$ is additively indecomposable, it belongs to $L_{\delta}$. The proof that (3) holds is similar. ∎

The proof of Proposition 7 shows:

Corollary 8**.**

Suppose $\sigma$ is $\Sigma^{1}_{1}$-reflecting and $(\delta_{\sigma}+1)$-stable. Then, it is a limit of $\Sigma^{1}_{1}$-reflecting ordinals.

One cannot improve the conclusion of Proposition 7—every $\Sigma^{1}_{1}$-reflecting ordinal is stable to the supremum of its recursive wellorderings:

Proposition 9**.**

Suppose $\sigma$ is $\Sigma^{1}_{1}$-reflecting. Then $\sigma$ is $\delta_{\sigma}$-stable.

Proof.

Since $\delta$ is a limit ordinal, it suffices to consider arbitrary $\gamma<\delta$ and show that

[TABLE]

Let $a\in L_{\sigma}$ and $\phi$ be a $\Sigma_{1}$-formula such that $L_{\gamma}\models\phi(a)$. Without loss of generality, assume that $a$ is an ordinal. Let $R$ be a $\sigma$-recursive wellordering of length $\gamma$. In particular, $R$ is $\sigma$-r.e., so there is a $\Sigma_{1}$ formula $\psi$, such that for all $x,y\in L_{\sigma}$,

[TABLE]

Let us assume for notational simplicity that $\psi$ is defined without parameters. Given an ordinal ${\sigma^{\prime}}$, let $R_{\sigma^{\prime}}$ be the binary relation given by

[TABLE]

Since $\psi$ is $\Sigma_{1}$, we have $R_{\sigma^{\prime}}\subset R_{\tau}$ whenever ${\sigma^{\prime}}\leq\tau\leq\sigma$. In particular, $R_{\sigma^{\prime}}$ is wellfounded for all $\sigma^{\prime}<\sigma$.

Because $L_{\gamma}\models\phi(a)$, there is a subset $A$ of $L_{\sigma}$ such that

1. (1)

$A$ codes a model $(M,E)$ of ${\mathsf{KP}}+V=L$; 2. (2)

$M$ has a largest admissible ordinal $\tau$ and $(\tau,E)$ is isomorphic to $(\sigma,\in)$; 3. (3)

there is an ordinal $\beta$ of $M$ and a function $f\in M$ which is an isomorphism between $R_{\tau}^{M}$ (i.e., $R_{\tau}$ computed within $M$) and $\beta$, and $L_{\beta}^{M}\models\phi(a)$.

The existence of such an $A$ can be expressed by a set-theoretic $\Sigma^{1}_{1}$ formula over $L_{\sigma}$ with parameter $a$ (as well as any other parameters involved in the definition of $R$).

By $\Sigma^{1}_{1}$-reflection, there is some ${\sigma^{\prime}}<\sigma$ and some $A_{\sigma^{\prime}}\subset L_{\sigma^{\prime}}$ such that $a\in L_{\sigma^{\prime}}$ and

1. (1)

$A_{\sigma^{\prime}}$ codes a model $(N,F)$ of ${\mathsf{KP}}+V=L$; 2. (2)

$N$ has a largest admissible ordinal $\eta$ and $(\eta,F)$ is isomorphic to $({\sigma^{\prime}},\in)$; 3. (3)

there is an ordinal $b$ of $N$ and a function $g\in N$ which is an isomorphism between $R_{\eta}^{N}$ and $b$, and $L_{b}^{N}\models\phi(a)$.

Here and for the rest of our lives, let us identify the wellfounded part of $N$ with its transitive collapse. Condition (2) implies that $L_{\sigma^{\prime}}\in N$. By (3), there is an ordinal $b$ of $N$ and an isomorphism $g\in N$ from $R_{\sigma^{\prime}}$ to $b$. Because $R_{\sigma^{\prime}}\subset R$, it is wellfounded, and so $b$ really is an ordinal. Now, $L_{b}\models\phi(a)$, and $N$ has no admissible ordinals above ${\sigma^{\prime}}$, so $b<({\sigma^{\prime}})^{+}<\sigma$. Since $\phi$ is $\Sigma_{1}$, we conclude that $L_{\sigma}\models\phi(a)$, as was to be shown. ∎

We have shown that $\sigma^{1}_{1}$ is $\delta_{\sigma^{1}_{1}}$-stable and not $(\delta_{\sigma^{1}_{1}}+1)$-stable. The proof of Proposition 9 illustrates how one derives consequences of an ordinal being $\Sigma^{1}_{1}$-reflecting. We shall carry out many similar arguments in the future, perhaps omitting some of the details that show up repeatedly. We note the following consequence of Proposition 9:

Proposition 10**.**

There is a $\Sigma^{1}_{1}$-sentence $\phi$ such that for every countable, locally countable $\sigma$, $L_{\sigma}\models\phi$ if, and only if, $\sigma$ is $\Sigma^{1}_{1}$-reflecting or $\Pi^{1}_{1}$-reflecting.

Proof.

Let $\phi$ be the sentence that asserts the existence of some $A\subset L_{\sigma}$ coding a model $(M,E)$ of ${\mathsf{KP}}+V=L$ containing $\sigma$ and such that

[TABLE]

Clearly, every $\Pi^{1}_{1}$-reflecting ordinal satisfies this sentence, as does every $\Sigma^{1}_{1}$-reflecting ordinal, by Proposition 9.

Suppose that $L_{\sigma}\models\phi$, as witnessed by $(M,E)$. Suppose moreover that $\sigma$ is not $\Sigma^{1}_{1}$-reflecting, so that $\delta_{\sigma}=\sigma^{+}$ by Gostanian’s characterization. Since $\sigma\in M$, a well-known theorem of F. Ville (see e.g., Barwise [5] for a proof) implies that $L_{\sigma^{+}}\subset M$. Given an arbitrary $\beta<\sigma^{+}$, we then have $\beta\in M$ and $\beta<\delta_{\sigma}^{M}$, for otherwise $\delta_{\sigma}^{M}<\sigma^{+}$, which is impossible, since any $\sigma$-recursive wellordering of a subset of $\sigma$ of length $\delta_{\sigma}^{M}$ would belong to $M$. By choice of $M$,

[TABLE]

and so $M\models L_{\sigma}\prec_{1}L_{\beta}$. However, being $\Sigma_{1}$-elementary is absolute, so we really do have $L_{\sigma}\prec_{1}L_{\beta}$ and, since $\beta$ was arbitrary, we have $L_{\sigma}\prec_{1}L_{\sigma^{+}}$, so $\sigma$ is $\Pi^{1}_{1}$-reflecting. ∎

An immediate consequence is Aanderaa’s classical result:

Corollary 11** (Aanderaa).**

$\pi^{1}_{1}<\sigma^{1}_{1}$.

Corollary 11 holds in a strong form:

Corollary 12**.**

$\sigma^{1}_{1}$* reflects $\Sigma^{1}_{1}$ sentences on $\Pi^{1}_{1}$-reflecting ordinals.*

Proof.

Let $\phi$ be the sentence from Proposition 10. Then, if $\psi$ is another $\Sigma^{1}_{1}$ sentence, so is the conjunction $\phi\wedge\psi$. ∎

Corollary 12 is not new; it also follows from the proof of Corollary 11 written down in Simpson [11]. Our method for analyzing the reflection order is to prove results akin to Corollary 12. Now that we know the degree of stability of $\sigma^{1}_{1}$, it is natural to ask what the least ordinal $\alpha$ which is $(\delta_{\alpha}+1)$-stable is. We shall eventually see that it is rather small and in fact smaller than the successor of $\sigma^{1}_{1}$ in the reflection order. We finish this section with some related results that will not be used in future sections.

Proposition 13**.**

Suppose $\sigma$ is locally countable and $\Sigma^{1}_{1}$-reflecting on $\Sigma^{1}_{1}$-reflecting ordinals. Then $\sigma$ is $(\delta_{\sigma}+1)$-stable.

Proof.

This is similar to the proof of Proposition 9. Again, it is easy to see that $\delta$ is a limit. Let $\gamma=\delta_{\sigma}+1$ and $a\in L_{\sigma}$ be such that $L_{\gamma}\models\phi(a)$, for some $\Sigma_{1}$ formula $\phi$. Let $\psi$ be the $\Sigma^{1}_{1}$ formula expressing that $\sigma$ is locally countable and there is a set $A\subset L_{\sigma}$ coding a model $(M,E)$ of ${\mathsf{KP}}+V=L$ with $\sigma\in M$ and such that

[TABLE]

Then $L_{\sigma}\models\psi$. By hypothesis, there is a $\Sigma^{1}_{1}$-reflecting $\tau<\sigma$ such that $L_{\tau}\models\psi$. Thus, $\tau$ is locally countable and there is a model $(M,E)$ of ${\mathsf{KP}}+V=L$ with $\tau\in M$ and such that

[TABLE]

By Ville’s theorem, $L_{\tau^{+}}\subset M$ and, since $\tau$ is $\Sigma^{1}_{1}$-reflecting, $\delta_{\tau}<\tau^{+}$. Hence, $M$ computes $\delta_{\tau}$ and $\delta_{\tau}+1$ correctly and so we really have $L_{\delta_{\tau}+1}\models\phi(a)$. Since $\phi$ is $\Sigma_{1}$, we conclude $L_{\sigma}\models\phi(a)$, as desired. ∎

The preceding proof shows that if $\sigma$ is as in Proposition 13, then $\sigma$ is $(\delta_{\sigma}+2)$-stable, $(\delta_{\sigma})^{\omega}$-stable, etc. It shows that if $f$ is a function on ordinals which is uniformly $\Sigma_{1}$-definable (with parameters in $L_{\sigma}$) on e.g., multiplicatively indecomposable levels of $L$ containing all parameters, then $\sigma$ is $f(\delta_{\sigma})$-stable.

Definition 14**.**

We denote by $\sigma^{1,\ell}_{1}$ the least $\Sigma^{1}_{1}$-reflecting ordinal which is a limit of $\Sigma^{1}_{1}$-reflecting ordinals.

Proposition 15**.**

$\sigma^{1,\ell}_{1}$* is smaller than the least $\alpha$ which is $(\delta_{\alpha}+1)$-stable.*

Proof.

Let $\alpha$ be as in the statement. We claim that $\alpha$ is $\Sigma^{1}_{1}$-reflecting. Otherwise, $\delta_{\alpha}=\alpha^{+}$, and so $\alpha$ is $(\alpha^{+}+1)$-stable. But surely $\alpha$ is locally countable, and thus $\Sigma^{1}_{1}$-reflecting, by Gostanian’s result mentioned in the introduction. Thus, $\alpha$ is $\Sigma^{1}_{1}$-reflecting. By Corollary 8, $\alpha$ is a limit of $\Sigma^{1}_{1}$-reflecting ordinals. However, this is expressible in $L_{\alpha}$; thus, the proof of Proposition 7 shows that $\alpha$ is a limit of ordinals which are both $\Sigma^{1}_{1}$-reflecting and limits of $\Sigma^{1}_{1}$-reflecting ordinals. ∎

As a consequence, we obtain a negative answer to one of the questions posed in the introduction.

Corollary 16**.**

$\sigma^{1,\ell}_{1}$* does not reflect $\Sigma^{1}_{1}$ statements onto $\Sigma^{1}_{1}$-reflecting ordinals.*

We state without proof a result implying that $\sigma^{1,\ell}_{1}<\pi^{1}_{1}(\sigma^{1}_{1})$. Its proof is similar to that of the more powerful Theorem 26 below.

Proposition 17**.**

For every $\alpha<\pi^{1}_{1}(\sigma^{1}_{1})$, there is some $\sigma<\pi^{1}_{1}(\sigma^{1}_{1})$ which is both $\Sigma^{1}_{1}$-reflecting and $(\delta_{\sigma}+\alpha)$-stable.

Figure 1 summarizes the relationships between the ordinals considered so far. We shall also see that

[TABLE]

3. Reflection Transfer Theorems

In this section, we will present some results on the transfer of reflection properties, i.e., results of the form

[TABLE]

where $s$ and $t$ are reflection patterns. Recall our convention that every ordinal considered is countable and locally countable. The first five reflection transfer results we present are rather elementary:

Lemma 18**.**

Let $s$ be a reflection pattern and $\sigma$ be an ordinal.

1. (1)

If $\sigma$ is $\Sigma^{1}_{1}(\sigma^{1}_{1}(s))$-reflecting, then $\sigma$ is $\Sigma^{1}_{1}(s)$-reflecting. 2. (2)

If $\sigma$ is $\Pi^{1}_{1}(\pi^{1}_{1}(s))$-reflecting, then $\sigma$ is $\Pi^{1}_{1}(s)$-reflecting.

Proof.

If $\sigma$ is $\Sigma^{1}_{1}(\sigma^{1}_{1}(s))$-reflecting and $L_{\sigma}$ satisfies a $\Sigma^{1}_{1}$ sentence $\phi$, then, by definition, there is a $\Sigma^{1}_{1}(s)$-reflecting $\tau<\sigma$ such that $L_{\tau}\models\phi$. By $\Sigma^{1}_{1}(s)$-reflection, there is an $s$-reflecting $\eta<\tau$ such that $L_{\tau}\models\phi$. Hence, $\sigma$ is $\Sigma^{1}_{1}(s)$-reflecting. The argument for $\Pi^{1}_{1}(\pi^{1}_{1}(s))$-reflection is similar. ∎

Lemma 19**.**

Let $s$ and $t$ be reflection patterns and $\sigma$ be an ordinal.

1. (1)

If $\sigma$ is $\sigma^{1}_{1}(s)\wedge\pi^{1}_{1}(t)$-reflecting, then it is $\pi^{1}_{1}(\sigma^{1}_{1}(s)\wedge t)$-reflecting. 2. (2)

If $\sigma$ is $\sigma^{1}_{1}(s)\wedge\pi^{1}_{1}(t)$-reflecting, then it is $\sigma^{1}_{1}(s\wedge\pi^{1}_{1}(t))$-reflecting. 3. (3)

If $\sigma$ is $\sigma^{1}_{1}(s)$-reflecting, then it is $\sigma^{1}_{1}(s\wedge\pi^{1}_{1})$-reflecting.

Proof.

Recall that if an ordinal $\sigma$ is $s$-reflecting, for any nontrivial reflection pattern $s$, then it is recursively inaccessible and, in fact, a limit of recursively inaccessible ordinals. (1) then follows from the simple observation that being $\Sigma^{1}_{1}(s)$-reflecting is expressible by a $\Pi^{1}_{1}$ sentence $\psi$. Thus, if $\sigma$ is $\sigma^{1}_{1}(s)\wedge\pi^{1}_{1}(t)$-reflecting and satisfies some $\Pi^{1}_{1}$-sentence $\phi$, then the conjunction $\phi\wedge\psi$ is also $\Pi^{1}_{1}$, and any ordinal satisfying it must be $\Sigma^{1}_{1}(s)$-reflecting.

(2) is similar. For (3), there are two cases: if $\sigma$ is $\Pi^{1}_{1}$-reflecting, then the result follows from (2). If $\sigma$ is not $\Pi^{1}_{1}$-reflecting, it is not $\sigma^{+}$-stable. Hence, there is a least $\gamma<\sigma^{+}$ such that $\sigma$ is not $\gamma$-stable, i.e., there is a $\Sigma_{1}$-formula $\psi$ and some parameter $\alpha<\sigma$ such that $L_{\gamma}\models\psi(\alpha)$, but $L_{\sigma}\not\models\psi(\alpha)$. The remainder of the proof is an adaptation of the proof of Corollary 11 presented in Simpson [11]:

Let $\phi$ be the $\Sigma^{1}_{1}$-statement expressing that there is a model $(M,E)$ of ${\mathsf{KP}}+V=L$ end-extending $L_{\sigma+1}$ such that for some $\gamma^{\prime}\in M$ with $\gamma^{\prime}<\sigma^{+M}$, $L_{\gamma^{\prime}}^{M}\models\psi(\alpha)$ and, moreover, if $\gamma^{\prime}$ is least such, then $M\models$“$\sigma$ is $({<}\gamma^{\prime})$-stable.” Then $L_{\sigma}\models\phi$. By choice of $\sigma$, there is an $s$-reflecting ordinal $\tau<\sigma$ such that $L_{\tau}\models\phi$. This means that there is a model $(M,E)$ of ${\mathsf{KP}}+V=L$ end-extending $L_{\tau+1}$ such that for some $\gamma^{\prime}\in M$ with $\gamma^{\prime}<\tau^{+M}$, $L^{M}_{\gamma^{\prime}}\models\psi(\alpha)$ and, for the least such $\gamma^{\prime}$, we have $M\models$“$\tau$ is $({<}\gamma^{\prime})$-stable.”. Since $\psi$ is $\Sigma_{1}$ and $L_{\sigma}\not\models\psi(\alpha)$, $\gamma^{\prime}$ must belong to the illfounded part of $M$. So $\tau$ is $s$-reflecting and, as in the proof of Proposition 10, $\tau$ is $\Pi^{1}_{1}$-reflecting. By taking conjunctions as before, one sees that every $\Sigma^{1}_{1}$ satisfied by $L_{\sigma}$ is satisfied by some $(s\wedge\pi^{1}_{1})$-reflecting $\tau<\sigma$, as was to be shown. ∎

Example 20*.*

We claim that

[TABLE]

To see this, notice that Lemma 19(1) and 19(2) imply that

[TABLE]

and thus that

[TABLE]

On the other hand, Lemma 19(3) implies that

[TABLE]

and so

[TABLE]

as claimed. ∎

A natural question is whether one can strengthen $\pi^{1}_{1}$ in the statement of Lemma 19(3) and, in particular, whether $\sigma^{1}_{1}$ is $\sigma^{1}_{1}(\pi^{1}_{1}(\pi^{1}_{1}))$-reflecting. By generalizing the proof of Lemma 19(3), we see that the answer is “yes.”

Definition 21**.**

Let $s$ be a reflection pattern. An ordinal $\alpha$ is $\beta$-stable on $s$ if whenever $L_{\beta}$ satisfies a $\Sigma_{1}$ sentence $\phi(L_{\alpha})$ with additional parameters in $L_{\alpha}$, there is an $s$-reflecting $\gamma<\alpha$ such that $L_{\gamma^{+}}\models\phi(L_{\gamma})$.

We caution the reader that an ordinal $\alpha$ being $\beta$-stable on $\varnothing$ is not the same as it being $\beta$-stable, for the first definition allows $L_{\alpha}$ as a parameter. We do have the following:

Lemma 22**.**

Let $s$ be a reflection pattern. The following are equivalent:

1. (1)

$\alpha$* is $\alpha^{+}$-stable on $s$;* 2. (2)

$\alpha$* is $\Pi^{1}_{1}(s)$-reflecting.*

We omit the proof of Lemma 22, which is a simple adaptation of Aczel and Richter’s characterization of $\Pi^{1}_{1}$-reflection.

Theorem 23**.**

Let $s$ be a reflection pattern. Suppose $\sigma$ is $\sigma^{1}_{1}(s)$-reflecting. Then, it is $\sigma^{1}_{1}(s\wedge\pi^{1}_{1}(s))$-reflecting.

Proof.

The conclusion of the theorem follows from Lemma 19 if $\sigma$ is $\pi^{1}_{1}(s)$-reflecting, so we may assume that it is not.

Since $\sigma$ is not $\pi^{1}_{1}(s)$-reflecting, it is not $\sigma^{+}$-stable on $s$, so there is a least $\beta<\sigma^{+}$ and a $\Sigma_{1}$-formula $\exists x\,\phi(y,x)$ such that $L_{\beta}\models\exists x\,\phi(L_{\sigma},x)$ and whenever $\gamma<\sigma$ and $\gamma$ is $s$-reflecting, then $L_{\gamma^{+}}\not\models\exists x\,\phi(L_{\gamma},x)$. Let $\psi$ be the formula expressing that there is a model $M$ of ${\mathsf{KP}}+V=L$ such that

1. (1)

$M$ contains $\sigma$. 2. (2)

$M\models$ “$\exists x\,\phi(L_{\sigma},x)$ and, letting $\gamma$ be least such that $M\models\phi(L_{\sigma},a)$ for some $a\in L_{\gamma}$, $\sigma$ is ${<}\gamma$-stable on $s$.”

By reflection, there is $\tau<\sigma$ with $L_{\tau}\models\psi$, as witnessed by some model $N$ which end-extends $L_{\tau^{+}}$. Since $\sigma$ is $\sigma^{1}_{1}(s)$-reflecting, we may assume that $\tau$ is $s$-reflecting. Let $\gamma$ be $N$-least such that $N\models\exists x\in L_{\gamma}\,\phi(L_{\tau},x)$. Then, we cannot have $\gamma<\tau^{+}$, for otherwise $\tau$ is an $s$-reflecting ordinal such that $L_{\tau^{+}}\models\exists x\,\phi(L_{\tau},x)$, contradicting the choice of $\phi$. Thus, $\gamma$ belongs to the illfounded part of $N$ and, in $N$, $\tau$ is ${<}\gamma$-stable on $s$. Since $\tau$ is recursively inaccessible (this can be assumed also if $s=\varnothing$), $N$ is correct about $s$-reflection below $\tau$, so an argument as before shows that $\tau$ is $\tau^{+}$-stable on $s$ and thus $\Pi^{1}_{1}(s)$-reflecting. ∎

Example 24*.*

By repeatedly applying Theorem 23, we obtain

[TABLE]

This implies the sequence of inequalities

[TABLE]

which strengthens Corollary 11.

The following strengthening of Proposition 9 is proved similarly:

Lemma 25**.**

Suppose $\sigma$ is $\Sigma^{1}_{1}(s)$-reflecting. Then, it is $\delta_{\sigma}$-stable on $s$.

Proof.

Let $\theta(L_{\sigma})$ be a $\Sigma_{1}$ sentence with parameters in $L_{\sigma}$, say, of the form $\exists x\,\theta_{0}(x,L_{\sigma})$. Let $\eta<\delta_{\sigma}$ and $b\in L_{\eta}$ be such that $L_{\delta_{\sigma}}\models\theta_{0}(b,L_{\sigma})$. Since $\eta<\delta_{\sigma}$, there is a $\sigma$-recursive wellorder $R$ of length $\eta$. Let $\psi$ be the sentence asserting the existence of a model $M$ of ${\mathsf{KPi}}$ such that111${\mathsf{KPi}}$ is the extension of ${\mathsf{KP}}$ by an axiom asserting that every set is contained in an admissible set.

1. (1)

$M$ end-extends $L_{\sigma+1}$; 2. (2)

in $M$, $R$ is isomorphic to an ordinal $\eta_{0}$ and there is $b_{0}\in L_{\eta_{0}}^{M}$ such that $L_{\eta_{0}}^{M}\models\theta_{0}(b_{0},L_{\sigma})$.

Then $L_{\sigma}\models\psi$. Moreover, $\psi$ is $\Sigma^{1}_{1}$ so, by reflection, there is an $s$-reflecting $\tau<\sigma$ such that $L_{\tau}\models\psi$, as witnessed by some model $N$ which end-extends $L_{\tau^{+}}$. Now, in $N$, $L_{\tau^{+}}^{N}\models\theta_{0}(b_{0},L_{\tau})$ for some $b_{0}\in L^{N}_{\eta_{0}}$, where $\eta_{0}$ is some $N$-ordinal isomorphic to $R_{\tau}$. However, $R_{\tau}\subset R$, since $R$ is $\sigma$-recursive, and $\theta_{0}$ is $\Sigma_{0}$, so we really have $L_{\tau^{+}}\models\theta(L_{\tau})$, as desired. ∎

The following theorem, although perhaps odd-looking at first, is crucial for our analysis of the reflection order.

Theorem 26**.**

Let $s$ be a reflection pattern. Suppose $\sigma$ is $\Pi^{1}_{1}(\sigma^{1}_{1}(s))$-reflecting but not $\Sigma^{1}_{1}$-reflecting. Then $\sigma$ is $\Pi^{1}_{1}(s)$-reflecting.

Proof.

Suppose $\sigma$ is $\Pi^{1}_{1}$-reflecting on $\Sigma^{1}_{1}(s)$-reflecting ordinals but not $\Sigma^{1}_{1}$-reflecting. Let $\psi$ be the statement expressing that whenever $(M,E)$ is an end-extension of $L_{\sigma+1}$ satisfying ${\mathsf{KPi}}$, then $M\models$“$\sigma$ is not $\Sigma^{1}_{1}(s)$-reflecting.” This sentence is $\Pi^{1}_{1}$ and thus cannot be satisfied by $L_{\sigma}$, for otherwise it would be reflected to a $\Sigma^{1}_{1}(s)$-reflecting ordinal. But clearly $L_{\alpha}$ cannot satisfy $\psi$ if $\alpha$ is $\Sigma^{1}_{1}(s)$-reflecting.

Thus, $L_{\sigma}\not\models\psi$, so there is a model $M$ of ${\mathsf{KPi}}$ end-extending $L_{\sigma+1}$ such that

[TABLE]

For ordinals $\tau<\sigma$, whether $\tau$ is $t$-reflecting is computed correctly by, say, $\tau^{++}$, and thus too by $M$, for any reflection pattern $t$. By Lemma 25 applied within $M$,

[TABLE]

Let $\phi$ be a $\Pi^{1}_{1}$ statement, and $a\in L_{\sigma}$ be a parameter such that $L_{\sigma}\models\phi(a)$. By Barwise-Gandy-Moschovakis [6], there is a $\Sigma_{1}$ formula $\phi^{*}$ such that for all admissible $\alpha$ with $a\in L_{\alpha}$, $L_{\alpha}\models\phi(a)$ if, and only if, $L_{\alpha^{+}}\models\phi^{*}(a,L_{\alpha})$; thus, $L_{\sigma^{+}}\models\phi^{*}(a,L_{\sigma})$. Let $b\in L_{\sigma^{+}}$ be a witness for $\phi^{*}$ and let $\gamma<\sigma^{+}$ be large enough so that $b\in L_{\gamma}$. Since $\sigma$ is not $\Sigma^{1}_{1}$-reflecting (in the real world), $\delta_{\sigma}=\sigma^{+}$, and thus

[TABLE]

Since $\phi^{*}$ is $\Sigma_{1}$,

[TABLE]

so by the $\delta_{\sigma}$-stability of $\sigma$ on $s$ within $M$, there is an $s$-reflecting $\tau<\sigma$ such that

[TABLE]

Since $\tau^{+}<\sigma$, we really do have

[TABLE]

and so $L_{\tau}\models\phi(a)$. This completes the proof of the theorem. ∎

Remark 27*.*

The assumption that $\sigma$ is not $\Sigma^{1}_{1}$-reflecting cannot be removed from the statement of Theorem 26. To see this, let $s$ be the trivial pattern. By Lemma 19(1), $\sigma^{1}_{1}\wedge\pi^{1}_{1}$ is $\Pi^{1}_{1}(\sigma^{1}_{1})$-reflecting. However, $\sigma^{1}_{1}\wedge\pi^{1}_{1}$ is not $\Pi^{1}_{1}(\pi^{1}_{1})$-reflecting, for being $\Sigma^{1}_{1}$-reflecting is expressible by a $\Pi^{1}_{1}$-formula, and thus every ordinal which is $(\sigma^{1}_{1}\wedge\pi^{1}_{1}(\pi^{1}_{1}))$-reflecting is also $\pi^{1}_{1}(\sigma^{1}_{1}\wedge\pi^{1}_{1})$-reflecting and, in particular, a limit of $(\sigma^{1}_{1}\wedge\pi^{1}_{1})$-reflecting ordinals.∎

Remark 28*.*

One cannot improve the statement of Theorem 26 to conclude that $\sigma$ is $\Pi^{1}_{1}(\sigma^{1}_{1}(s)\wedge s)$-reflecting, for let $s=\pi^{1}_{1}$. Then, $\pi^{1}_{1}(\sigma^{1}_{1}(\pi^{1}_{1}))=\pi^{1}_{1}(\sigma^{1}_{1})$ by Lemma 19(3). However, as in Example 20,

[TABLE]

This is in contrast to Theorem 23.∎

Example 29*.*

By combining Theorems 26 and 23, one sees that

[TABLE]

Since we have seen that these reflection patterns are all smaller than $\sigma^{1}_{1}(\sigma^{1}_{1})$ and $\sigma^{1}_{1}\wedge\pi^{1}_{1}$, it follows that $\sigma^{1}_{1}$ and $\pi^{1}_{1}(\sigma^{1}_{1})$ have order-types $\omega$ and $\omega+1$ in the reflection order, respectively.∎

Example 30*.*

Let us present a proof of the inequality

[TABLE]

First, apply Theorem 23 to see that

[TABLE]

Then, apply Lemma 19 to see that

[TABLE]

so that

[TABLE]

Finally, by Theorem 23, every $\sigma^{1}_{1}(\sigma^{1}_{1}\wedge\pi^{1}_{1}(\sigma^{1}_{1}\wedge\pi^{1}_{1}))$-reflecting ordinal is also $\sigma^{1}_{1}(\pi^{1}_{1}(\sigma^{1}_{1}\wedge\pi^{1}_{1}(\sigma^{1}_{1}\wedge\pi^{1}_{1})))$-reflecting, so that $\sigma^{1}_{1}(\sigma^{1}_{1})$ is a limit of $\pi^{1}_{1}(\sigma^{1}_{1}\wedge\pi^{1}_{1}(\sigma^{1}_{1}\wedge\pi^{1}_{1}))$-reflecting ordinals. ∎

We finish this section with a final reflection transfer theorem. It is a strengthening of Theorem 26 which clarifies the hypothesis on $\sigma$ not being $\Sigma^{1}_{1}$-reflecting. We state it separately, however, since the proof is longer and the result is not used afterwards.

Theorem 31**.**

Suppose $\sigma$ is $\pi^{1}_{1}(\sigma^{1}_{1}(t\wedge\pi^{1}_{1}(s)))$-reflecting. Then, one of the following holds:

1. (1)

$\sigma$* is $\pi^{1}_{1}(s)$-reflecting; or* 2. (2)

$\sigma$* is $\sigma^{1}_{1}(t\wedge\pi^{1}_{1}(s))$-reflecting.*

Proof.

Suppose $\sigma$ is $\pi^{1}_{1}(\sigma^{1}_{1}(t\wedge\pi^{1}_{1}(s)))$-reflecting but not $\pi^{1}_{1}(s)$-reflecting. Let $\theta$ be a $\Sigma^{1}_{1}$ sentence with parameters in $L_{\sigma}$ such that

[TABLE]

we need to find a $(t\wedge\pi^{1}_{1}(s))$-reflecting $\tau<\sigma$ such that

[TABLE]

By Barwise-Gandy-Moschovakis [6], there is a $\Pi_{1}$ formula $\theta^{*}(a)$ such that for every admissible $\alpha$ containing the parameters of $\theta$, $L_{\alpha}\models\theta$ if, and only if, $L_{\alpha^{+}}\models\theta^{*}(L_{\alpha}).$ In particular,

[TABLE]

Since $\sigma$ is not $\pi^{1}_{1}(s)$-reflecting, there is a least $\beta<\sigma^{+}$ such that $\sigma$ is not $\beta$-stable on $s$. Because $\theta^{*}$ is $\Pi_{1}$,

[TABLE]

Let $\phi$ be the sentence asserting the non-existence of a model $M$ of ${\mathsf{KPi}}+V=L$ end-extending $L_{\sigma+1}$ in which $\sigma$ is $\sigma^{1}_{1}(t\wedge\pi^{1}_{1}(s))$-reflecting. This is a $\Pi^{1}_{1}$ sentence and thus cannot be satisfied by any $\pi^{1}_{1}(\sigma^{1}_{1}(t\wedge\pi^{1}_{1}(s)))$-reflecting ordinal and, in particular, by $\sigma$. Thus, there is a model $M$ of ${\mathsf{KPi}}+V=L$ end-extending $L_{\sigma+1}$ and such that

[TABLE]

Let $\chi$ be the sentence asserting the existence of a model $N$ of ${\mathsf{KP}}+V=L$ such that

1. (1)

$N$ contains $\sigma$; 2. (2)

in $N$, letting $\beta$ be least such that $\sigma$ is not $\beta$-stable on $s$, we have

[TABLE]

Since $\beta<\sigma^{+}$ and $M$ must end-extend $L_{\sigma^{+}}$, $\beta\in M$ and $M$ is correct about $\beta$ being the least ordinal at which $\sigma$ fails to be stable on $s$. Thus, we have

[TABLE]

as witnessed, say, by $L^{M}_{\sigma^{+}}$. Within $M$, $\sigma$ is $\sigma^{1}_{1}(t\wedge\pi^{1}_{1}(s))$-reflecting and thus there is some $\tau<\sigma$ such that,

[TABLE]

and so we really do have

[TABLE]

Moreover, $M$ is correct about reflection below $\sigma$, so we may assume that $\tau$ is $(t\wedge\pi^{1}_{1}(s))$-reflecting. By the definition of $\chi$, there is a model $N$ of ${\mathsf{KP}}+V=L$ such that

1. (1)

$N$ contains $\tau$; 2. (2)

in $N$, letting $\gamma$ be least such that $\tau$ is not $\gamma$-stable on $s$, we have

[TABLE]

Since $\tau$ is $\pi^{1}_{1}(s)$-reflecting, it is $\tau^{+}$-stable on $s$, and thus $\gamma$ cannot be a true ordinal smaller than $\tau^{+}$. By Ville’s Theorem, $N$ must end-extend $L_{\tau^{+}}$. Because $\theta^{*}$ is $\Pi_{1}$, it follows that

[TABLE]

and thus, that

[TABLE]

as was to be shown. ∎

4. Patterns Below $\sigma^{1}_{1}(\sigma^{1}_{1})$

In this section, we describe the reflection order below $\sigma^{1}_{1}(\sigma^{1}_{1})$. The remaining sections do not depend on this one, so the reader who so desires should feel free to skip ahead. To ease notation, we shall omit subscripts and superscripts and simply write $\sigma$ for $\sigma^{1}_{1}$ and $\pi$ for $\pi^{1}_{1}$. We shall also sometimes omit parentheses; thus, e.g., we will write

[TABLE]

instead of

[TABLE]

We will also express concatenation by direct juxtaposition, so that e.g., if $s=\sigma\wedge\pi$, then

[TABLE]

We remind the reader one last time of our convention on all ordinals being countable and locally countable. The following notation will be useful:

Definition 32**.**

Let $s$ and $t$ be reflection patterns. We write $s\equiv t$ if for every ordinal $\alpha$, $\alpha$ is $s$-reflecting if, and only if, it is $t$-reflecting.

Definition 33**.**

Let $k\in\mathbb{N}$ and $s$ be a reflection pattern. We write $c^{k}_{0}s=s$; inductively,

[TABLE]

We write $c^{k}_{n}$ for $c^{k}_{n}s$, where $s$ is the empty pattern.

We remark that, in particular, $c^{0}_{n}=(\sigma\pi)^{n}$.

Lemma 34**.**

For every $n,k\in\mathbb{N}$ and every reflection pattern $s$, every $(\sigma\wedge\pi^{k+1}s)$-reflecting ordinal is $(\pi c^{k}_{n}s)$-reflecting.

Proof.

We first show by induction on $n$ that

[TABLE]

Suppose that every $(\sigma\wedge\pi^{k+1})$-reflecting ordinal is $(\pi c^{k}_{n})$-reflecting, i.e., that

[TABLE]

After some applications of Lemma 19, we have

[TABLE]

as desired. The argument given also shows that every every $(\sigma\wedge\pi^{k+1}s)$-reflecting ordinal is $(\sigma\wedge\pi c^{k}_{n}s)$-reflecting, although it does not prove the converse (which, incidentally, is not true). ∎

Corollary 35**.**

For every $n,k\in\mathbb{N}$, every $l\leq k$, and every reflection pattern $s$, every $(\sigma\wedge\pi^{k+1}s)$-reflecting ordinal is $(\pi^{k+1}\sigma\pi c^{l}_{n}s)$-reflecting.

Proof.

Suppose $\alpha$ is $(\sigma\wedge\pi^{k+1}s)$-reflecting. By Lemma 34,

[TABLE]

By definition,

[TABLE]

in particular,

[TABLE]

We may thus apply Lemma 18 $n$ times to see that every $c^{k}_{n}s$-reflecting ordinal is also $c^{l}_{n}$-reflecting, from which the result follows. ∎

Lemma 36**.**

For every $k\in\mathbb{N}$, every $n_{0},\ldots,n_{k}\in\mathbb{N}$, and every reflection pattern $s$, every $(\sigma\wedge\pi^{k+1}s)$-reflecting ordinal is $(\pi^{k+1}\sigma\pi c^{0}_{n_{0}}c^{1}_{n_{1}}\dots c^{k}_{n_{k}}s)$-reflecting.

Proof.

This follows from applying Corollary 35 repeatedly. ∎

Definition 37**.**

A reflection pattern is in $2$-normal form if it is of the form

[TABLE]

for some natural numbers $k$, $n_{0},n_{1},\ldots,n_{k}$. If $w$ is the reflection pattern above, we define

[TABLE]

For now, we shall simply refer to patterns in $2$-normal form as being in normal form. We shall see that patterns in normal form have very nice properties.

Lemma 38**.**

*Suppose $s$ is a reflection pattern in normal form. Then, every $\sigma\sigma$-reflecting ordinal is $\sigma(s)$-reflecting. *

Proof.

By Theorem 23, any such ordinal is $\sigma(\sigma\wedge\pi^{k})$-reflecting for any $k$. The lemma now follows from Lemma 36. ∎

We also have the following “contraction” lemma, which will be crucial:

Lemma 39**.**

Suppose $s$ is a reflection pattern. Then, every $\sigma\pi\sigma\pi s$-reflecting ordinal is $\sigma\pi s$-reflecting.

Proof.

Suppose $\alpha$ is $\sigma\pi\sigma\pi s$-reflecting. By an argument as in Lemma 18, applying Lemma 19, for every $\Sigma^{1}_{1}$-sentence $\phi$ satisfied by $L_{\alpha}$, one can find some $\pi\sigma\pi s$-reflecting $\beta<\alpha$ such that $L_{\beta}\models\phi$ and $\beta$ is not $\Sigma^{1}_{1}$-reflecting. By Theorem 26, $\beta$ is $\pi\pi s$-reflecting. By Lemma 18, $\beta$ is $\pi s$-reflecting, as desired. ∎

Lemma 40**.**

Suppose $\pi t$ and $\pi s$ are reflection patterns in normal form such that $o(\pi t)\leq o(\pi s)$. Then, every $(\pi s)$-reflecting ordinal is either $(\pi t)$-reflecting or $\sigma$-reflecting.

Proof.

Let

[TABLE]

and

[TABLE]

where $n$ and $m$ are nonzero. Without loss of generality, we assume that $n_{k}$ and $m_{l}$ are also nonzero. It follows that $k\leq l$. Suppose that $o(\pi t)<o(\pi s)$. It will be convenient, for illustrative purposes, to consider the case that $k<l$ first. If so, it suffices to show that every $\pi(\sigma\wedge\pi^{l})$-reflecting ordinal which is not $\sigma$-reflecting is $(\pi t)$-reflecting, for then the result follows from Theorem 26. By Lemma 36, every $\pi(\sigma\wedge\pi^{l})$-reflecting ordinal is

[TABLE]

By Lemma 19, every such ordinal is

[TABLE]

and, by Theorem 23 or Lemma 18, according as $n\leq l$ or $l\leq n$, it is

[TABLE]

so that, if it is not $\sigma$-reflecting, then it is

[TABLE]

by Theorem 26, i.e., $\pi(t)$-reflecting.

The general case is similar: let $i\leq k$ be greatest such that $n_{i}<m_{i}$ and notice that

[TABLE]

Thus,

[TABLE]

where $t^{\prime}=c^{i}_{n_{i}}\dots c^{k}_{n_{k}}$; and

[TABLE]

It suffices to show that every $\pi(\sigma\wedge\pi^{i}\sigma\pi t^{\prime})$-reflecting ordinal which is not $\sigma$-reflecting is $(\pi t)$-reflecting, for then the result follows from Theorem 26. Lemma 36 (with $\sigma\pi t^{\prime}$ being the $s$ in the statement) shows that every such ordinal is

[TABLE]

As before, by Lemma 19, every such ordinal is

[TABLE]

By Theorem 23 and Lemma 18, every such ordinal is

[TABLE]

By contraction (Lemma 39), it is

[TABLE]

and by an argument like the one for Lemma 39, it is

[TABLE]

so that if it is not $\sigma$-reflecting, then it is

[TABLE]

by Theorem 26, as desired. ∎

Lemma 41**.**

Suppose $t$ is a reflection pattern in normal form. Then $\sigma\wedge t$ is equivalent to a reflection pattern in normal form.

Proof.

Put $t=\pi^{k}c^{0}_{m_{0}}c^{1}_{m_{1}}\dots c^{l}_{m_{l}}$. The lemma is immediate unless $k\neq 0$ and there is some least $i\leq l$ such that $m_{i}\neq 0$. Thus,

[TABLE]

Let

[TABLE]

so that

[TABLE]

If all the indicated $m_{j}$ are zero, then the result follows easily; otherwise, by Lemma 19 and Lemma 39,

[TABLE]

By Lemma 19,

[TABLE]

By Lemma 36 on the one hand and Lemma 18 and Lemma 39 on the other,

[TABLE]

but the reflection pattern on the left-hand side is readily seen to be equivalent to one in normal form. ∎

Theorem 42**.**

Let $s$ be a reflection pattern in which the string $\sigma\sigma$ does not occur. Then, it is equivalent to a reflection pattern in $2$-normal form.

Proof.

This is proved by induction on the construction of $s$. Clearly, if $s$ is equivalent to a reflection pattern in normal form, then so too is $\pi s$. We only need consider the string $\sigma s$ in the case that $s$ is of the form

[TABLE]

where $m$ is nonzero. Then, it is easy to see that

[TABLE]

Now, let $s$ be as above, and let

[TABLE]

We need to show that $s\wedge t$ is equivalent to a reflection pattern in normal form. If both $n$ and $m$ are nonzero, then the result follows from Lemma 40; so suppose that one of $m$ and $n$ is zero, so that

[TABLE]

Write $n=n_{-1}$ and $m=m_{-1}$ and let $i$ and $j$ be least such that $n_{i}$ and $m_{j}$ are nonzero, respectively. There are four cases to consider. The first one is that in which both $i$ and $j$ are equal to [math]. Then, there are reflection patterns $u$ and $v$, both in normal form, such that

[TABLE]

and

[TABLE]

Suppose without loss of generality that $o(\pi u)<o(\pi v)$. By Lemma 40, every $(\pi v)$-reflecting ordinal which is not $\sigma$-reflecting is $(\pi u)$-reflecting. Thus, every $s$-reflecting ordinal is either $t$-reflecting or $\sigma\sigma$-reflecting, in which case it is also $t$-reflecting by Lemma 38.

The second case is that in which $i=0$ but $j\neq 0$. Then, there are reflection patterns $u$ and $v$, both in normal form, such that

[TABLE]

and

[TABLE]

By direct computation,

[TABLE]

Now, it is easily seen that

[TABLE]

where the last equivalence follows from Lemma 39, and so

[TABLE]

Since each of $\pi u$ and $\pi^{m_{j}}\sigma\pi v$ is a reflection pattern in normal form, Lemma 40 implies that their conjunction is equivalent to one of $\pi u$ and $\pi^{m_{j}}\sigma\pi v$ in this context. Let us denote this conjunct by $w$. Then, by an argument as before,

[TABLE]

Since $w$ is in normal form and of the form $\pi w^{\prime}$, so too is $\pi^{m_{j}}\sigma w$, so the result follows from Lemma 41. The case in which $i\neq 0$ and $j=0$ is analogous.

The remaining case is that in which both $i$ and $j$ are nonzero. Suppose without loss of generality that $n_{i}<m_{j}$ By replacing $n_{i}$ and $m_{j}$ by larger numbers if necessary (this might need to be done in the case $i=-1$ and $m_{0}=0$—a situation similar to the one in Lemma 41) we may assume that there are reflection patterns $u$ and $v$ such that

[TABLE]

and

[TABLE]

Then, we have

[TABLE]

By contraction,

[TABLE]

so, because $n_{i}<m_{j}$, it follows that every ordinal which is

[TABLE]

is also

[TABLE]

Similarly, we have

[TABLE]

and so every ordinal which is $(\sigma\wedge\pi^{m_{j}}\sigma\pi^{m_{j}}\sigma\pi v)$-reflecting is also

[TABLE]

From these two observations, we see that

[TABLE]

Since both $m_{j}$ and $n_{i}$ are nonzero, Lemma 40 implies that in this context the conjunction

[TABLE]

is equivalent to one of $\pi^{n_{i}}\sigma\pi u$ or $\pi^{m_{j}}\sigma\pi v$. Denote this conjunct by $w$. Then, $w$ is in normal form. Since $w$ begins with the symbol $\pi$, $\pi^{m_{j}}\sigma w$ is also equivalent to a pattern in normal form. The result then follows from Lemma 41. This proves the theorem. ∎

Theorem 43**.**

$\sigma\sigma$* has order-type $\omega^{\omega}$ in the reflection order.*

Proof.

By Theorem 42, every reflection pattern in which the string $\sigma\sigma$ does not occur is equivalent to one in $\sigma\sigma$-normal form. By Lemma 38, this implies that $\sigma\sigma$ is strictly bigger than each reflection pattern in which the string $\sigma\sigma$ does not occur. Conversely, if a reflection pattern does contain the string $\sigma\sigma$, then naturally, it cannot be strictly smaller than $\sigma\sigma$. An easy induction using Lemma 36 shows that, for reflection patterns $u$ and $v$ in $2$-normal form, $u<v$ if, and only if, $o(u)<o(v)$, so the result follows. ∎

5. Linear Patterns

Our first result in this section concerns the length of the linear reflection order; its proof induces a simple algorithm for comparing two arbitrary linear reflection patterns.

Theorem 44**.**

The length of the linear fragment of the reflection order is $\omega^{\omega}$.

Proof.

Let us employ the simplified notation from the previous section. Recursively, we assign ordinals to reflection patterns without conjunction: we assign the ordinal $\omega^{n}$ to the pattern

[TABLE]

In particular, the ordinal $1$ is assigned to the empty pattern. If $s$ and $t$ are patterns to which ordinals $o(s)$ and $o(t)$ have been assigned, we assign the ordinal

[TABLE]

to the pattern

[TABLE]

Note that if $n<m$, then, on the one hand,

[TABLE]

while, on the other,

[TABLE]

Thus, every $\sigma^{m}\pi s$-reflecting ordinal is also $\sigma^{m}\pi\sigma^{n}\pi s$-reflecting when $n<m$; the converse is also true, by Lemma 18 and Theorem 26 (cf. the argument of Lemma 39 on p. 39). It follows that this assignment of ordinals is well-defined. By Lemma 18 and Theorem 23, we have

[TABLE]

for any pair of nonzero numbers $n$ and $m$, so every conjunction-free reflection pattern is equivalent to one to which an ordinal has been assigned. It should be clear by now that, for conjunction-free reflection patterns, $s<t$ if, and only if, $o(s)<o(t)$, which completes the proof. ∎

The second result of this section is that the linear patterns are cofinal in the reflection order.

Theorem 45**.**

The sequence $\{(\sigma^{1}_{1})^{n}:n\in\mathbb{N}\}$ is cofinal in the reflection order.

Proof.

To prove the theorem, we shall prove by induction on the construction of a reflection pattern $s$ that if $s$ contains no occurrence of $\sigma^{n+1}$, then every $\sigma(\sigma^{n}\wedge t)$-reflecting ordinal is also $\sigma(\sigma^{n}\wedge t\wedge s)$-reflecting, for every reflection pattern $t$ (cf. Lemma 38 on p. 38). We may assume that $n\neq 0$, for otherwise the conclusion follows from Theorem 23. Note that the case that $s$ is a conjunction is immediate from the induction hypothesis and the case that $s$ is of the form $\pi s^{\prime}$ is also immediate from the induction hypothesis and Theorem 23, thus, we suppose that $s$ is of the form $\sigma s^{\prime}$. The pattern $s^{\prime}$ might be a conjunction, say,

[TABLE]

where each $s_{i}$ is of the form $\sigma^{m_{i}}s_{i}^{\prime}$ for some $m_{i}<n$ and some $s_{i}^{\prime}$ which is a conjunction of patterns of the form $\pi s^{*}$, and each $r_{i}$ is of the form $\pi r_{i}^{\prime}$. Instead of proving that every $\sigma(\sigma^{n}\wedge t)$-reflecting ordinal is $\sigma(\sigma^{n}\wedge t\wedge\sigma s^{\prime})$-reflecting, we shall prove the stronger fact that it is

[TABLE]

By the induction hypothesis applied to $s^{\prime}$, and the fact that each of $s^{\prime}_{i}$ and

[TABLE]

is a conjunction of patterns of the form $\pi s^{*}$, we obtain:

[TABLE]

where the last two equivalences follow from Lemma 19. This completes the proof of the theorem. ∎

6. Concluding Remarks

We have not given any bounds on the length of the reflection order. Let us say something about this:

Proposition 46**.**

Let $s$ be a reflection pattern and suppose $\alpha$ is a countable, locally countable ordinal such that

[TABLE]

Then, $\alpha$ is $s$-reflecting.

Proof.

Since

[TABLE]

it follows that $\alpha$ is $\pi^{1}_{1}$-reflecting. By Gostanian’s theorem [8] mentioned in the introduction, $\alpha$ is $\sigma^{1}_{1}$-reflecting. Inductively, suppose it is

[TABLE]

and let $\psi$ be a $\Sigma^{1}_{1}$ sentence such that

[TABLE]

Choose a $\Pi_{1}$ sentence $\psi^{*}$ such that for all admissible $\beta$ containing all relevent parameters,

[TABLE]

if, and only if,

[TABLE]

so that, in particular,

[TABLE]

Then, from the point of view of $L_{\alpha^{+}+1}$, there are admissible sets $L_{\alpha}$ and $L_{\alpha^{+}}$ such that

1. (1)

$\alpha$ is $(\sigma^{1}_{1})^{n}$-reflecting, i.e., for every $\Pi_{1}$ sentence $\phi^{*}$, if $L_{\alpha^{+}}\models\phi^{*}(L_{\alpha})$, then there is a $(\sigma^{1}_{1})^{n-1}$-reflecting $\beta<\alpha$ such that $L_{\beta^{+}}\models\phi^{*}(L_{\beta})$. (The quantification over $\phi^{*}$ is bounded.) 2. (2)

$L_{\alpha^{+}}\models\psi^{*}(L_{\alpha})$.

Thus, by stability, there are admissible sets $L_{\beta}$ and $L_{\beta^{+}}$ in $L_{\alpha}$ such that $\beta$ is $(\sigma^{1}_{1})^{n}$-reflecting and $L_{\beta^{+}}\models\psi^{*}(L_{\beta})$. Hence, $\alpha$ is $(\sigma^{1}_{1})^{n+1}$-reflecting. Therefore, a simple induction on the construction of a reflection pattern $s$, using the proofs of Theorem 26 and Theorem 45 shows that $\alpha$ is $s$-reflecting. ∎

The length of the reflection order is thus at most the least ordinal $\alpha$ such that

[TABLE]

Moreover, surely each inequality between reflection patterns is provable in any theory that proves the existence of the corresponding ordinals. This suggests strongly that the length of the reflection order is smaller than the proof-theoretic ordinal of the subsystem $\bm{\Pi}^{1}_{2}$-CA0 of analysis and in fact smaller than the ordinal described in Rathjen [10], though we do not have a proof of this. The reader may consult [4] for an example of a chain of length $\varepsilon_{0}$ in the reflection order.

An interesting question is that of the structure of the “higher” reflection order, defined in terms of iterated $\Pi^{1}_{n}$ and $\Sigma^{1}_{n}$-reflection and conjunctions. The situation there is very different and involves set-theoretic considerations; it will be the subject of a forthcoming article.