Semantics for Provability Logic

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Semantics for Provability Logic

Selma Boeke

July 13, 2017

Bachelor Thesis Mathematics

Supervisor: dr. Paula Henk and dr. Nick Bezhanishvili

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Abstract

In this thesis we overview the relational and topological semantics of the G¨odel-L¨ob provability logic GL and the bimodal provability logic GLB, as well as their arith-metical interpretations. GL is a system of propositional modal logic which is sound an complete with respect to transitive, converse well-founded Kripke frames. If we in-terpret the modality of GL as the provability predicate of Peano Arithmetic PA, then GL is arithmetically sound and complete with respect to PA. The bimodal provability logic GLB is the system where a second GL modality is added to GL. GLB is also arithmetically sound and complete, when the second modality is interpreted as a certain stronger provability predicate in PA. However, unlike GL, GLB is not complete with respect to any class of Kripke frames. In fact, there are no non-trivial Kripke frames of GLB. Because of this, in the last part of the thesis we turn to topological semantics of provability logic. We show that GL is sound and complete with respect to the class of scattered spaces. We will also examine the bitopological space ω + 1 where τ1 is the

upset topology and τ2 the interval topology. We show that this bitopological space is a

(non-trivial) model of GLB. While we show that GLB is not complete with respect to this space, by a recent result of Beklemishev and Gabelaia it is topologically complete.

Title: Semantics for Provability Logic

Authors: Selma Boeke, selmaboeke@hotmail.com, 10508236 Supervisor: dr. Paula Henk and dr. Nick Bezhanishvili Second grader: prof. dr. Yde Venema

Date: July 13, 2017

Korteweg-de Vries Instituut voor Wiskunde Universiteit van Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.science.uva.nl/math

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1 Introduction

The language of modal logic results from adding the modal operator2 to the language of propositional logic. The weakest normal modal logic is K. Gödel-Löb provability logic GL is a system of modal logic where the so-called Löb’s axiom

2(2p → p) → 2p (1) is added to K. We will first look at the Kripke semantics. A Kripke frame consists of a set W of possible worlds and a binary relation R on W. If p is true in all worlds that are accessible from a world w, then 2p is true in w. We will see that GL characterizes the class of transitive, converse well-founded frames. The classes of frames that L¨ob’s rule

2ϕ → ϕ/ϕ (2)

characterizes will also be studied. GL is important due to its relation to Peano Arith-metic. This relation was established by Solovay and will be discussed in the thesis.

Peano Arithmetic, PA, is a first-order theory in which we can prove statements about the natural numbers N. If there is a proof of ϕ in PA, we write PA ` ϕ. PA is sound: PA ` ϕ ⇒ N |= ϕ. (3) As N does not have any contradictions, such as 0 = 1, it follows from (3) that PA does not prove any contradiction either, we write PA 6` ⊥. Now, we wonder if the other direction of (3) also holds. Does PA prove every true statement in N? By G¨odel’s First Incompleteness Theorem, this is not the case: There are statements about the natural numbers such that PA can neither prove these statements, nor their negations. Since one of these statements must be true, this means that PA cannot prove every true statement. G¨odel’s Second Incompleteness Theorem gives an example of such a statement, namely the consistency of PA. To formulate this theorem, we need the so-called provability predicate Pr(x), which expresses facts about provability in PA. We will see that the axioms and rules of GL can be used, when proving statements about the provability predicate in PA. Solovay proved that GL captures in fact everything that PA can prove about its provability predicate. We say that GL is arithmetically sound and complete with respect to PA.

We can add an extra modality to GL and interpret this as a certain stronger prov-ability predicate in PA. This system is called bimodal provprov-ability logic GLB and is arithmetically sound and complete. However, in contrast to GL, GLB is not complete

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with respect to any class of Kripke frames. There are in fact no non-trivial Kripke frames of GLB.

An alternative way to study the semantics of GLB, as well as of GL, is to look at the topological semantics. If we interpret the dual of 2, 3, as the set of limit points, then on a topological space the axioms of GL are valid if and only if the space is scattered. This means that every nonempty subset of the space has an isolated point.

As for GLB, we will examine an example of a space with two topologies. Namely (ω + 1, τ1, τ2), where τ1 is the upset topology and τ2 is the interval topology. It turns

out that GLB is sound with respect to this space. However, the so called linearity axiom also holds on (ω + 1, τ1, τ2). Therefore, GLB is not complete with respect to this

topological space. Despite this, GLB is in fact topologically complete.

In summary, in this thesis we study two important systems of provability logic: GL and GLB. Their Kripke and topological semantics are thoroughly discussed, as well as their relation to PA.

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2 Provability logic

In this chapter we will study the syntax and semantics of the G¨odel-L¨ob provability logic GL and bimodal provability logic GLB. In order to do so, we will first examine basic modal logics.

2.1 Modal logics

We will begin by discussing the syntax of modal logic. After that, we will continue with the semantics.

2.1.1 Syntax

We assume the reader to be familiar with propositional logic. The language of propo-sitional modal logic, L₂, is an extension of the language of propositional logic with a unary operator2. The operator has different interpretations in various systems of modal logic. For example, in provability logic2ϕ is read as ‘ϕ is provable’ whereas in epistemic logic it is interpreted as ‘it is known that ϕ’.

Modal logic is a useful tool for describing and working with relational structures and is widespread in artificial intelligence, mathematics and computer science.

Definition 2.1.1. The set of well-formed formulas of L₂ is constructed in the following way:

1. The propositional variables are well-formed formulas.

2. If ϕ and ψ are well-formed formulas, then so are (ϕ ∨ ψ) and ¬ψ. 3. If ϕ is a well-formed formula, then so is2ϕ.

We use the following standard abbreviations: ¬(¬ϕ ∨ ¬ψ) := (ϕ ∧ ψ), (¬ϕ ∨ ψ) := (ϕ → ψ), (¬ϕ ∨ ψ) ∧ (ϕ ∨ ¬ψ) := (ϕ ↔ ψ) and3ϕ := ¬2¬ϕ.

Definition 2.1.2. The minimal normal modal logic K is given by all formulas in L₂ with the form of a tautology of propositional logic, and by the following axiom and derivation rules:

1. K-Axiom: all formulas of the form2(ϕ → ψ) → (2ϕ → 2ψ). 2. Modus Ponens: If ϕ and ϕ → ψ, then ψ.

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Definition 2.1.3. We say that ϕ is derivable in K, and write K ` ϕ, if there is a sequence of formulas of which ϕ is the last such that every formula in the sequence is either an axiom of K or else is derived by means of one of the rules from previous formulas in the sequence.

As usual we use ϕ1...ϕn/ψ as shorthand for: if ` ϕi for all i ≤ n, then ` ψ.

We will now give some examples of derivations in K. Example 2.1. ϕ → ψ, ψ → χ/ϕ → χ

1. ϕ → ψ Premise 2. ψ → χ Premise 3. (ϕ → ψ) → ((ψ → χ) → (ϕ → χ)) Tautology

4. (ψ → χ) → (ϕ → χ) Modus Ponens (MP) on 1 and 3 5. ϕ → χ Modus Ponens on 2 and 4 Example 2.2. ϕ → ψ/2ϕ → 2ψ

1. ϕ → ψ Premise

2. 2(ϕ → ψ) Necessitation on 1 3. 2(ϕ → ψ) → (2ϕ → 2ψ) K-Axiom

4. 2ϕ → 2ψ Modus Ponens on 2 and 3 Proposition 2.1.4. K `2(ϕ ∧ ψ) ↔ (2ϕ ∧ 2ψ)

Proof. Left to right:

1. (ϕ ∧ ψ) → ϕ Tautology

2. 2(ϕ ∧ ψ) → 2ϕ Example 2.2 on 1

We can use the same format to show that2(ϕ ∧ ψ) → 2ψ. It follows from propositional logic that 2(ϕ ∧ ψ) → (2ϕ ∧ 2ψ). Right to left: 1. ϕ → (ψ → (ϕ ∧ ψ)) Tautology 2. 2ϕ → 2(ψ → (ϕ ∧ ψ)) Example 2.2 on 1 3. 2(ψ → (ϕ ∧ ψ)) → (2ψ → 2(ϕ ∧ ψ)) K-Axiom 4. 2ϕ → (2ψ → 2(ϕ ∧ ψ)) Example 2.1 on 2 and 3 5. 2ϕ ∧ 2ψ → 2(ϕ ∧ ψ) Propositional logic on 4

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For a sentence ϕ, derivability in K + ϕ is the same as in Definition 2.1.3, except we now have ϕ as an extra axiom at our disposal. If S = K + ϕ and ψ is derivable in S, we write S ` ψ.

The following modal logics will be used later on: K4 = K +2p → 22p

GL = K +2(2p → p) → 2p.

The axiom2p → 22p is called Axiom 4 and 2(2p → p) → 2p is called Löb’s axiom. GL is shorthand for Gödel and Löb and represents the provability logic. We will come back to this modal logic in the next section.

2.1.2 Kripke semantics

After having defined the syntax, we move on to the Kripke semantics for modal logic. Definition 2.1.5. A frame F is a tuple hW, Ri where

1. W is a non-empty set of possible worlds,

2. R ⊆ W × W is a binary relation that we call the accessibility relation. For w, v ∈ W, if wRv we say that v is accessible from w.

Definition 2.1.6. A model M is a triple hW, R, |=i where 1. hW, Ri is a frame,

2. |= is a relation (a valuation) between the worlds of W and propositional variables. We use the following conditions to extend ‘|=’ to a relation between worlds and all modal formulas:

a) w |= ¬ϕ ⇔ w 6|= ϕ,

b) w |= ϕ ∨ ψ ⇔ w |= ϕ or w |= ψ,

c) w |=2ϕ ⇔ for all w0 such that wRw0, w0 |= ϕ.

It follows from the definition of3 that w |= 3ϕ if and only if there is a w0 with wRw0 such that w0 |= ϕ.

If w |= ϕ we say that ϕ is true in w.

Definition 2.1.7. The validity of ϕ can be defined in three different ways:

1. M |= ϕ if and only if for all w ∈ W, w |= ϕ. We say that ϕ is valid in the model M.

2. F |= ϕ if and only if for all models M based on F , M |= ϕ. We say that ϕ is valid on F .

3. C |= ϕ if and only if for all F ∈ C, F |= ϕ. We say that ϕ is valid in the class of frames C.

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Below, we will work with validity on classes of frames. We will discuss an example of a modal formula that corresponds to a class of frames with a certain property.

Definition 2.1.8. A set of formulas ∆ characterizes a class of frames C if and only if C = {F | F |= ϕ for all ϕ ∈ ∆}.

Theorem 2.1.9. 2p → 22p characterizes the class of transitive frames.

Proof. Let F = hW, Ri. Assume F |=2p → 22p. We want to show that R is transitive. Take any world w ∈ W and suppose that wRx and xRy. Let ‘|=’ be the valuation such that for all z ∈ W, z |= p if and only if wRz. Because of this valuation we get w |=2p. Since we have assumed that2p → 22p is valid on F, it follows that w |= 22p. Because wRx we have x |=2p and because of xRy we get y |= p. Finally, by our choice of the valuation, we have wRy. Therefore, R is transitive.

For the other direction, suppose that R is transitive and let ‘|=’ be any valuation. Assume that w |= 2p. To show that w |= 22p, take any successor x of w, so we have wRx. If xRy, then it follows by transitivity that wRy. Because of our assumption that w |=2p we have y |= p. So x |= 2p because xRy. Finally, as a consequence of wRx, we get w |=22p. So it follows that F |= 2p → 22p. We now can distinguish validity ‘|=’ from derivability ‘`’. It turns out to be the case that derivability in K coincides with validity on all frames. K is sound: if ϕ is derivable in K, then ϕ is valid in the class of all frames. Also, K is complete: if ϕ is valid on all frames, then there is a proof of ϕ in K. Soundness and completeness also hold for K4 with respect to the class of transitive frames.

Theorem 2.1.10.

1. K is sound and complete with respect to the class of all Kripke frames.

2. K4 is sound and complete with respect to the class of transitive Kripke frames. Soundness of K can be proven with induction in a straightforward way [10, Thm. 3]. Theorem 2.1.9 tells us that if we have a transitive frame F , then Axiom 4 is valid on F . From this, soundness of K4 follows. As both K and K4 are canonical, completeness of these systems is proven by constructing the canonical model [10, p. 34].

2.2 G¨

odel-L¨

ob provability logic

The provability logic GL, named after logicians Gödel and Löb, is the modal logic where Löb’s axiom is added to K:

GL = K +2(2p → p) → 2p. In this section we will discuss the syntax and semantics of GL.

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2.2.1 Syntax

We will now show the well-known fact that GL contains K4. We will do this by deriving Axiom 4 in GL [4, Thm. 18].

Lemma 2.2.1. GL `2ϕ → 22ϕ

Proof. 1. ϕ → ((22ϕ ∧ 2ϕ) → (2ϕ ∧ ϕ)) Tautology

2. ϕ → (2(2ϕ ∧ ϕ) → (2ϕ ∧ ϕ)) Proposition 2.1.4 and Example 2.1 3. 2ϕ → 2(2(2ϕ ∧ ϕ) → (2ϕ ∧ ϕ)) Example 2.2 on 2 4. 2(2(2ϕ ∧ ϕ) → (2ϕ ∧ ϕ)) → 2(2ϕ ∧ ϕ) L¨ob’s axiom 5. 2ϕ → 2(2ϕ ∧ ϕ) Example 2.1 on 3 and 4 6. 2(2ϕ ∧ ϕ) → (22ϕ ∧ 2ϕ) Proposition 2.1.4 7. 2ϕ → (22ϕ ∧ 2ϕ) Example 2.1 on 5 and 6 8. 2ϕ → 22ϕ Propositional logic on 7 We can conclude that GL contains all the theorems of K4.

2.2.2 Kripke semantics

We will now discuss the Kripke semantics of GL.

Definition 2.2.2. A frame F is converse well-founded if and only if for all nonempty X ⊆ W, there exists a w ∈ X such that wRz for no z ∈ X. We call w the R-greatest element.

In other words, F is converse well-founded if and only if it does not contain an infinite, ascending chain.

Lemma 2.2.3. Any converse well-founded frame F is irreflexive.

Proof. Suppose that F is not irreflexive, then there is a w ∈ W such that wRw. Then the set {w} has no R-greatest element as w ∈ {w} and wRw. This contradicts converse well-foundedness, so F must be irreflexive. Theorem 2.2.4. L¨ob’s axiom characterizes the class of transitive, converse well-founded frames.

Proof. First suppose that F |=2(2p → p) → 2p. We want to show that R is transitive and converse well-founded. Let us assume that R is not transitive, so there are w, x, y ∈ W such that wRx and xRy but not wRy. Let ‘|=’ be the valuation such that p is true in every world except for x and y. Let z be a successor of w, we distinguish three cases. If z = x, then z |= ¬p. If z 6= x and neither zRy nor zRx, then z |=2p. At last, if z 6= x and zRy or zRx, then z |= (¬2p ∧ p). Therefore, for all z ∈ W such that wRz we have z |= ¬(2p ∧ ¬p), which is equivalent to z |= 2p → p. Hence, we get w |= 2(2p → p).

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But because x 6|= p we have w 6|= 2p, which contradicts L¨ob’s axiom. So R has to be transitive.

Now suppose that R is not converse well-founded, so there is some X ⊆ W that does not have an R-greatest element. Let ‘|=’ be the valuation such that for all v ∈ W, v |= p if and only if v 6∈ X. Let w ∈ X be arbitrary and suppose that wRx and x 6|= p. By our choice of the valuation, we have that x ∈ X. Because there is no R-greatest element in X, there is a y ∈ X such that xRy and y 6|= p. Hence, x 6|= 2p. Because we already had x 6|= p we now get x |= ¬p → ¬2p, which is equivalent to x |= 2p → p. For all wRz such that z 6∈ X, z |= p by choice of our valuation, so also z |= 2p → p. Hence, w |=2(2p → p). But, because x 6|= p we have w 6|= 2p which contradicts the assumption that L¨ob’s axiom was valid on F . Therefore, R is converse well-founded.

For the other direction, suppose that R is transitive and converse well-founded. We will prove that L¨ob’s axiom is valid on F by contraposition. Assume that w 6|= 2p. From this, it follows that the set X = {x ∈ W : wRx and x 6|= p} is nonempty. Since R is converse well-founded, there is an R-greatest element y ∈ X, so if a ∈ W and yRa then a 6∈ X. We want to show that y |=2p. Let a be any successor of y (note that if y does not have a successor, then 2p is always true in y). Now, wRy and yRa, so by transitivity we get wRa with a |= p, because a 6∈ X. Since a is an arbitrary successor of y, we get y |= 2p. It follows by propositional logic that y 6|= 2p → p, and hence w 6|=2(2p → p). We have shown that w |= ¬2p → ¬2(2p → p), which is equivalent to

w |=2(2p → p) → 2p.

If F is transitive and converse well-founded, we say that F is a GL frame.

Theorem 2.2.5. GL is sound and complete with respect to the class of transitive, converse well-founded frames.

Theorem 2.2.4 tells us that if L¨ob’s axiom is valid on F , then F is transitive and converse well-founded. From this, soundness of GL follows. As for completeness of GL, the proof does not have the same format as for K and K4. Namely, GL is not compact and is therefore not canonical [3, p. 211]. The completeness of GL was first proven by Segerberg [12].

Characterization by a rule We have seen that modal formulas can characterize a class of frames. In this paragraph we will look at a rule, instead of an axiom, that characterizes a class of frames.

Definition 2.2.6. A rule ϕ/ψ characterizes a certain class of frames C if and only if ∀F ∈ C and ∀M based on F , if M |= ϕ then M |= ψ.

The following rule

2ϕ → ϕ/ϕ

is called L¨ob’s rule (LR). The theorem below tells us that L¨ob’s rule characterizes the class of converse well-founded frames.

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Theorem 2.2.7. Löb’s rule characterizes the class of converse well-founded frames. Proof. First suppose that Löb’s rule is valid on F . We want to show that R is converse well-founded. Let us assume that R is not converse well-founded, so there is some X ⊆ W that does not have an R-greatest element. Let ‘|=’ be the valuation such that for all a ∈ W, a |= p if and only if a 6∈ X. Take w ∈ X, so w |= ¬p. Because there is no R-greatest element there is a v ∈ X such that wRv, so v |= ¬p and therefore w |= ¬2p. We now have w |= ¬p → ¬2p which is equivalent to w |= 2p → p. As w was chosen arbitrary in X, 2p → p holds for every world in X. Because of the valuation we have for every world z 6∈ X that z |= p, and therefore also z |= 2p → p. Now, for every world y ∈ W we have y |=2p → p. By Löb’s rule, it follows that for all y ∈ W, y |= p. But by the choice of our valuation, p is not true in any world of X, so this leads to a contradiction. Therefore, R is converse well-founded.

For the other direction, suppose that R is converse well-founded and let ‘|=’ be any valuation. Now assume that 2p → p is true in every world, but p is not true in every world. Then, by converse well-foundedness there is an R-greatest element w such that w 6|= p and for all y if wRy, then y |= p. Now we have w |=2p and therefore w |= ¬p∧2p, which is equivalent to w |= ¬(2p → p) by propositional logic. However, this contradicts the assumption that2p → p is true in every world. We can conclude that L¨ob’s rule is

valid on F .

From this theorem, it follows that a transitive frame on which Löb’s rule is valid, is a GL frame. Therefore, the system where Löb’s rule is added to K4 (K4+LR) characterizes the same class of frames as GL. Löb’s axiom is thus semantically equivalent to Axiom 4 and Löb’s rule. Moreover, GL and K4+LR also contain the same theorems: they are syntactically equivalent [4, p. 59].

2.3 Bimodal provability logic GLB

In the following section we will discuss a certain system of bimodal provability logic, GLB, where a second modality, 4, is added to the language of propositional modal logic L₂.

Definition 2.3.1. GLB has the same axioms and inference rules as GL for 2 and 4 separately, and furthermore:

1. 2ϕ → 4ϕ, 2. 3ϕ → 43ϕ.

We will show below that GLB is not complete with respect to any class of Kripke frames.

A frame for a modal logic with two modalities is of the form hW, R, Si where W is a nonempty set and R and S are binary relations. R is used as a relation to interpret 2 and S as a relation to interpret 4. We have that w |= 4p if and only if for all x with wSx, x |= p. Furthermore, w |=2p is defined in the same way as in Definition 2.1.6. We denote bimodal models as hW, R, S, |=i.

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Lemma 2.3.2.

1. 2ϕ → 4ϕ characterizes the class of frames where S ⊆ R.

2. 3ϕ → 43ϕ characterizes the class of frames where for all w, x, y ∈ W, if wSx and wRy, then xRy.

Proof. 1. Let us first assume that 2p → 4p is valid and wSx. We want to show that wRx. Let ‘|=’ be the valuation such that p is valid in every world except for x, so we have x 6|= p. Because wSx it follows that w |= ¬4p. From our assumption that 2p → 4p is valid, we get w |= ¬2p. By the choice of our valuation it follows that wRx.

For the other direction, we assume that for all w, x ∈ W if wSx then wRx. By contraposition we will show that 2p → 4p is valid. Suppose that w |= ¬4p. This means that there is an x ∈ W such that wSx and x |= ¬p. By our assumption, we get wRx. But as x |= ¬p, it follows that w |= ¬2p.

2. First assume that 3ϕ → 43p is valid and suppose that wSx and wRy. Let ‘|=’ be the valuation such that p is only true in y, so we have y |= p. Because wRy, we have w |=3p. From our assumption it follows that w |= 43p. As wSx, we have x |= 3p. By the choice of the valuation it follows that xRy.

For the other direction we assume that for all w, x, y ∈ W, if wSx and wRy, then xRy. Fix w and x such that w |= 3p and wSx. Now, because w |= 3p, there is a y such that wRy and y |= p. As we also have that wSx, it follows from our assumption that xRy, and therefore x |=3p. Because wSx together with the fact that x is chosen arbitrary, we get w |= 43p. Furthermore, we know from Lemma 2.2.3 that if L¨ob’s axiom is valid on a frame hW, Ri, then R is irreflexive.

Theorem 2.3.3. GLB is not sound and complete with respect to any class of Kripke frames.

Proof. We will first show by contradiction that if all the axioms of GLB are valid on a frame hW, R, Si, then S is the empty relation. Let us assume that all the axioms of GLB are valid and that there are w, x ∈ W such that wSx. Then we know from Lemma 2.3.2 (1) that wRx, and from Lemma 2.3.2 (2) we get that xRx, which contradicts the fact that R is irreflexive. It follows that 4⊥ is valid on all frames that make all GLB axioms valid. However, we will see in the last paragraph of the next chapter (p. 19) that GLB 6` 4⊥. Therefore, GLB cannot be Kripke complete. We will see in Chapter 4 that GLB is complete with respect to a topological semantics.

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3 Arithmetical interpretations of

provability logic

In this chapter we will discuss the system of Peano Arithmetic and its relation to GL and GLB.

3.1 Peano Arithmetic

The language of PA, LPA, is a first-order language, where the non-logical symbols are

(0,S,≤, +, ·). Here, 0 is a constant symbol and S is a function symbol that is interpreted as the successor function on the natural numbers N. Furthermore, ≤ is the less than relation, + stands for addition and · stands for multiplication. The numeral of a number n, n, is given by the constant 0 followed by n occurrences of the successor function. If ϕ is a theorem of PA, we write PA ` ϕ.

The formula (∃x ≤ y)ϕ is an abbreviation for ∃x(x ≤ y ∧ ϕ), and (∀x ≤ y)ϕ is an abbreviation for ∀x(x ≤ y → ϕ), where x and y are distinct variables [7, Para. 0.30]. A ∆0-formula is an arithmetical sentence that has only bounded quantifiers, i.e. occur in

a context as above. If an arithmetical formula is of the form ∀xϕ, where ϕ is ∆0, we call

it a Π1-formula. A Σ1-formula is of the form ∃xϕ where ϕ is ∆0.

Definition 3.1.1. The system of PA contains the axioms and rules of predicate logic. We assume an axiomatization with Modus Ponens as the only rule of inference [6, Sec. 2]. Furthermore, it has the following axioms:

1. ∀x(Sx 6= 0) 2. ∀x, y(Sx = Sy → x = y) 3. ∀x(x + 0 = x) 4. ∀x, y(x + Sy = S(x + y)) 5. ∀x(x · 0 = 0) 6. ∀x, y(x · Sy = x · y + x)

7. For each formula α ∈ LPA, (α(0) ∧ ∀x(α(x) → α(Sx))) → ∀xα(x)

Axiom 7 is called the induction axiom and is a not a single axiom, but a schema. Therefore, PA has infinitely many axioms.

The rules and axioms of PA are chosen in such a way that if a statement is provable in PA, then it is true in the natural numbers. We say that PA is sound and write: if

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PA ` ϕ, then N |= ϕ. Now, we wonder if the other direction also holds: Does PA prove all true statements about N? By G¨odel’s First Incompleteness Theorem, this is not the case.

Theorem 3.1.2 (G¨odel’s First Incompleteness Theorem). If PA is consistent, there is a ϕ ∈ LPA such that:

PA 6` ϕ and PA 6` ¬ϕ

Every sentence ϕ in the language of PA is about the natural numbers. G¨odel’s First Incompleteness Theorem tells us that there is ϕ, which is thus a statement about N, such that PA can neither prove ϕ, nor its negation. However, one of the cases must be true about the natural numbers. Therefore, there is a true statement that is not provable.

For G¨odel’s Second Incompleteness Theorem, we need to introduce a formula that ex-presses facts about provability in PA. Through the method of arithmetization developed by G¨odel we assign natural numbers to the formulas of LPA in a specific way. We write

pϕq (G¨odel number) for the code of the formula ϕ.

A small example of arithmetization of propositional logic is as follows: the connectives have fixed codes and the propositional variables are assigned to uneven numbers. In order to construct the code of a formula, prime numbers are raised to the power of the codes of these connectives or propositional variables. For example, suppose that ppq = 1, pqq = 3 and p→ q = 6. Then p → q is coded by 21· 36_{· 5}3_.

Via the coding we can construct the so-called provability predicate Pr(x). For a formula ϕ with G¨odel number pϕq, Pr(pϕq) expresses in PA that ϕ is provable in PA. We have:

PA ` ϕ ⇔ PA ` Pr(pϕq).

In order to define the provability predicate, we first have to introduce the proof predicate Prf(y, x). This is defined as a formula saying that there is a finite sequence y = (y1, .., yn)

such that yn= x and for all i ≤ n, yi is one of the following possibilities:

1. The code of an axiom in PA. 2. An axiom in predicate logic.

3. There are j, k ≤ i with yk = pyj → yiq where yi and yj are codes of sentences,

and yk is the code of the implication. So yi is obtained from previous elements by

Modus Ponens.

Prf(y, x) formalizes that y is a proof of x. Being a theorem of PA means that there is a proof of this theorem in PA. Therefore, the provability predicate is defined in the following way: Pr(x) := ∃yPrf(y, x). The provability predicate is a Σ1-formula [8, p. 20].

The following conditions concerning the provability predicate are derived by Hilbert and Bernays from Gödel’s work and are of great importance when proving Gödel’s Second Incompleteness Theorem. Löb simplified these conditions and they are now known as the Hilbert-Bernays-Löb (HBL) derivability conditions. From now on, we will write ϕ

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(H1) If PA ` ϕ, then PA ` ϕ (H2) PA ` (ϕ → ψ) → (ϕ → ψ) (H3) PA ` ϕ → ϕ

Together with the conditions above, the following theorem is needed to prove Gödel’s Second Incompleteness Theorem. Moreover, it is also used to prove Gödel’s First In-completeness Theorem. Carnap isolated the theorem from Gödel’s work and formulated it in the so-called Diagonalization lemma [5].

Theorem 3.1.3 (Diagonalization Lemma). For each arithmetical formula ϕ(x), with x being the only free variable, there is an arithmetical sentence ψ, such that

PA ` ψ ↔ ϕ(pψq).

For Gödel’s First Incompleteness Theorem, the Dianogalization Lemma is used to construct the sentence G ↔ ¬G, such that this bi-implication is provable in PA. Gödel used this to show that PA cannot prove G, nor its negation. We will see in the next section how Gödel’s Second Incompleteness Theorem follows from the Dianogalization Lemma and the HBL conditions.

Theorem 3.1.4 (G¨odel’s Second Incompleteness Theorem). If PA is consistent, then PA 6` ¬⊥.

This theorem is an extension of Gödel’s First Incompleteness Theorem in the sense that it gives an example of a statement such that PA cannot prove it, nor its negation. Namely, the statement of its own consistency. Gödel’s Second Incompleteness Theorem tells us that PA cannot prove its own consistency, unless it is inconsistent. Gödel’s theorems are a very important result in logic and the philosophy of mathematics. They tell us that every consistent arithmetical theory leaves out some true statements, and cannot prove its own consistency. Because of this, one could argue that it is impossible to find an arithmetical theory to formalize all mathematics.

3.2 Peano Arithmetic and G¨

odel-L¨

ob provability logic

In this section, we will discuss the relation between PA and GL. In order to do so, we need a so-called realization to map modal sentences to arithmetical sentences.

Definition 3.2.1. A realization * is a function from propositional variables into arith-metical sentences. It is extended to a function from all sentences of the modal language by requiring that:

1. (ϕ ∨ ψ)* = ϕ* ∨ ψ* 2. ⊥* = ⊥

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4. (2ϕ)* = ϕ*

We say that ϕ* is the translation of the modal sentence ϕ.

Each realization determines to which arithmetical sentence each propositional vari-able is mapped. How a sentence is translated depends only on the translation of the propositional variables. If we have a sentence ϕ such that for all propositional variables p in this sentence p*=p?, for two realizations * and ?, then we have ϕ*=ϕ?.

It follows from the HBL conditions that if K4 ` ϕ, then for all realizations *, PA ` ϕ*. We say that K4 is arithmetically sound.

Theorem 3.2.2 (L¨ob’s theorem). For any formula ϕ, if PA ` ϕ → ϕ, then PA ` ϕ. Proof. We will give a K4 modal derivation, using 2ϕ → ϕ and χ ↔ (2χ → ϕ) as premises. 1. 2ϕ → ϕ Premise 2. χ ↔ (2χ → ϕ) Premise 3. χ → (2χ → ϕ) Propositional logic on 2 4. 2χ → 2(2χ → ϕ) Example 2.2 on 3 5. 2(2χ → ϕ) → (22χ → 2ϕ) K-Axiom 6. 2χ → (22χ → 2ϕ) Example 2.1 on 4 and 5 7. 2χ → 22χ Axiom 4

8. 2χ → 2ϕ Propositional logic on 6 and 7 9. 2χ → ϕ Example 2.1 on 1 and 8

10. 2χ MP on 3 and 9 and Necessitation 11. ϕ Modus Ponens on 9 and 10

We have shown that2ϕ → ϕ, χ ↔ (2χ → ϕ)/ϕ. Now, if we set ψ(x) as ψ(x) := x → ϕ, then by the Diagonalization Lemma we have PA ` χ ↔ (χ → ϕ) for some sentence χ. Using this, together with arithmetical soundness of K4, Löb’s theorem follows. By substituting ⊥ for ϕ in Löb’s theorem, Gödel’s Second Incompleteness Theorem will follow: Suppose that PA is consistent, i.e. PA 6` ⊥. We want to prove the theorem by contradiction, so we suppose that PA ` ¬ ⊥. This is the same as PA ` ⊥ → ⊥. But from Löb’s theorem we get PA ` ⊥, which contradicts the assumption that PA is consistent.

It follows from L¨ob’s theorem together with the arithmetical soundness of K4 that K4+LR is arithmetically sound too. As mentioned in Section 2.2 (p. 12), GL and K4+LR are syntactically equivalent. From this, arithmetical soundness of GL follows. Solovay ([13]) proved that the other direction also holds: if all translations of a modal sentence ϕ are theorems of PA, then ϕ is a theorem of GL. This is called arithmetical

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Theorem 3.2.3 (Arithmetical soundness and completeness). For all modal sentences ϕ,

GL ` ϕ ⇔ for all realizations *, PA ` ϕ*

Because of arithmetical soundness we can use the axioms and rules of GL when reasoning about the provability predicate in PA. Solovay proved that GL captures in fact everything that PA can prove about its provability predicate.

3.3 Peano Arithmetic and bimodal provability logic GLB

In this section, a new notion of provability will be introduced and its interaction with ordinary provability will be discussed. Recall that a Π1-sentence is of the form ∀xϕ,

where all quantifiers of ϕ are bounded (ϕ is ∆0).

Definition 3.3.1.

1. A formula ϕ is 1-provable if it is provable in PA together with all true Π1-sentences.

2. A formula ϕ is 1-consistent if ¬ϕ is not 1-provable.

We write PA1 for the system obtained by adding to PA all true Π1-sentences as extra

axioms.

By G¨odel’s Second Incompleteness Theorem (Theorem 3.1.4), the consistency of PA cannot be proved in PA; PA 6` ¬⊥. As the provability predicate is Σ1 (p. 15), the

sentence ¬⊥, which is shorthand for ∀x : ¬Pr(x, p⊥q), is Π1. Moreover, ¬⊥ is a

true Π1-sentence as PA is consistent. Therefore, ¬⊥ is 1-provable. From this example

we can conclude that provability is a stronger notion than ordinary provability: 1-provability of ϕ does not always imply ordinary 1-provability of ϕ.

There is a formula TrΠ1(x), expressing in the language of arithmetic that x is a true Π1

-sentence [7, Thm. I.2.55]. Now, similarly like the provability predicate expresses inside PA that something is provable, the 1-provability predicate expresses inside PA that something is 1-provable. The 1-provability predicate is defined as follows: Pr1(ϕ) :=

∃π(TrΠ1(π) ∧ Pr(π → ϕ)). If we change Definition 3.2.1 of a realization such that

(2ϕ)*= Pr1(pϕ* q), then GL stays arithmetically sound [8, p. 20].

We extend Definition 3.2.1 by adding the condition that (4ϕ)* = Pr1(pϕ* q). We can

now interpret the modal symbols2 and 3 as provability and consistency and 4 and 5 as 1-provability and 1-consistency, respectively. It is interesting to study how this new notion of provability interacts with ordinary provability. As stated in Definition 2.3.1, the axioms of GLB which contain both modalities are:

1. 2ϕ → 4ϕ 2. 3ϕ → 43ϕ

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These axioms are arithmetically valid. Namely, if ϕ is provable in PA, then it is also provable in PA1. As for the second implication,3ϕ is shorthand for ∀x : ¬Pr(x, p¬ϕq). If 3ϕ is true, then it is a true Π1-sentence, and therefore 1-provable. For arithmetical

soundness of GLB, these two statements have to be provable in PA.

Now, there is an analogue of Solovay’s theorem for GLB proven by Japaridze [9]. Theorem 3.3.2 (Arithmetical soundness and completeness of GLB). For all modal sentences ϕ,

GLB ` ϕ ⇔ for all realizations * , PA ` ϕ*

By using arithmetical soundness of GLB, we can now complete the proof of Theorem 2.3.3.

Kripke incompleteness GLB We have seen in Section 2.3 that 4⊥ is valid on all frames that make all GLB axioms valid. In order to show that GLB is not Kripke complete, we want to prove that GLB 6` 4⊥. Due to arithmetical soundness of GLB, it is enough to show that PA 6` Pr1(p⊥q). Suppose we have PA ` Pr1(p⊥q). Because

of soundness of PA this implies PA1 ` ⊥. As PA 6` ⊥, there are finitely many true Π₁ -sentences ∀xϕ1, ..., ∀xϕn such that adding these to PA makes it inconsistent. Adding

finitely many sentences is the same as adding their conjunction. By predicate logic, ∀xϕ₁ ∧ ... ∧ ∀xϕ_n is the same as ∀x(ϕ1 ∧ ... ∧ ϕn). If we write ψ for the conjunction

of all the ϕ’s, we get the true Π1-sentence ∀xψ. We have PA ` ∀xψ → ⊥, which is

equivalent to PA ` ¬∀xψ. It follows from soundness of PA that N 6|= ∀xψ. However, this contradicts the fact that ∀xψ is a true Π1-sentence, so we get PA 6` Pr1(p⊥q). As

stated above, it follows from arithmetical soundness that GLB 6` 4⊥. Therefore, GLB is not Kripke complete.

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4 Topological semantics of provability

logic

In this chapter we will examine an example of a topological space on which the axioms of GLB are valid. But first, we will introduce the needed topological definitions and study the topological semantics of GL.

4.1 Topological semantics of G¨

odel-L¨

ob provability logic

In order to study the topological semantics of GL, we need the following definitions. Definition 4.1.1. Let (X, τ ) be a topological space.

1. The derivative of A ⊆ X, dτ(A), is the set of limit points of A. That is,

x ∈ dτ(A) ↔ ∀U ∈ τ (x ∈ U → A ∩ U \ {x} 6= ∅).

2. The set i(A) is the set of isolated points of A ⊆ X. That is,

x ∈ i(A) ↔ ∃U ∈ τ : {x} = A ∩ U ↔ x ∈ A \ dτ(A).

Definition 4.1.2. (X, τ ) is scattered if every nonempty subspace A ⊆ X has an isolated point.

Definition 4.1.3. Consider the topological space (X, τ ). A topological model is of the form (X, τ, |=) where ‘|=’ is a relation (valuation) between the elements of X and propositional formulas such that:

1. w |= ¬ϕ ⇔ w 6|= ϕ

2. w |= ϕ ∨ ψ ⇔ w |= ϕ or w |= ψ

3. w |=3ϕ ⇔ ∀U ∈ τ(w ∈ U → ∃v 6= w, v ∈ U and v |= ϕ) We write v(ϕ) for the set of all x ∈ X such that x |= ϕ.

We say that a formula ϕ is valid in (X, τ ) if for all valuations w |= ϕ for all w ∈ X. If C is a class of spaces, then ϕ is valid in C if it is valid in every member (X, τ ) of C. Log(C) is the set of formulas that is valid in C.

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Theorem 4.1.4. (X, τ ) is scattered if and only if (X, τ ) |=3ϕ → 3(ϕ ∧ ¬3ϕ).

Proof. Note that3ϕ → 3(ϕ ∧ ¬3ϕ) is equivalent to L¨ob’s axiom. This can be derived using contraposition and the definition of 3.

First suppose that (X, τ ) is scattered. We want to show that if x |= 3ϕ then x |= 3(ϕ ∧ ¬3ϕ). Assume that x |= 3ϕ and take a U ∈ τ such that x ∈ U. Because of Definition 4.1.3 v(ϕ) ∩ U \ {x} is nonempty. Therefore, also v(ϕ) ∩ U is nonempty. Because of scatterdness, it has an isolated point, say y, so there is a V ∈ τ such that {y} = V ∩ (v(ϕ) ∩ U ). As both U and V are open, its intersection is also open, W := V ∩ U . We now have, {y} = v(ϕ) ∩ W , so y is an isolated point of v(ϕ). Note that if y = x, this would contradict the fact that x |= 3ϕ, so y 6= x. From Definition 4.1.1 we get y |= ϕ \3ϕ, which is equivalent to y |= ϕ ∧ ¬3ϕ. As y ∈ U we get y ∈ v(ϕ ∧ ¬v(ϕ)) ∩ U \ {x}. It follows from Definition 4.1.3 that x |=3(ϕ ∧ ¬3ϕ).

For the other direction we assume that (X, τ ) |=3ϕ → 3(ϕ ∧ ¬3ϕ). Take a nonempty U ⊆ X and let ‘|=’ be the valuation such that x |= p if and only if x ∈ U . We have that v(p) = U . If there is no world w such that w |=3p, then all the points of U are isolated. Now, suppose that there is a w such that w |=3p, then because of our assumption, we have w |=3(p ∧ ¬3p). This means that w is a limit point of the isolated points of v(p), so there has to be at least one isolated point of v(p). As v(p) = U , we can conclude that

(X, τ ) is scattered.

We have seen that GL is sound an complete with respect to the class of scattered topological spaces. It is in fact sufficient to take converse well-founded spaces with the upset topology. Recall that the open sets of the right topology τ→ are of the form ↑x.

Lemma 4.1.5. Let (X, ≺) be a strict partial ordering. Then (X, ≺) is converse well-founded if and only if (X, τ→) is scattered.

Proof. First suppose that (X, ≺) is converse well-founded. We want to show that every nonempty A ⊆ X has an isolated point. Take such A ⊆ X. Then, because of converse well-foundedness, there is an x ∈ A such that if x ≺ y then y 6∈ A. Now, take ↑x ∈ τ→.

Then ↑x ∪ {x} is open and only intersects A in the point x, so x is an isolated point of A. Therefore, (X, τ→) is scattered.

For the other direction, suppose that (X, τ→) is scattered. We want to show that

every subset of X has a greatest element. Take A ⊆ X, then because of scatterdness, it has an isolated point x such that there is a V ∈ τ→ with A ∩ V = {x}. Therefore, x is

the greatest element.

Corollary 4.1.6. If C is the class of all converse well-founded spaces with the upset topology, then Log(C) = GL.

4.2 Topological semantics of bimodal provability logic GLB

In this section we will consider the ordinal space (ω + 1, τ1, τ2). Here, τ1 is the upset

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standard ordinal ordering on ω + 1. The topology τ2 is generated by singletons {n}

and the sets ω + 1 \ {n} for n ∈ ω. We can interpret the space ω + 1 as the interval [0, ω] of ordinals. Similarly, the set ω is interpreted as [0, ω). Now, we have two distinct modalities 31 and 32, which we define as follows:

1. x |=31ϕ ↔ ∃y ∈ X such that x ≺ y and y |= ϕ,

2. x |=32ϕ ↔ ∀U ∈ τ2(x ∈ U → ∃y 6= x, y ∈ U and y |= ϕ).

The modality 31 satisfies the GL axioms, as (ω + 1, ≺) is transitive and there is no

infinite, ascending chain (see Theorem 2.2.4). Note that {ω} has every other point as a successor. As for the other topology, every point except for {ω} is open and therefore isolated in every nonempty set. Now, take a nonempty set U ⊆ ω + 1. If there is an n ∈ ω such that n ∈ U , then n is an isolated point of U . If there is no n ∈ ω such that n ∈ U , then U = {ω}. In this case, for every V ∈ τ2 we have {ω} = U ∩ V , so the point

ω is isolated in U . We can conclude that ω + 1 is scattered. It follows from Theorem 4.1.4 that the GL axioms are valid on (ω, τ2).

We have shown that both modalities satisfy the GL axioms. In the following propo-sitions we will show that also the GLB axioms hold on (ω + 1, τ1, τ2).

Proposition 4.2.1. 32ϕ →31ϕ is valid on (ω + 1, τ1, τ2).

Proof. Let ‘|=’ be any valuation on (ω + 1, τ1, τ2). Suppose that x ∈ ω, then x is an

isolated point of ω + 1. So, {x} is an open neighborhood that does not intersect any other set. Therefore, x cannot be a limit point of any nonempty set U . Now, suppose that y |= 32ϕ, so y is a limit point of v(ϕ). By the reasoning above, it follows that

y = ω. Now, if v(ϕ) = {ω}, then ω is an isolated point of v(ϕ) and therefore not a limit point. So v(ϕ) contains at least one point n ∈ ω. As w ≺ n and n |= ϕ, it follows that ω |=31ϕ. Therefore, 32ϕ →31ϕ is valid on (ω + 1.τ1, τ2).

Proposition 4.2.2. 31ϕ →2231ϕ is valid on (ω + 1, τ1, τ2).

Proof. First, note that31ϕ →2231ϕ is equivalent to31ϕ → ¬32¬31ϕ.

Let us suppose that v(ϕ) = {ω}, i.e. ω |= ϕ. Then, there is no x ∈ X such that x |=31ϕ. Therefore, the implication holds.

Now, assume that v(ϕ) ∩ ω 6= ∅. Then, because of transitivity, v(31ϕ) is an infinite

set, so v(¬31ϕ) is finite. As seen in the proof of Proposition 4.2.1, only {ω} could be a

limit point of this set. However, there will always be an open neighborhood of ω that does not intersect v(¬31ϕ), as this set is finite. Hence, ω cannot be a limit point of

v(¬31ϕ). Therefore, v(32¬31ϕ) = ∅ and v(¬32¬31ϕ) = ω + 1. So,31ϕ → ¬32¬31ϕ

is valid on (ω, τ1, τ2).

We have shown that GLB is sound with respect to ω + 1. However, not only the axioms of GL and GLB are valid on ω + 1, the so-called linearity axiom:

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also holds for 31. Namely, suppose that x |= 31p ∧31q, then there are y and z such

that x ≺ y with y |= p and x ≺ z with z |= q. We can now distinguish three cases. If z = y, then x |= 31(p ∧ q). If y ≺ z, then x |= 31(p ∧31q). Finally, if z ≺ y, then

x |=31(31p ∧ q).

The linearity axiom is not a theorem of GLB [4, Ch. 15]. Hence, GLB is not complete with respect to ω + 1. Despite this, Beklemishev and Gabelaia showed that GLB is in fact topologically complete [1, Thm. 11].

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5 Conclusion

In this thesis we have studied two important systems of provability logic: GL and GLB. We have seen that GL is sound and complete with respect to both its Kripke and topological semantics. If the modality of GL is interpreted as the provability predicate in PA, then GL is arithmetically sound and complete. As for GLB, it is not complete with respect to any class of Kripke frames. However, it is topologically complete. If its second modality is interpreted as the 1-provability predicate, then GLB is arithmetically sound and complete too.

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6 Popular summary

In this thesis we have studied provability logic. In order to do so, we first examined modal logic. Modal logic is a tool for describing and working with relational structures and is widespread in artificial intelligence, mathematics and computer science. We write 2 for the so-called modality. It can be interpreted in many different ways. Let p stand for the sentence ‘It rains’. Examples of the interpretation of2p are ‘it is necessary that it rains’ and ‘it will be the case that it rains’. The logical symbol ¬ stands for negation, i.e. ¬p is read as ‘not p’. Now, 3p is defined as ¬2¬p. So using the first example used above, 3p means ‘it is not necessary that it does not rain’, which is equivalent to ‘it is possible that it rains’.

In order to study these sentences formally we use the so-called relational semantics. We consider a set W of possible worlds w1, w2 etc. On W, there is an accessibility

relation R defined: if w2 is accessible from w1 we draw an arrow from w1 to w2. Now,

we say that 2p is true in a certain world w1, if p is true in every world accessible from

w1 (see Figure 1). As 3p = ¬2¬p, 3p is true when it is not the case that in all worlds

accessible from w1, ¬p is true. In other words, there is at least one world accessible from

w1 where p is true (see Figure 2). Note that in Figure 2, 2p is not true in w1, as p is

not true in w2. w1 w2 w3 p p 2p Figure 1. w1 w2 w3 p ¬p 3p, ¬2p Figure 2. The relation R can have different properties. We will now discuss two of them, which are of great importance in this thesis. First, suppose that there are three worlds w1, w2

and w3, such that w2 is accessible from w1 and w3is accessible from w2 (see Figure 3.1).

Then, we say that the relation is transitive, if w3 is accessible from w1 (see Figure 3.2).

w1 w2 w3 w1 w2 w3

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Secondly, if we have worlds which are related as in Figure 4,we say that this is an ascending chain. An ascending chain is infinite if there are infinitely many worlds in the chain.

w1 w2 w3 w4 w5

Figure 4.

Now, provability logic GL, named after logicians G¨odel and L¨ob, is a branch of modal logic. Here, 2p is interpreted as ‘it is provable that p’. We have seen in this thesis that in provability logic, the relation on the set of possible worlds W is transitive and does not have any infinite ascending chains.

Another result we have studied is the relation of provability logic with Peano Arith-metic PA. PA is a theory about the natural numbers:

0, 1, 2, 3, 4, ...

Within the system of PA we can prove statements about the natural numbers. For example, that there are infinitely many prime numbers. Now, let ϕ be a statement about the natural numbers. There is a formula, the so-called provability predicate Pr(ϕ), which expresses that ϕ is provable in PA. The mathematician Solovay proved that if we reason about Pr(ϕ) in PA, we can use the theorems of GL. In other words, we can study the system of PA through GL. This is a convenient result, as it can sometimes be easier to prove statements in GL.

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Semantics for Provability Logic