Proof Theoretic Analysis by Iterated Reflection

Abstract

Progressions of iterated reflection principles can be used as a tool for the ordinal analysis of formal systems. Moreover, they provide a uniform definition of a proof-theoretic ordinal for any arithmetical complexity $\Pi _{n}^{0}$. We discuss various notions of proof-theoretic ordinals and compare the information obtained by means of the reflection principles with the results obtained by the more usual proof-theoretic techniques. In some cases we obtain sharper results, e.g., we define proof-theoretic ordinals relevant to logical complexity $\Pi _{1}^{0}$. We provide a more general version of the fine structure relationships for iterated reflection principles (due to Ulf Schmerl). This allows us, in a uniform manner, to analyze main fragments of arithmetic axiomatized by restricted forms of induction, including $I\Sigma _{n}$, $I\Sigma _{n}^{-}$, $I\Pi _{n}^{-}$ and their combinations. We also obtain new conservation results relating the hierarchies of uniform and local reflection principles. In particular, we show that (for a sufficiently broad class of theories T) the uniform $\Sigma _{1}$-reflection principle for T is $\Sigma _{2}$-conservative over the corresponding local reflection principle. This bears some corollaries on the hierarchies of restricted induction schemata in arithmetic and provides a key tool for our generalization of Schmerl’s theorem.

AMS Classifications: 03F15, 03F30, 03F40, 03F45

This article is a reprint of [6] with a new sect. 1 “Preliminary Notes” added.

Appendices

Appendix 1

Let $(D,\prec )$ be an elementary well-ordering. Define

$$ \displaystyle\begin{array}{rcl} \alpha [x]&:=& \max _{\prec }\{\beta \leq x:\beta \prec \alpha \} {}\\ \beta \prec _{x}\alpha &:\Leftrightarrow & (\beta \leq x\wedge \beta \prec a). {}\\ \end{array} $$

Recall that the functions F_α are defined as follows:

$$\displaystyle{F_{\alpha }(x):=\max \{ 2_{x}^{x} + 1\} \cup \{ F_{\beta }^{(v)}(u) + 1:\beta \prec \alpha,\ u,v,\beta \leq x\}.}$$

For technical convenience we also define F₋₁(x) = 2^x and α[x] = −1, if there is no $\beta \prec _{x}\alpha$.

Lemma 43

For all α,β,x,y,

• (i) $x \leq y \rightarrow F_{\alpha }(x) \leq F_{\alpha }(y);$

• (ii) $\beta \prec \alpha \rightarrow F_{\beta }(x) \leq F_{\alpha }(x).$

Proof

Part (i) is obvious. Part (ii) follows from the fact that

$$\displaystyle{\gamma \prec _{x}\beta \prec \alpha \Rightarrow \gamma \prec _{x}\alpha,}$$

$\boxtimes$

Lemma 44

For all α,x, F _α (x) = F _α[x] ^(x) (x) + 1.

Proof

This is obvious for α[x] = −1. Otherwise, from Part (i) of the previous lemma we obtain

$$\displaystyle{u,v \leq x \rightarrow F_{\beta }^{(v)}(u) \leq F_{\beta }^{(x)}(x).}$$

Part (ii) a nd Part (i) by an elementary induction on y then yield

$$\displaystyle{\beta \prec \alpha \rightarrow F_{\beta }^{(y)}(x) \leq F_{\alpha }^{(y)}(x).}$$

Hence, if u, v ≤ x and $\beta \prec _{x}\alpha$, then $\beta \prec _{x}\alpha [x]$ or β = α[x], and in both cases

$$\displaystyle{F_{\beta }^{(v)}(u) \leq F_{\alpha [x]}^{(x)}(x),}$$

$\boxtimes$

We now define evaluation trees. An evaluation tree is a finite tree labeled by tuples of the form $\langle \alpha,x,y\rangle$ satisfying the following conditions:

• 1. If there is no $\beta \prec _{x}\alpha$, then $\langle \alpha,x,y\rangle$ is a terminal node and $y = 2_{x}^{x} + 1$.

• 2. Otherwise, there are x immediate successors of $\langle \alpha,x,y\rangle$, and their labels have the form

  $$\displaystyle{\langle \alpha [x],x,y_{1}\rangle,\ \langle \alpha [x],y_{1},y_{2}\rangle,\ldots,\langle \alpha [x],y_{x-1},y_{x}\rangle }$$

  and y = y_x + 1.

Obviously, the relation x is a code of an evaluation tree is elementary.

Lemma 45

• (i) If a node of an evaluation tree is labeled by $\langle \alpha,x,y\rangle$, then F_α(x) = y.

• (ii) If F _α (x) = y, then there is an evaluation tree with the root labeled by $\langle \alpha,x,y\rangle$.

Proof

Part (i) is proved by transfinite induction on α. If α[x] = −1, the statement is obvious. Otherwise, $\alpha [x] \prec \alpha$, hence by the induction hypothesis at the immediate successor nodes one has $F_{\alpha [x]}(y_{i}) = y_{i+1}$ for all i < x (where we also put y₀: = x). It follows that $y_{x} = F_{\alpha [x]}^{(x)}(x)$, whence $y = y_{x} + 1 = F_{\alpha }(x)$.

Part (ii) obviously follows from the definition of F_α, $\boxtimes$

Now we observe that, for any evaluation tree T, whose root is labeled by $\langle \alpha,x,y\rangle$, the value $\max (\alpha,y)$ is a common bound to the following parameters:

• (a) each γ, u, v such that $\langle \gamma,u,v\rangle$ occurs in T;

• (b) the number of branches at every node of T;

• (c) the depth of T.

Ad (a): If $\langle \gamma,u,v\rangle$ is an immediate successor of $\langle \alpha,x,y\rangle$, then γ = α[x] ≤ x, and u, v ≤ y by the monotonicity of F.

Statement (b) follows from (a) and the fact that the number of branches at a node $\langle \gamma,u,v\rangle$ equals u.

Statement (c) follows from the observation that if $\langle \gamma,u,v\rangle$ is an immediate successor of $\langle \alpha,x,y\rangle$, then y > v.

An immediate corollary of (a)–(c) is that the code of the evaluation tree T is bounded by the value $g(\max (\alpha,y))$, for an elementary function g. Hence we obtain

Proof of Lemma 7

Using Lemma 45 the relation F_α(x) = y can be expressed by formalizing the statement that there is an evaluation tree with the code$\leq g(\max (\alpha,y))$, whose root is labeled by$\langle \alpha,x,y\rangle$. All quantifiers here are bounded, hence the relation F_α(x) = y is elementary. $\boxtimes$

Inspecting the definition of the relation F_α(x) = y notice that the proofs of the monotonicity properties and bounds on the size of the tree only required elementary induction (transfinite induction is not used). Hence, these properties together with the natural defining axioms for F_α can be verified in EA. This yields a proof of Lemma 8. Here we just formally state the required properties of F_α formalizable in EA.

• F1. $(\forall \beta \leq x\:\neg \beta \prec \alpha ) \rightarrow [F_{\alpha }(x) \simeq y \leftrightarrow y = 2_{x}^{x} + 1]$

• F2. $F_{\alpha }(x) \simeq y\wedge \beta \leq x\wedge \beta \prec \alpha \rightarrow \exists z \leq y\:\exists u,v \leq x\ F_{\beta }^{(v)}(u) \simeq z$

• F3. $\forall \beta,u,v \leq x(\beta \prec \alpha \rightarrow \exists y \leq z\ F_{\beta }^{(v)}(u) \simeq y) \rightarrow \exists y \leq z\ F_{\alpha }(x) \simeq y$.

Here, as usual, $F_{\beta }^{(x)}(x) \simeq y$ abbreviates

$$\displaystyle{\exists s \in Seq\:[(s)_{0} = x \wedge \forall i <x\ F_{\beta }((s)_{i}) \simeq (s)_{i+1} \wedge (s)_{x} = y].}$$

Appendix 2

One can roughly classify the existing definitions of proof-theoretic ordinals in two groups, which I call the definitions “from below” and “from above”. Informally speaking, proof-theoretic ordinals defined from below measure the strength of the principles of certain complexity $\Gamma$ that are provable in a given theory T. In contrast, the ordinals defined from above measure the strength of certain characteristic for T unprovable principles of complexity $\Gamma$. For example, Con(T) is such a characteristic principle of complexity $\Pi _{1}^{0}$.

The standard $\Pi _{1}^{1}$- and $\Pi _{2}^{0}$-ordinals are defined from below, and so are the $\Pi _{n}^{0}$-ordinals introduced in this paper. The notorious ordinal of the shortest natural primitive recursive well-ordering $\prec$ such that $TI_{p.r.}(\prec )$ proves Con(T) (apart from the already discussed feature of logical complexity mismatch) is a typical definition from above.

All the usual definitions of proof-theoretic ordinals can also be reformulated in the form “from above”. Let a natural elementary well-ordering be fixed. For the case of $\Pi _{n}^{0}$-ordinals the corresponding approach would be to let

$$\displaystyle{\vert T\vert _{\Pi _{n}^{0}}^{\vee }:=\min \{\alpha:\mathrm{ EA}^{+} +\mathrm{ RFN}_{ \Pi _{n}}((\mathrm{EA})_{\alpha }^{n}) \vdash \mathrm{ RFN}_{ \Pi _{n}}(T)\}.}$$

(Notice that for n > 1 the theory on the left hand side of $\vdash$ can be replaced by $(\mathrm{EA})_{\alpha +1}^{n}$.)

In a similar manner one can transform the definition of the $\Pi _{2}^{0}$-ordinal via the Fast Growing hierarchy into a definition “from above”. The class of p.t.c.f. of T has a natural indexing, e.g., we can take as indices of a function f the pairs $\langle e,p\rangle$ such that e is the usual Kleene index (= the code of a Turing machine) of f, and p is the code of a T-proof of the $\Pi _{2}^{0}$-sentence expressing the totality of the function {e}. With this natural indexing in mind one can write out a formula defining the universal function φ_T(e, x) for the class of unary functions in $\mathcal{F}(T)$. Then the $\Pi _{2}^{0}$-sentence expressing the totality of φ_T would be the desired characteristic principle. (It is not difficult to show that the totality of φ_T formalized in this way is EA⁺-equivalent to $\mathrm{RFN}_{\Pi _{2}}(T)$.) The $\Pi _{2}^{0}$-ordinal of T can then be defined as follows:

$$\displaystyle{\vert T\vert _{\Pi _{2}^{0}}^{\vee }:=\min \{\alpha:\varphi \in \mathcal{F}_{\alpha +1}\}.}$$

Notice that the proof-theoretic ordinals of T defined “from above” not only depend on the externally taken set of theorems of T, but also on the way T is formalized, that is, essentially on the provability predicate or the proof system for T. For example, in the above definition the universal function φ_T(e, x) depends on the Gödel numbering of proofs in T. In practice, for most of the natural(ly formalized) theories the ordinals defined “from below” and those “from above” coincide:

Proposition 46

If T is EA⁺-provably$\Pi _{n}^{0}$-regular and contains EA⁺, then$\vert T\vert _{\Pi _{n}^{0}}^{\vee } = \vert T\vert _{\Pi _{n}^{0}}$.

Proof

Let $\alpha = \vert T\vert _{\Pi _{n}^{0}}$. By provable regularity,

$$\displaystyle{\mathrm{EA}^{+} \vdash \mathrm{ RFN}_{ \Pi _{n}}(T) \leftrightarrow \mathrm{ RFN}_{\Pi _{n}}((\mathrm{EA})_{\alpha }^{n}),}$$

hence $\mathrm{EA}^{+} + (\mathrm{EA})_{\alpha +1}^{n} \vdash \mathrm{ RFN}_{\Pi _{n}}(T)$. On the other hand, by Gödel’s theorem

$$\displaystyle{\mathrm{EA}^{+} + (\mathrm{EA})_{\alpha }^{n} \subseteq T \nvdash \mathrm{ RFN}_{ \Pi _{n}}(T),}$$

$\boxtimes$

The following example demonstrates that, nonetheless, there are reasonable (and naturally formalized) proof systems for which these ordinals are different, so sometimes the ordinal defined from above bears essential additional information.

Consider some standard formulation of PA, it has a natural provability predicate $\square _{\mathrm{PA}}$. The system PA^∗ is obtained from PA by adding Parikh’s inference rule:

$$\displaystyle{\frac{\square _{\mathrm{PA}}\varphi } {\varphi },}$$

where φ is any sentence. For the reasons of semantical correctness, Parikh’s rule is admissible in PA, so PA^∗ proves the same theorems as PA. However, as is well known, the equivalence of the two systems cannot be established within PA (otherwise, PA^∗ would have a speed-up over PA bounded by a p.t.c.f. in PA, which was disproved by Parikh [24]).⁹ Below we analyze the situation from the point of view of the proof-theoretic ordinals.

Notice that PA^∗ is a reasonable proof system, and it has a natural $\Sigma _{1}$ provability predicate $\square _{\mathrm{PA}^{{\ast}}}$. Lindström [21] proves the following relationship between the provability predicates in PA and PA^∗:

Lemma 47

$\mathrm{EA} \vdash \forall x\ (\square _{\mathrm{PA}^{{\ast}}}(x) \leftrightarrow \exists n\ \square _{\mathrm{PA}}\square _{\mathrm{PA}}^{n}(\dot{x})),$ where $\square _{\mathrm{PA}}^{n}$ means n times iterated $\square _{\mathrm{PA}}$.

Notation: The right hand side of the equivalence should be understood as the result of substituting in the external $\square _{\mathrm{PA}}$ the elementary term for the function $\lambda n,x.\ \ulcorner \square _{\mathrm{PA}}^{n}(\bar{x})\urcorner$.

Proof (Sketch)

The implication $(\leftarrow )$ holds, because PA^∗ is provably closed under Parikh’s rule, that is,

$$\displaystyle{\mathrm{PA} \vdash \square _{\mathrm{PA}}^{n}\varphi \quad \Rightarrow \quad \mathrm{PA}^{{\ast}}\vdash \varphi,}$$

by n applications of the rule, and this is obviously formalizable.

The implication $(\rightarrow )$ holds, because the predicate $\exists n\ \square _{\mathrm{PA}}\square _{\mathrm{PA}}^{n}(\dot{x})$ is provably closed under PA, modus ponens and Parikh’s rule:

$$\displaystyle\begin{array}{rcl} \mathrm{PA} \vdash \varphi & \Rightarrow & \mathrm{PA} \vdash \square _{\mathrm{PA}}^{1}\varphi, {}\\ \mathrm{PA} \vdash \square _{\mathrm{PA}}^{n}\varphi,\quad \mathrm{PA} \vdash \square _{\mathrm{ PA}}^{m}(\varphi \rightarrow \psi )& \Rightarrow & \mathrm{PA} \vdash \square _{\mathrm{ PA}}^{\max (n,m)}\psi, {}\\ \mathrm{PA} \vdash \square _{\mathrm{PA}}^{n}(\square _{\mathrm{ PA}}\varphi )& \Rightarrow & \mathrm{PA} \vdash \square _{\mathrm{PA}}^{n+1}\varphi, {}\\ \end{array}$$

and this is formalizable. $\boxtimes$

Corollary 48

$\mathrm{EA} \vdash \mathrm{ Con}(\mathrm{PA}^{{\ast}}) \leftrightarrow \mathrm{ Con}(\mathrm{PA}_{\omega })$.

Proof

By induction, Con(PA_n) is equivalent to $\neg \square _{\mathrm{PA}}\square _{\mathrm{PA}}^{n}\perp$, moreover this equivalence is formalizable in EA (with n a free variable). Hence, Con(PA_ω) is equivalent to $\forall n\ \neg \square _{\mathrm{PA}}\square _{\mathrm{PA}}^{n}\perp$, which yields the claim by the previous lemma. $\boxtimes$

Applying this to $\Pi _{1}^{0}$-ordinals defined from above, we observe

Proposition 49

• (i) $\vert \mathrm{PA}\vert _{\Pi _{1}^{0}}^{\vee } =\epsilon _{0}$;

• (ii) $\vert \mathrm{PA}^{{\ast}}\vert _{\Pi _{1}^{0}}^{\vee } =\epsilon _{0}\cdot \omega$.

Proof

Statement (i) follows from the EA⁺-provable $\Pi _{1}^{0}$-regularity of PA.

To prove (ii) we first obtain

$$\displaystyle{\mathrm{PA}_{\omega } \equiv _{\Pi _{1}}\mathrm{EA}_{\epsilon _{0}\cdot \omega }}$$

by Proposition 40(i). Formalization of this in EA⁺ yields

$$\displaystyle{\mathrm{EA}^{+} +\mathrm{ EA}_{\epsilon _{ 0}\cdot \omega +1} \vdash \mathrm{ Con}(\mathrm{PA}_{\omega }) \vdash \mathrm{ Con}(\mathrm{PA}^{{\ast}}),}$$

by Corollary 48. By Schmerl’s formula it is also clear that

$$\displaystyle{\mathrm{EA}^{+} +\mathrm{ EA}_{\epsilon _{ 0}\cdot \omega }\subseteq ((\mathrm{EA})_{1}^{2})_{\epsilon _{ 0}\cdot \omega } \nvdash \mathrm{ Con}(\mathrm{PA}^{{\ast}}),}$$

which proves Statement (ii). $\boxtimes$

Observe that the $\Pi _{1}^{0}$-ordinals of PA and PA^∗ defined from below both equal ε₀, because PA^∗ is deductively equivalent to PA. Hence, for PA^∗ the $\Pi _{1}^{0}$-ordinals defined from below and from above are different—this reflects the gap between the power of axioms of this theory and the effectiveness of its proofs.

Despite the fact that, as we have seen, the proof-theoretic ordinals defined from above may have some independent meaning, it seems that those from below are more fundamental and better behaved.

Appendix 3

Here we discuss the role of the metatheory EA⁺ that was taken as basic in this paper. On the one hand, it is the simplest choice, and if one is interested in the analysis of strong systems, there is no reason to worry about it. Yet, if one wants to get meaningful ordinal assignments for theories not containing EA⁺, such as $\mathrm{EA} + I\Pi _{1}^{-}$ or $\mathrm{EA} +\mathrm{ Con}(I\Sigma _{1})$, the problem of weakening the metatheory has to be addressed. For example, somewhat contrary to the intuition, it can be shown (see below) that $\mathrm{EA} +\mathrm{ Con}(I\Sigma _{1})$ is not a $\Pi _{1}^{0}$-regular theory in the usual sense.

These problems can be handled, if one reformulates the hierarchies of iterated consistency assertions using the notion of cut-free provability and formalizes Schmerl’s formulas in EA using cut-free conservativity. Over EA⁺ these notions provably coincide with the usual ones, so they can be considered as reasonable generalizations of the usual notions in the context of the weak arithmetic EA. The idea of using cut-free provability predicates for this kind of problems comes from Wilkie and Paris [39]. Below we briefly sketch this approach and consider some typical examples.

A formula φ is cut-free provable in a theory T (denoted $T \vdash ^{\text{cf}}\varphi$), if there is a finite set T₀ of (closed) axioms of T such that the sequent $\neg T_{0},\varphi$ has a cut-free proof in predicate logic. Similarly, φ is rank k provable, if for some finite $T_{0} \subseteq T$, the sequent $\neg T_{0},\varphi$ has a proof with the ranks of all cut-formulas bounded by k.

If T is elementary presented, its cut-free $\square _{T}^{\text{cf}}$ and rank k provability predicates can be naturally formulated in EA. It is known that EA⁺ proves the equivalence of the ordinary and the cut-free provability predicates. On the other hand, EA can only prove the equivalence of the cut-free and the rank k provability predicates for any fixed k, but not the equivalence of the cut-free and the ordinary provability predicates.

The behavior of $\square _{T}^{\text{cf}}$ in EA is very much similar to that of $\square _{T}$, e.g., $\square _{T}^{\text{cf}}$ satisfies the EA-provable $\Sigma _{1}$-completeness and has the usual provability logic—this essentially follows from the equivalence of the bounded cut-rank and the cut-free provability predicates in EA.

Visser [38], building on the work H. Friedman and P. Pudlák, established a remarkable relationship between the predicates of ordinary and cut-free provability: if T is a finite theory, then¹⁰

$$\displaystyle{ \mathrm{EA} \vdash \forall x\:(\square _{T}\varphi (\dot{x}) \leftrightarrow \square _{\mathrm{EA}}^{\text{cf}}\square _{ T}^{\text{cf}}\varphi (\dot{x})). }$$

(13)

In particular, for EA itself Visser’s formula (13) attains the form

$$\displaystyle{\mathrm{EA} \vdash \forall x\:(\square _{\mathrm{EA}}\varphi (\dot{x}) \leftrightarrow \square _{\mathrm{EA}}^{\text{cf}}\square _{\mathrm{ EA}}^{\text{cf}}\varphi (\dot{x})).}$$

This can be immediately generalized (by reflexive induction) to progressions of iterated cut-free consistency assertions. Let $\mathrm{Con}^{\text{cf}}(T)$ denote $\neg \square _{T}^{\text{cf}}\perp$ and let a nice well-ordering be fixed.

Proposition 50

The following holds provably in EA :

• (i) If$\alpha \prec \omega$, then$\mathrm{EA}_{\alpha } \equiv \mathrm{ Con}^{\text{cf}}(\mathrm{EA})_{2\cdot \alpha };$

• (ii) If α is a limit ordinal, then$\mathrm{EA}_{\alpha } \equiv \mathrm{ Con}^{\text{cf}}(\mathrm{EA})_{\alpha };$

• (iii) If α = ω ⋅β + n + 1, where$\beta \succ 0$and$n \prec \omega$, then

  $$\displaystyle{\mathrm{EA}_{\alpha } \equiv \mathrm{ Con}^{\text{cf}}(\mathrm{EA})_{\omega \cdot \beta +2n+1}.}$$

We omit a straightforward proof by Visser’s formula. We call a theory T$\Pi _{1}^{0}$-cf-regular, if for some α,

$$\displaystyle{ T \equiv _{\Pi _{1}}\mathrm{Con}^{\text{cf}}(\mathrm{EA})_{\alpha }. }$$

(14)

The situation with the cut-free reflection principles of higher arithmetical complexity is even easier.

Proposition 51

Let T be a finite extension of EA . Then for any n > 1,

$$\displaystyle{\mathrm{EA} \vdash \mathrm{ RFN}_{\Pi _{n}}^{\text{cf}}(T) \leftrightarrow \mathrm{ RFN}_{ \Pi _{n}}(T).}$$

Proof

We only show the implication $(\rightarrow )$, the opposite one is obvious. For any $\varphi (x) \in \Pi _{n}$ the formula $\square _{T}\varphi (\dot{x})$ implies $\square _{T}^{\text{cf}}\square _{T}^{\text{cf}}\varphi (\dot{x})$. Applying $\mathrm{RFN}_{\Pi _{n}}^{\text{cf}}(T)$ twice yields φ(x). $\boxtimes$

This equivalence carries over to the iterated principles. Let $(T)_{\alpha }^{n,\text{cf}}$ denote $\mathrm{RFN}_{\Pi _{n}}^{\text{cf}}(T)_{\alpha }$. Using the fact that for successor ordinals α the theories $(T)_{\alpha }^{n,\text{cf}}$ are finitely axiomatizable we obtain by reflexive induction in EA using Proposition 51 for the induction step:

Proposition 52

For any n > 1, provably in EA,

$$\displaystyle{\forall \alpha \ \ (T)_{\alpha }^{n,\text{cf}} \equiv (T)_{\alpha }^{n}.}$$

We say that T is cut-free$\Pi _{n}$-conservative over U, if for every $\varphi \in \Pi _{n}$, $T \vdash ^{\text{cf}}\varphi$ implies $U \vdash ^{\text{cf}}\varphi$. Let $T \equiv _{\Pi _{n}}^{\text{cf}}U$ denote a natural formalization in EA of the mutual cut-free $\Pi _{n}$-conservativity of T and U. Externally $\equiv _{\Pi _{n}}^{\text{cf}}$ is the same as $\equiv _{\Pi _{n}}$, so the difference between the two notions only makes sense in formalized contexts.

Analysis of the proof of Schmerl’s formula reveals that we deal with an elementary transformation of a cut-free derivation into a derivation of a bounded cut-rank. To see this, the reader is invited to check the ingredient proofs of Theorem 2 and Propositions 19 and 25. All these elementary proof transformations are verifiable in EA, which yields the following formalized variant of Schmerl’s formula (we leave out all the details).

Proposition 53

For all n ≥ 1, if T is an elementary presented $\Pi _{n+1}$ -axiomatized extension of EA, the following holds provably in EA:

$$\displaystyle{\forall \alpha \succeq 1\ (T)_{\alpha }^{n+m} \equiv _{ \Pi _{n}}^{\text{cf}}(T)_{\omega _{ m}(\alpha )}^{n}}$$

We notice that this relationship holds for the ordinary as well as for the cut-free reflection principles, because, even for n = 0, the ordinal on the right hand side of the equivalence is a limit (if m > 0).

Now we consider a few examples. The following proposition shows that the theory $\mathrm{EA} + I\Pi _{1}^{-}$ is both $\Pi _{1}^{0}$-cf-regular and $\Pi _{1}^{0}$-regular with the ordinal ω.

Proposition 54

Provably in EA,

$$\displaystyle{\mathrm{EA} + I\Pi _{1}^{-}\equiv _{ \Pi _{1}}\mathrm{Con}^{\text{cf}}(\mathrm{EA})_{\omega } \equiv _{ \Pi _{1}}\mathrm{EA}_{\omega }.}$$

Proof

The logics of the ordinary and the cut-free provability for EA coincide, so by the usual proof the schema of local reflection w.r.t. the cut-free provability is $\Pi _{1}$-conservative over $\mathrm{Con}^{\text{cf}}(\mathrm{EA})_{\omega }$. But the former contains $\mathrm{EA} + I\Pi _{1}^{-}$ by the comment following (E3)(c), moreover this inclusion is easily formalizable in EA (both in the usual and in the cut-free version). This proves the first equivalence. The second one follows from Proposition 50(ii). $\boxtimes$

Consider the theories $\mathrm{EA} +\mathrm{ Con}^{\text{cf}}(I\Sigma _{1})$ and $\mathrm{EA} +\mathrm{ Con}(I\Sigma _{1})$. We show that the first one is $\Pi _{1}^{0}$-cf-regular with the ordinal ω^ω + 1, and the second is $\Pi _{1}^{0}$-cf-regular with the ordinal ω^ω + 2. Notice, however, that only the first theory is $\Pi _{1}^{0}$-regular in the usual sense: by Proposition 50(iii), $\mathrm{EA}_{\omega ^{\omega }+1} \equiv \mathrm{ Con}^{\text{cf}}(\mathrm{EA})_{\omega ^{\omega }+1}$, whereas

$$\displaystyle{\mathrm{EA}_{\omega ^{\omega }+2} \equiv \mathrm{ Con}^{\text{cf}}(\mathrm{EA})_{\omega ^{\omega }+3}\not\equiv \mathrm{Con}^{\text{cf}}(\mathrm{EA})_{\omega ^{\omega }+2}.}$$

Proposition 55

Provably in EA ,

• (i) $\mathrm{EA} +\mathrm{ Con}^{\text{cf}}(I\Sigma _{1}) \equiv \mathrm{ Con}^{\text{cf}}(\mathrm{EA})_{\omega ^{\omega }+1}$;

• (ii) $\mathrm{EA} +\mathrm{ Con}(I\Sigma _{1}) \equiv \mathrm{ Con}^{\text{cf}}(\mathrm{EA})_{\omega ^{\omega }+2}$.

Proof

Part (i) follows from the (obvious) formalizability of the equivalence of $I\Sigma _{1}$ and $(\mathrm{EA})_{1}^{3}$ in EA and Proposition 53. One can show that the two systems are also cut-free equivalent, provably in EA.

To prove Part (ii) recall that by Visser’s formula (13), provably in EA, $\mathrm{Con}(I\Sigma _{1})$ is equivalent to $\mathrm{Con}^{\text{cf}}(\mathrm{EA} +\mathrm{ Con}^{\text{cf}}(I\Sigma _{1}))$, and hence to $\mathrm{Con}^{\text{cf}}(\mathrm{EA})_{\omega ^{\omega }+2}$ by Part (i). $\boxtimes$

The following facts are also worth noticing. Statement (i) below implies that $I\Sigma _{n}$ is not EA-provably $\Pi _{1}^{0}$-regular (and thus the original Schmerl’s formula is not formalizable in EA). In contrast, Statement (ii) implies that incidentally PA itself is EA-provably $\Pi _{1}^{0}$-regular.

Proposition 56

• (i) $\mathrm{EA} \nvdash \mathrm{ Con}(\mathrm{EA}_{\omega _{n+1}}) \rightarrow \mathrm{ Con}(I\Sigma _{n})$;

• (ii) $\mathrm{PA} \equiv _{\Pi _{1}}\mathrm{EA}_{\epsilon _{0}}$, provably in EA, hence$\mathrm{EA} \vdash \mathrm{ Con}(\mathrm{EA}_{\epsilon _{0}}) \rightarrow \mathrm{ Con}(\mathrm{PA})$.

Proof

Fact (i) has just been proved for $I\Sigma _{1}$: $\mathrm{Con}(I\Sigma _{n})$ is $\Pi _{1}^{0}$-conservative over $\mathrm{Con}^{\text{cf}}(\mathrm{EA})_{\omega _{n+1}+2}$, whereas $\mathrm{Con}(\mathrm{EA}_{\omega _{n+1}})$ is equivalent to $\mathrm{Con}^{\text{cf}}(\mathrm{EA})_{\omega _{n+1}+1}$ by Proposition 50(iii). Hence, (i) follows by Löb’s principle for the cut-free provability.

To prove (ii) we formalize the following reasoning in EA: Assume $\pi \in \Pi _{1}$ and $\mathrm{PA} \vdash \pi$. Then for some n, $I\Sigma _{n} \vdash \pi$. Since $I\Sigma _{n}$ is finitely axiomatized, by (13) we obtain that $\exists n\ \mathrm{EA} \vdash ^{\text{cf}}\square _{I\Sigma _{n}}^{\text{cf}}\pi$, therefore by Proposition 53

$$\displaystyle{\exists n\ \mathrm{EA} \vdash ^{\text{cf}}\square _{\mathrm{EA}_{\omega _{ n+1}}}^{\text{cf}}\pi,}$$

which can be weakened to

$$\displaystyle{ \exists n\ \mathrm{EA}_{\epsilon _{0}} \vdash \square _{\mathrm{EA}_{\omega _{ n+1}}}^{\text{cf}}\pi. }$$

(15)

On the other hand, we notice that (provably in EA) for every fixed $\beta \prec \epsilon _{0}$ and $\pi \in \Pi _{1}$,

$$\displaystyle{\mathrm{EA}_{\epsilon _{0}} \vdash \square _{\mathrm{EA}_{\beta }}^{\text{cf}}\pi \rightarrow \pi,}$$

by the cut-free version of the $\Sigma _{1}$-completeness principle, and applying this to (15) yields $\mathrm{EA}_{\epsilon _{ 0}} \vdash \pi$. $\boxtimes$

Corollary 57

$\vert \mathrm{EA} +\mathrm{ Con}(\mathrm{PA})\vert _{\Pi _{1}^{0}} =\epsilon _{0} + 1$.