A Refutation of the Equivalence Between the Multiplication, Pigeonhole, Induction, and Well- Ordering Principles

A refutation of the equivalence between the multiplication, pigeonhole, induction, and well- ordering principles

-with a main focus on the often claimed equivalence between the induction principle and the well-ordering principle when studying the natural numbers.

Malene Sandvær Lunnergård

Vt 2016 Kandidatexamensarbete i matematik, 15 hp Handledare: Lars-Daniel Öhman

A refutation of the equivalence between the multiplication, pigeonhole, induction, and well-ordering principles

-with a main focus on the often claimed equivalence between the induction principle and the well-ordering principle when studying the natural numbers.

Malene Sandvær Lunnerg˚ard

June 21, 2016

Bachelor’s Thesis in Mathematics, 15 hp. Department of Mathematics and Mathematical Statistics Ume˚aUniversity

1 Sammanfattning

Att induktionsprincipen och välordningsprincipen ärekvivalenta ärett vanligt p˚ast˚aendenärde naturliga talen studeras. Utg˚angspunktenfördetta examensarbete ärartikeln The equivalence of the multiplication, pigeonhole, induction, and well ordering principles, av Swanson och Hansen. Därp˚ast˚asatt multiplikationsprincipen , l˚adprincipen, induktionsprincipen och välordningsprincipen ärekvivalenta. Detta bevisas gällaförde naturliga talen i Zermelo-Fraenkel-Skolem mängdlära,däralla dessa principer ärsatser. Detta examensarbete inneh˚alleren presentation av generella axiomatiska system, och den logiska strukturen i s˚adanasystem. Syftet med denna presentation äratt redogöraförvad det innebärförprinciper, som ärsatser i ett axiomatiskt system, att vara ekvivalenta, samt att se p˚askillnaden mellan denna typ av ekvivalens och ekvivalens relativt till ett axiomatiskt system därprinciperna inte ärsatser. De naturliga talen, Peano axiomen och Zermelo-Fraenkel-Skolem mängdlärapresenteras som ett underlag förfortsättningenp˚aexamensarbetet. Huvudsyftet med detta examensarbete äratt motbevisa ekvivalens mellan induktionsprincipen och välordninsprincipen relativt till ett axiomatiskt system där dessa principer inte ärsatser. Därmedmotbevisas även ekvivalens mellan alla de fyra principerna i artikeln av Swanson och Hansen relativt ett s˚adant system. Detta utförsgenom att konstruera ett axiomatiskt system därvarken induktionsprincipen eller välordningsprincipen ärsatser, och däreftervisa att induktionsprincipen och välordninsprincipen inte ärekvivalenta relativt detta system.

Abstract

It is a common claim that the induction principle and the well-ordering principle are equivalent to each other when the natural numbers are studied. The starting point for this thesis is the article The equivalence of the multiplication, pigeonhole, induction, and well ordering principles, by Swanson and Hansen, in which it is claimed that all of these four principles are equivalent. This equivalence is proven for the natural numbers in Zermelo-Fraenkel-Skolem set theory, in which all of these principles are theorems. This thesis contains a presentation of axiomatic systems in general, and the logical structure of such systems. The purpose of this presentation is to account for what equivalence between theorems of an axiomatic system entails, and look at the distinction between this kind of equivalence and equivalence relative to an axiomatic system in which the principles are not theorems. The natural numbers, the Peano axioms and the Zermelo-Fraenkel-Skolem system are presented as a foundation for the continuation of the thesis. The main purpose of this thesis is to refute the equivalence between the induction principle and the well-ordering principle relative to an axiomatic system where the principles are not theorems, and thereby refute the equivalence of all four principles in the article by Swanson and Hansen relative to such a system. This refutation proceeds by constructing an axiomatic system where neither the induction principle nor the well-ordering principle are theorems, and showing that the induction principle and the well-ordering principle are not equivalent relative to this system.

2 Acknowledgements

I would like to thank my advisor, Lars-Daniel Ohman,¨ for suggesting the subject of this thesis, and for being my sounding board throughout the process of writing this thesis.

Thanks to Joel Larson for providing me with a good template and for some additional help regarding writing a thesis in LATEX.

A big part of the work behind this thesis has been to gain knowledge on the subject of logic, and in particular the logic of an axiomatic system. It would have been beneﬁcial to have taken a course on the basics of logics before starting working on this thesis, as I have used a lot of time to gain a decent understanding of this. It would have been preferred to have that time to focus on the more speciﬁc topics of the subject.

3 Contents

Contents

1Introduction 5 1.1 Aim and objectives ...... 6 1.2 The structure of the thesis ...... 6

2 Thestructureofaxiomaticsystems 8 2.1 The logic of an axiomatic system ...... 9 2.2 Diﬀerent kinds of equivalence ...... 10 2.3 On the logical equivalence relative to an axiomatic system ...... 11 2.4 The notion of equivalence relative to an axiomatic system illustrated by axiomatic geometry ...... 13

3Naturalnumbers 18 3.1 The Peano axioms ...... 18

4 TheZermelo-Fraenkel-Skolemsystem 20 4.1 The natural numbers deﬁned in ZFS set theory ...... 22

5Whatisorder? 25

6 An axiomatic system, P , weak enough for the principles inquestionnottobetheoremsofthesystem 28 6.1 The axiomatic system P with the induction principle added as an axiom 30 6.2 The axiomatic system P with the well-ordering principle added as an axiom 32 6.3 Ordinal numbers ...... 33 6.4 The refutation of equivalence between the induction principle and the well-ordering principle relative to a theory where non of the principles are theorems ...... 34

7Discussion 37

References 40

4 1 Introduction

There seems to be a widespread misconception relating to the equivalence between the induction principle and the well-ordering principle. The so called Peano axioms constitute an axiomatic system founding the natural numbers [5, pp. 22]. In this axiomatic system the induction principle is one of the axioms. The claim that these two principles are equivalent relative to the Peano axioms is for example stated on Wikipedia [20].

The point of departure of this thesis was the article The equivalence of the multiplication, pigeonhole, induction, and well ordering principles which was published by Leonard G. Swanson and Rodney T. Hansen in 1988 [15]. In this article the authors work in Zermelo-Fraenkel-Skolem (ZFS) set theory, and claim to prove the equivalence of the four principles mentioned in the title of the article.

In different axiomatic systems different principles may apply all depending on the axioms of the system. This might cause some difficulties in certain matters, in particular when it comes to the concept of equivalence between principles.

The principles, as stated in the article by Swanson and Hansen, are as follows: The multiplication principle (MP). If there exist n ways of performing one operation and m ways of performing another operation (independent of the ﬁrst), then there exist mn ways of performing both operations, one followed by the other. The pigeonhole principle (PH). Suppose that m pigeons are to be distributed into n pigeonholes. If m > n, then at least one pigeonhole must contain more than one pigeon. The well-ordering principle (WO). Every non-empty subset of the natural numbers has a least element. The principle of mathematical induction (MI). If S is a subset of the natural numbers such that: (a) 1 ∈ S, and (b) n + 1 ∈ S whenever n ∈ S, then S is the set of natural numbers. Swanson and Hansen claim to prove the equivalence between the four principles working in ZFS set theory and using the natural numbers deﬁned in this particular axiomatic structure. Within this theory all of the principles in question are theorems, as they point out themselves in the article. They claim to prove the equivalence of the four principles by proving (MP) ⇒ (PH) ⇒ (WO) ⇒ (MI) ⇒ (MP).

In the introduction of the article by Swanson and Hansen they write the following: It is interesting to note that these two basic concepts(the MP and PH) are really more basic than they might at ﬁrst appear. They are, in fact, equivalent to each other and they are a part of the axiomatic structure of the natural numbers in the sense that they are equivalent to the following principles(the WO and MI). [15]

5 1.1 Aim and objectives

This is a sentence that is to imprecise to be completely sure of what is meant by it, but when they mention ”the axiomatic structure of the natural numbers” it is assumed in this thesis that it is the Peano axiomatic system they refer to. If looking at the Peano axiomatic system independently from ZFS set theory the induction principle, which in ZFS theory is a deduced theorem, is not a deduced theorem but is one of the axioms of the system. Where a theorem of a system is called a deduced theorem if it is deducible from the axioms of the system. This means that if a theorem of a system is not a deduced theorem it must be an axiom of the system that is not deducible from the remaining axioms of the system. The claim by Swanson and Hansen that the multiplication principle and the pigeonhole principle are ”part of the axiomatic structure of the natural numbers” in the sense that they are equivalent to the well-ordering principle and the induction principle is interpreted in this thesis as a claim that the three ﬁrst mentioned principles could each separately replace the induction principle in the Peano axioms with no alterations to the resulting axiomatic system.

If this is what is meant by that statement, they seem to draw the conclusion, based on equivalence between the four principles in one speciﬁc axiomatic system in which all of the principles are theorems, that this implies equivalence between the same four principles in another axiomatic system in which the principles are not theorems. But is this really the case? It is pointed out in a review by Smith that this conclusion can not be drawn on the basis of equivalence relative to a system in which the principles are theorems [14].

1.1 Aim and objectives One purpose of this thesis is to explain what it means that principles are theorems of an axiomatic system, and to ﬁgure out what it means for theorems of an axiomatic system to be equivalent. The main purpose of this thesis is to refute that because of equivalence between the multiplication, pigeonhole, induction, and well-ordering principles relative to Zermelo-Fraenkel-Skolem set theory, the same principles are equivalent to each other in a system where the principles are not theorems. This will be done by refuting the common claim that the induction principle is equivalent to the well-ordering principle in such a system.

1.2 The structure of the thesis To fulfil the purpose of this thesis it is important to establish a good understanding of how an axiomatic system is constructed and what it really means for two statements to be equivalent, this is the purpose of section 2, The structure of axiomatic systems. This is done by introducing some ground rules about the structure of axiomatic systems, and looking at both axiomatic systems and equivalence from a logical point of view. What it means for principles to be equivalent under different circumstances is defined and named to avoid misunderstandings when using this term. Subsection 2.3 consists of an analysis of what it means for two principles that are both theorems of a system to be equiva-

6 1.2 The structure of the thesis lent, which fulﬁls the ﬁrst purpose of this thesis. The well known Euclidean geometry is presented with the purpose of presenting a concrete example of an axiomatic system as previously described in this section. What it means for a principle to be equivalent to another principle that is part of the axioms and not a deduced theorem of a system is illustrated in this system.

In section 3, Natural numbers, the natural numbers are described, and the arithmetic theory formed by the Peano axioms is presented since this is an axiomatic structure of the natural numbers. Since Swanson and Hansen say they work in ZFS set theory a presentation of this axiomatic structure and how the natural numbers relate to this theory is presented in section 4, The Zermelo-Fraenkel-Skolem system.

One of the four principles in question in this thesis, the well-ordering principle, requires knowledge of what order is. Diﬀerent kinds of ordering, and in particular what a well- ordering is, is presented in section 5, What is order?.

Section 6, An axiomatic system, P, weak enough for the principles in question not to be theorems of the system, is dedicated to fulﬁl the main purpose of the thesis; to refute that because of equivalence between the multiplication, pigeonhole, induction, and well-ordering principles relative to ZFS set theory, the same principles are all equivalent to each other relative to a system where the principles are not theorems. This will be done by refuting that the induction principle and the well-ordering principle are equivalent relative to an axiomatic system in which the two principles are not theorems. To do this a new axiomatic system, with this property, is constructed. With this axiomatic system in place it will be showed that the induction principle and the well-ordering principle are not equivalent to each other relative to this axiomatic system. In showing this, it is also implied that all four principles can not be equivalent relative to such a system, where the principles are not theorems, if two of them are not.

Section 7, Discussion, consists of reﬂections on the results of the thesis and some ideas for continued work on this subject.

7 2 The structure of axiomatic systems

An important part of this thesis is to obtain a good understanding of the structure of axiomatic systems and what equivalence between principles in such a system really is.

An axiomatic system consists of primitive notions, definitions, axioms and theorems. When defining a concept, other concepts previously defined are used to do this. But this process must start somewhere, and when the first concept of a system is to be defined there are no available concepts to use in the definition. This is why some concepts can not be defined in terms of other concepts. These concepts are called primitive notions and are the ultimate foundation of an axiomatic system. With the primitive notions in place new concepts can be defined in terms of them, and this process can continue by defining new concepts from already defined terms.

The axioms of an axiomatic system are statements, often intuitive ones, giving properties to the primitive notions and the deﬁned terms. Every axiomatic system has an underlying logic that constitutes the rules of reasoning in the system, these rules are called rules of inference. In this thesis classical logic is the underlying logic in all axiomatic systems mentioned. Theorems are statements that can be deduced with help of the logic of the system from the given deﬁnitions and axioms. The theorems can either be directly deduced from the axioms or by using other theorems which have already been deduced [5, pp.20-21]. An axiomatic system constitutes a theory.

The language of an axiomatic system may be given in completely logical terms, which is a very formal language, more information on this will be presented in the next subsection. This might not be very useful if it is not possible to assign the formal language a more concrete meaning. An interpretation may be applied to the formal language. This interpretation can provide a way to determine if the statements in the axiomatic system are true or false. If every statement in an axiomatic system is true when applying a certain interpretation this interpretation is called a model of the axiomatic system. A model must realize all of the axioms in the system it is to be a model of [11, pp. 60]. Often when making an axiomatic system the motivation is to formalize some subject by making an axiomatic system with that particular subject in mind. When this is the case that model is called the standard model of the system. Even though the axiomatic system were in- tended for some standard model there might be additional models of the system that are not isomorphic to the standard model, these are called non-standard models of the system.

One aim when making an axiomatic structure is to make the list of axioms as short as possible, i.e. to have no superﬂuous axioms, but still long enough to provide the system with all wanted properties. If an axiomatic system has a set of axioms with the property that no contradictions can be derived from them, the system is called consistent. Proving the consistency of a system is not easy, but ﬁnding a model that realizes the system is a good reassurance. Another wanted property of an axiomatic structure is that

8 2.1 The logic of an axiomatic system the list of axioms is long enough that every truth in the system is deducible. If this is the case the system is called complete [5, pp.20-22]. According to Gödel’sfirst incompleteness theorem neither The Peano arithmetic nor the ZFS set theory are complete theories because the natural numbers can be defined in both of the theories. Gödel’s second incompleteness theorem gives that the consistency of the Peano arithmetic can not be proven within the theory itself, and the same applies to the ZFS set theory [18, pp. 595-596].

Let’s take a look at a simple example of an axiomatic system: Let the axiomatic system W consist of the two primitive notions ◦ and −, so that the set {◦, −} consists of the symbols available in the theory. The axioms specify what statements are available in the system. Let U be the set of available statements, and let W have only one axiom.

Axiom. ◦ ◦ ◦◦ is in U.

Let the only rule of inference of W be Rule of inference. For any statement in U any symbol ◦ may be replaced by the symbol −, and the resulting statement is available, i.e in U.

Some of the resulting theorems of this system is:

Theorem 2.1. ◦ ◦ −◦ is in U.

Theorem 2.2. ◦ − −◦ is in U.

Theorem 2.3. − − −− is in U.

A model of this system could be the binary numbers. If deﬁning the number 1 as ◦ and the number 0 as −, the axiom says that 1111 is a available binary number, i.e. in U. All the theorems of W , with the binary numbers as a model, gives that all binary numbers with four digits are available. This is not a very useful theory since all it will consist of is this set U of binary numbers with only four digits.

2.1 The logic of an axiomatic system In order to understand what equivalence is it is necessary to review how an axiomatic structure is constructed in terms of logic.

An axiomatic structure, or theory T , in logical terms consists of a set of symbols that can be arranged in different ways, i.e. are the building blocks of the system. A finite sequence of symbols is called an expression. From the symbols of the theory expressions can be constructed by so called logical connectives such as ¬, ∧, ∨, ⇒, ⇔. Some of these expressions are called wff’s, an abbreviation for well formed formulas. The simplest wff’s are called atomic formulas. A wff can then be defined recursively as follows:

9 2.2 Diﬀerent kinds of equivalence

Deﬁnition 2.4. 1. Every atomic formula is a wﬀ.

2. If B and C are wff’s, then (¬B) and (B ⇐ C ) are wff’s. 3. An expression is a wff if and only if it can be shown to be a wff on the basis of conditions 1 and 2.

If assigning a model to a theory all wff’s are going to be either true or false in that model. The axioms of a system must be wff’s. The rules of inference in the theory, T , is a finite set of relations, R1, ..., Rn, among the wff’s. These relations decide what conclusions can be drawn within the theory. So every axiom of a theory is a wff, and if some expression follow from one or more established wff’s of the system by the rules of inference that expression is a new wff in that theory.

A proof in the theory T is a sequence of wff’s, B = B1, ..., Bk, such that each of the Bi’s is either an axiom in T , a previously proven theorem in T , or follows from some of the previous wff’s in the sequence by a rule of inference. If the sequence B above is a proof, then the last of the wff’s in the sequence, Bk, is a theorem of that theory. Thus the sequence B is a proof of Bk in the theory T . If B is a theorem in T it is denoted `T B. Often it is just written as ` B if it is taken for granted what theory B is a theorem of. This type of notation indicates that B is a theorem of T independently of any model of the theory, because there exists a proof in complete logical terms [11, pp. 34-35].

A wff that is always true, independent of the truth value of the symbols forming the wff, is called a tautology[11, pp.16]. This is a universal logical truth that is independent of what axiomatic system it is part of. An example of a tautology is (A ∨ (¬A)). This expression is true when A is true and when A is false. This tautology is called the law of the excluded middle[11, pp. 16]. In this thesis the expression a tautology relative to a theory will be used, which differs from a regular tautology. Let M be a model of a theory T . The notation |=M B means that the wff B is true in the model M. The expression |= B means that B is true in every model of T . If this is the case B is called a tautology relative T . Thus, two different models may both satisfy all axioms of a theory, but a principle might be true in one model but not be true in another. A principle that is true in every model of a theory is then called a tautology relative to that theory [11, pp. 60-61].

2.2 Different kinds of equivalence The term equivalence is widely used when implying that two things have the same meaning in some way, but there is not really a universal agreement of the deﬁnition of this term. Because of this a review of the meaning of three diﬀerent kinds of equivalence used in this thesis will be presented in this subsection.

10 2.3 On the logical equivalence relative to an axiomatic system

The ﬁrst kind of equivalence, logical equivalence, can according to Hamilton [7, pp. 9] be deﬁned as:

Definition 2.5. Let A and B be a wff. Then A is logically equivalent to B if and only if (A ⇔ B) is a tautology. The symbol ⇔ is a logical connective called a biconditional, and means that the expressions A and B must either both be true or both be false for (A ⇔ B) to be true [11, pp. 13]. This is a type of equivalence that is only defined in a completely logical language, and must be true regardless of any theory. An example of this is the tautology ((A ⇒ B) ⇔ (¬B ⇒ ¬A )), where (A ⇒ B) is a rule of inference called Modus Ponens and (¬B ⇒ ¬A ) is a rule of inference called Modus Tollens. Then according to definition 2.5 Modus Ponens is logically equivalent to Modus Tollens.

The second kind of equivalence will in this thesis be called logical equivalence relative to a theory. If two expressions, A and B, are to be logically equivalent relative to some theory, both of them must be true in every model of that theory. The deﬁnition used in this thesis is:

Deﬁnition 2.6. Let A and B be wﬀ’s of the theory T . A and B are logically equivalent relative to T if and only if (A ⇔ B) is a tautology relative to T . The third kind of equivalence will be called equivalence relative to a theory. This is a type of equivalence that determines what it means for two principles to be equivalent relative to a theory where non of them are theorems.

Definition 2.7. Let A and B be wff’s, and let T be a theory in which nether A nor B is a theorem. If A , B and T are such that if A is added as an axiom to the theory T then B becomes a theorem of T , and if B is added as an axiom to T then A becomes a theorem of T , then A and B are called equivalent relative to T . This kind of equivalence is sometimes just called logical equivalence, see for example [2, pp. 563], which might be misleading since this is not the same kind of equivalence as the logical equivalence in definition 2.5.

2.3 On the logical equivalence relative to an axiomatic system Considering the logical equivalence between principles relative to some axiomatic system in which the principles are theorems is one of the purposes of this thesis. Deﬁnition 2.6 says that for two wﬀ’s to be logical equivalent relative to a theory they must either both be true or both be false in every model of that theory.

The logical structure of an axiomatic system have some properties with importance relating Swanson’s and Hansen’s article. One property of consistent axiomatic systems is the following theorem, called The Soundness Theorem.

11 2.3 On the logical equivalence relative to an axiomatic system

Theorem 2.8. If B is a wﬀ in some theory T , and Γ is a sequence of wﬀ’s, then if Γ ` B, then Γ |= B. Or in other words; Every theorem of a theory T is a tautology relative to that theory.

Proof. Let B be a theorem in T , i.e. Γ ` B. We know that there is some sequence of wﬀ’s that is the proof of B, let this sequence be B1, B2, ..., B. This is a proof by induction on the number of wﬀ’s in the proof of B.

Base step: Assume there is only one wﬀ in the proof of B, i.e. only B. Then B must be an axiom of T . And all axioms are tautologies (You can not have a model of a theory without it fulﬁlling all the axioms of the theory, which means that each axiom is true in every model of T ). So B is a tautology.

Induction step: Assume there are n (n >1) wff’s in the proof of B, that is B1, B2, ..., Bn−1, B. Now let the induction hypothesis be that all theorems with a proof containing less than n number of wff’s are tautologies. The two alternatives for B is that B is either a tautology, or it follows from previous tautologies (since all of B1, B2, ..., and Bn−1 are tautologies by the induction hypothesis). For the wff’s in the proof to logically imply B, Bn−1 must be a conditional; Bn−1 = (Bn−2 ⇒ B). We know that Bn−2 and (Bn−2 ⇒ B) are tautologies because of the induction hypothesis. Since (Bn−2 ⇒ B) is only true if either Bn−2 is false or Bn−2 and B must both be true. From this follows that, since Bn−2 is always true, B must always be true, thus B is a tautology.

The next theorem gives information about the relationship between diﬀerent tautologies relative to a theory, and because of theorem 2.8 also about the relationship between theorems of a theory.

Theorem 2.9. B ⇒ C is false if and only if |= B and ¬ |= C . Proof. First assume that B ⇒ C is false. We know that for B ⇒ C to be true either B must be false or both B and C must be true. So the only remaining possibility must make B ⇒ C false (Since wﬀ’s can only be true or false), and that is B true and C false (this is the same as ¬C being true). Now assume that |= B and ¬ |= C this means that B is true in every model and C is false in every model. This does not make B ⇒ C true, thus it must be false.

Theorem 2.9 gives that if two wﬀ’s, A and B, are tautologies relative to a theory, T , both A ⇒ B and B ⇒ A must be true in every model of T . This implies that A ⇔ B must be true in every model of T , i.e. be a tautology relative to T . Deﬁnition 2.6 then gives that the tautologies A and B are logically equivalent relative to T . Since theorem 2.8 says that all theorems of a theory are tautologies, the same thing applies to the theorems of a theory. This gives the important consequence that picking any two theorems from a theory they are going to be logically equivalent to each other relative to that theory.

12 2.4 Axiomatic geometry

In the article by Swanson and Hansen they claim to show logical equivalence between the four principle relative to ZFS set theory in which all these principles are theorems. Because of this, theorem 2.8 and 2.9 assures that they are logically equivalent relative to ZFS set theory. So their result, that these four principles are logically equivalent to each other, is really no new information.

2.4 The notion of equivalence relative to an axiomatic system illustrated by axiomatic geometry Probably the most well-known axiomatic system is the one constituting the so called Euclidean geometry. This section is a presentation of this system which gives an op- portunity to apply the information given previous in this section to a more familiar environment, and to consider the concept of equivalence relative an axiomatic system. The book Elementa was written by the Greek mathematician Euclid about year 300 BCE [8, pp. 64]. In this book Euclid made an attempt to make an axiomatic system to establish geometry, a branch of mathematics considering the properties of points, lines and planes. This foundation has been modified by different mathematicians later on, such as David Hilbert. Hilbert discovered that Euclid sometimes used properties of geometry in his proofs without foundation in the axioms. In the book Foundations of Geometry which Hilbert published in 1899 he therefore refined some of the axioms of Euclid and added more axioms to the theory. Hilbert also added that the terms point, line, plane are primitive notions in Euclidean geometry, thus can not be defined, which was something Euclid attempted to do [2, pp. 147-148, 621].

The ﬁve axioms of Euclid, as given in Elementa, are as follows:

Axiom 1 (A1). Given two points, there is a straight line that joins them.

Axiom 2 (A2). A straight line segment can be prolonged indeﬁnitely.

Axiom 3 (A3). A circle can be constructed when a point for its centre and a distance for its radius are given.

Axiom 4 (A4). All right angles are equal.

13 2.4 Axiomatic geometry

Axiom 5 (A5). If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indeﬁnitely, will meet on that side on which the angles are less than the two right angles.

The last axiom, A5, is often called the parallel axiom. This axiom is not quite as intuitive as the four others, for that reason many have tried to prove this axiom on the basis of A1-A4, i.e to show that A5 is a deduced theorem of the theory. But these attempts all failed. It appeared that every time someone though they had proven the parallel axiom, some other property, that is not deducible from the axioms A1-A4, were used as a theorem in the proof. If adding one of these additional properties, a property not deducible from A1-A4, to the axioms A1-A4 this results in a new theory, in which the parallel axiom becomes deducible, thus becomes a theorem. If the same property is a theorem of the theory with A1-A5, i.e. with the parallel axiom, this means, by deﬁnition 2.7, that this property is equivalent to the parallel axiom relative to the theory consisting of the axioms A1-A4. The attempt to make the parallel axiom a deduced theorem has thus resulted in many properties that are equivalent to the parallel axiom relative to the theory consisting of A1-A4 [2, pp. 563-581]. One of these equivalent properties is called Playfair’s axiom, and it says:

Playfair’s axiom. Given a line, l, and a point, p, not on it, in the plane of l and p there exists one and only one line parallel to l that can be drawn through the point p.

Theorem 2.10. Playfair’s axiom and the parallel axiom are equivalent relative to Eu- clidean geometry.

The proof of theorem 2.10 in its entirety will be omitted, but some important properties of the proof will be mentioned. The proof consists of two parts: (i) ”Parallel axiom ⇒ Playfair’s axiom”, which means assuming the Parallel axiom to be true and showing that Playfair’s axiom must be true in every model. (ii) ”Playfair’s axiom ⇒ Parallel axiom”, which means assuming Playfair’s axiom to be true and showing that the parallel axiom must be true in every model. Both (i) and (ii) can be proven in complete logical terms, i.e. with just the information from the axioms together with logical connectives and the rules of inference of the system, with no properties that is not deducible from the axioms used. This means that if the terms ”point”, ”line” and ”right angle” used in the formulation of the four ﬁrst axioms, the parallel axiom and Playfair’s axiom were to be replaced with something completely diﬀerent like ”tree”, ”ocean” and ”book”, which have no connection between them, the result would be exactly the same; (i) and (ii) would still be provable. This means that the proof is independent of the model of the system which means it applies in all models, thus the parallel axiom and Playfair’s axiom are equivalent relative to Euclidean geometry. A complete proof of both (i) and (ii) is

14 2.4 Axiomatic geometry given by Greenberg [4, pp. 107-108].

Sinse the parallel axiom and Playfair’s axiom are equivalent relative to Euclidean geometry, Playfair’s axiom can replace the parallel axiom as the ﬁfth axiom and the result is the exact same theory. If the parallel axiom is assumed to be true, i.e. chosen as an axiom, then Playfair’s axiom is a theorem in this structure, and vice versa.

Some additional statements that have been proven to be equivalent to the parallel axiom relative to Euclidean geometry, according to Burton [2, pp. 564-565], are shown in the following list:

• A line that intersects one of two parallel lines intersects the other also.

• There exist lines that are everywhere equidistant from one another.

• The sum of the angles of a triangle is equal to two right angles.

• For any triangle, there exists a similar non-congruent triangle.

• Any two parallel lines have a common perpendicular.

• There exists a circle passing through any three non-collinear points.

• Two lines parallel to the same line are parallel to each other.

In the beginning of the 19th century a new type of geometry started to arise. Mathe- maticians began wondering what would happen if the parallel axiom was excluded from the axioms of geometry, or replaced with some non-equivalent statement. The time must have been ripe for this idea; maybe after so many years of people trying to prove the parallel axiom it was time to take another approach to the axiom. In the exact same time period the Russian Nicholai Lobatjevski and the Hungarian Janos Bolyai separately published their work on so called non-euclidean geometry. In correspondence between the German Carl Friedrich Gauss and other mathematicians it is evident that Gauss was actually before Bolyai and Lobatjevski in this discovery of a non-Euclidean geometry, but he never published his work on this subject [2, pp. 584]. The theorems derivable from a non-Euclidean geometry are diﬀerent from the ones in Euclidean geometry, so replacing the parallel axiom with some statement that is not equivalent to it relative to Euclidean geometry results in a completely diﬀerent structure [8, pp. 437-439].

When associating a line l and a point p not on that line with the lines trough p that is parallel to l, like in Playfair’s axiom, the following are three logical possibilities;

a. There are no lines that go through p that are parallel to l.

b. There is one line that goes through p that is parallel to l. (Playfair’s axiom.)

15 2.4 Axiomatic geometry

c. There are two or more lines that go through p that are parallel to l.

These logical cases give rise to three diﬀerent kinds of geometry, where case b. is Playfair’s axiom which is equivalent to the parallel axiom relative Euclidean geometry, thus gives rise to Euclidean geometry. If a. or c. replace the ﬁfth axiom two types of non-euclidean geometry rise, elliptic and hyperbolic geometry respectively [9, pp. 425-426].

The fact that this construction results in an axiomatic system that fulﬁls P1-P4 and an additional axiom, a. or c., which if one of them is true precludes the parallel axiom to be true, actually proves the independence of the parallel axiom from the four other axioms. One can therefore be sure of that no one would have never been able to prove the parallel axiom from P1-P4, and thus made it a theorem, so the eﬀorts to do so, are no longer ongoing.

A model of the elliptic geometry is a spherical model, like a globe. One can think of it as Euclidean geometry, but on the surface of a sphere, this is illustrated in ﬁgure 1:

Figure 1: An illustration of the spherical model of elliptic geometry, and an interpretation of a triangle in this model; inscribed on the spherical surface.

Figure 1 also shows what a triangle looks like in elliptic geometry with the spherical model. One can notice that the angles of the two lowermost vertices seems to be close to 90◦, and that the angle of the upper vertex is not at all close to zero degrees, this indicates that the sum of the angles of the triangle is more than 180◦. This demonstrates that elliptic geometry is a completely diﬀerent system than Euclidean geometry where the sum of all angles of a triangle always equals 180◦. This is actually a very applicable type of geometry, since the shape of the earth is a sphere. On earth Euclidean geometry

16 2.4 Axiomatic geometry can only be used as an approximation when working in small scales. When measuring for example a long straight line on the face of the earth the line is only straight with respect to the earth which is a sphere, with respect to Euclidean space the same line is curved.

A model of interpretation of hyperbolic geometry is Euclidean geometry on a saddle surface, see ﬁgure 2.

Figure 2: An illustration of the model of hyperbolic geometry. An interpretation of a triangle in this model is also shown; inscribed on the saddle surface.

Figure 2 also shows a triangle in this geometry, where it can be observed that the sum of the angles is less than 180◦, which demonstrates its diﬀerence from both Euclidean and elliptic geometry.

These diﬀerent systems, Euclidean, elliptic and hyperbolic geometry, show the importance of equivalence relative to a system when trying to make changes in the axioms of a system. To obtain the same system, an axiom can be replaced with some principle that is equivalent to that axiom relative to that system, but it can not be replaced by something that is not, without changing the system.

The rest of this thesis will be devoted to refuting Swanson’s and Hansen’s claim that the induction, well-ordering, multiplication, and pigeonhole principles are equivalent to each other relative to a system where the principles are not theorems. This will be done by showing a counterexample in a weaker system than ZFS set theory where this is the case.

17 3 Natural numbers

The natural numbers, 0, 1, 2, 3, 4, ..., have a quite intuitive meaning, through counting for example. They sometimes include the number zero, and sometimes not. Historically zero has often been omitted from the set of natural numbers. This might be because zero is not as intuitive as a number, and therefore not as easily accepted as a natural number as the others.1 Because of the intuition of the concept of natural numbers it is an easy concept for people to grasp, and for thousands of years the natural numbers have been an object of analysing and discovering. It is intuitive that you can have more or less of something. That one apple is one apple, and that you can have more than one apple. It makes sense to talk about some number of objects, the number of days in a week, or the number of students in a class and so on. This intuition gives rise to the natural numbers as something that exists regardless of any mathematical theory.

In the late 19th century mathematicians found interest in trying to rigorously found the natural numbers with an axiomatic system. And in 1889 the Italian mathematician Giuseppe Peano formed the axiomatic system for the natural numbers used today. Peanos work was based on axioms received in a letter from another mathematician, Richard Dedekind. These axioms are mostly called the Peano axioms, but sometimes the Dedekind-Peano axioms [5, pp. 22].

3.1 The Peano axioms The Peano axioms are the foundation of the theory of arithmetic. The primitive notions of this axiomatic system are natural number, 0 and the successor function, S(x), where S(x) is called the successor of x. Let ω be a set. The Peano axioms are:

Peano axiom 1 (P1). 0 is a member of ω; 0 ∈ ω.

Peano axiom 2 (P2). The successor of an element in ω is a member of ω; If x ∈ ω, then S(x) ∈ ω,

Peano axiom 3 (P3). 0 is not the successor of anything; S(x) 6= 0 for all x in ω.

1Even today it differs from book to book if the natural numbers include zero or not. In Swanson and Hanson’s article they do not include zero, but it makes no difference to the arguments given in this thesis if zero is defined as natural number or not.

18 3.1 The Peano axioms

Peano axiom 4 (P4). Two numbers of which the successors are equal are themselves equal; If x and y are in ω, and if S(x) = S(y), then x = y.

Peano axiom 5 (P5). If S ⊂ ω is such that (i) 0 ∈ S (ii) S(x) ∈ S whenever x ∈ S then S = ω.

Axiom P5 is called the induction principle where (i) is called the base step and (ii) the induction step.

The set of natural numbers is the standard model of this arithmetic theory, and every additional model is in fact going to be isomorphic to the standard model, thus there exists no non-standard models of this arithmetic theory.2 The natural numbers can be made a model of the Peano axioms by deﬁning the natural number 0, as 0, deﬁne the successor function S(x) as:

Deﬁnition 3.1. S(x) := x + 1, and deﬁne the natural number n + 1 as:

Definition 3.2. n + 1 := S(n), where n is a natural number. Then because of the definition of 0 as a natural number, definition 6.1 and 6.2, the natural number one is defined as 1 := S(0) = 0 + 1, two is defined as 2 := S(1) = 1 + 1, three is defined as 3 := S(2) = 2 + 1, and so on. Then the set of natural numbers, N, fulfils all the Peano axioms as the set ω.

2This is true under the condition that the induction principle is given in second-order logic, which is the case in this thesis. If the induction principle is to be formulated in first-order logic it has to be replaced with an axiom scheme consisting of countably many axioms. But according to Dedekind the induction principle in first-order logic does not suffice to prevent the existence of non-standard models of the Peano axiomatic system [18, pp. 98-103].

19 4 The Zermelo-Fraenkel-Skolem system

The Zermelo-Fraenkel-Skolem system, denoted ZFS, is a system of axiomatic set theory. The most used name of this axiomatic system is ZFC set theory which stands for Zermelo-Fraenkel set theory with the axiom of choice. The Zermelo-Fraenkel set theory, denoted ZF, is also a commonly mentioned theory, this is the theory consisting of the same axiomas as ZFC, but without the axiom of choice. In this thesis the name Zermelo- Fraenkel-Skolem, ZFS, is used because this is the name of the theory that Swanson and Hansen use in their article3.

The ZFS set theory can be written in complete logical terms, but here a more everyday language is used. The material on the ZFS axiomatic system is obtained from Halmos [6], Mendelson [12] and Tiles [16].

Sets are the elementary objects of the ZFS axiomatic system. The concept of sets and elements of a set are primitive notions in this theory. An attempt to explaining the terms ”set” and ”element” even though they can not be deﬁned is as follows: ”A set is formed by the grouping of single unique elements into a whole. A set is a plurality thought of as a unit” [16, pp. 99]. The relations of the ZFS set theory are the logical connectives plus a relation called belonging. If an element x belongs to the set A, or x is an element of A, it is written x ∈ A. The following seven axioms constitutes the ZFS set theory:

Axiom 1 ( Axiom of Extension). Given any two sets, A and B, if A and B have the same elements they are equal and we write A = B.

Axiom of Extension states that the identity of a set is only determined by its elements, not by the order of the elements.

Axiom 2 (Axiom of Speciﬁcation). To every set A and to every condition C(x) there corresponds a set B whose elements are exactly those elements x of A for which C(x) holds.

A condition is a logical sentence that selects those x ∈ A that fulﬁls the condition of the logical sentence. The Axiom of Speciﬁcation makes it possible to form subsets from sets already assured by the other axioms.

The assumption in the Axiom of Speciﬁcation that there must be a pre existing set from where the elements corresponding to the condition C(x) are chosen from, i.e the part ”To every set A”, eliminates the so called Russell’s paradox. If this part was not in the axiom, every possible element, which includes every possible set, would be available for the condition to choose, and Russell’s paradox would be:

3It is probably the ZF set theory they refer to and not ZFC set theory, but both of the systems have the same properties when relating to the arguments in this thesis.

20 Russell’s Paradox. Deﬁne the set R as

R = {x : x∈ / x}, which means that R is the set of every set that are not a member of itself.

Is R then a member of itself?

First assume that R/∈ R. Then by definition of R, R is a member of R, i.e. R ∈ R. Which is a contradiction. Then assume that R ∈ R. Then the definition of R gives that R can not be a member of R since it fulfils the property of the set. So R/∈ R. Which also is a contradiction. So both cases leads to contradictions which establishes the paradox;

R is neither not in R or in R.

The additional claim to have some pre existing set A with available elements, as in Axiom of Speciﬁcation, deals with the paradox. This property makes the set in question become R = {x ∈ A : A is a set and x∈ / x}. Assume now that R ∈ R, then it must be true that R ∈ A and R/∈ R, which is contradictory. Then R/∈ R. Before the additional claim to have a pre existing set A this would again lead to contradiction, since then R would be in R just because it fulﬁlled the condition of the set. But now the implication is that R/∈ A, and that is it. No more problem. So the conclusion in ZFS set theory is that R is not a member of itself. Thus in ZFS set theory there exists no set of all possible sets, and no set is a member of itself.

The first and second axioms ensures the first concrete set; Let A be a set, and apply the axiom of specification with the condition {x ∈ A : x 6= x}. This obviously gives a set with no elements, since every element equals itself. The axiom of extension imply that there can only be one set with no elements; Assume there are two such sets, E1 and E2. Then all their elements are the same since they have no elements, by the vacuous truth, thus E1 = E2. This set, with no elements, is called the empty set, and is denoted ∅.

Axiom 3 (Axiom of Pairing). For any two sets there exists a set that they both belong to.

The Axiom of Pairing assures the existence of sets with members. Since ∅ is a set the set {∅, ∅} may be formed. The elements of a set are unique, thus {∅, ∅} is equal to the set of the empty set {∅}. Now two sets are ensured. The axiom then assures the empty set, and the set of the empty set, put into a new set, {∅, {∅}}. This procedure can go on, and the set theory, with these three ﬁrst axioms, then contains a lot of possible sets.

21 4.1 The natural numbers deﬁned in ZFS set theory

Axiom 4 (Axiom of Unions). For every collection C there exists a set ∪ such that if s ∈ X for some X in C, then x ∈ ∪.

A collection is a set of sets. The set ∪ is called the union, and we say that we take the union of C when constructing the set described in the Axiom of Unions. This axiom establishes a new way of constructing sets. It assures sets with an assemble of elements by taking the union of a collection of sets already deﬁned.

Axiom 5 (Axiom of Powers). If E is a set, then there exists a set P(E) such that if X ⊂ E, then X ∈ P(E). P(E) is called the power set of E.

The Axiom of Powers is also a way of constructing new sets. The term X ⊂ E is pro- nounced ”X is a subset of E”, and it means that every element of X also is an element of E, but every element of E is not necessarily an element of X. If the latter is the case then X and E are equal, by the Axiom of Extension.

Let the successor function in ZFS set theory be denoted by S∗(x) and be deﬁned as;

Deﬁnition 4.1. For every set x the successor function S∗(x) is the set obtained by adjoining x to the elements of x, thus S∗(x) = x ∪ {x}.

∗ An example of a successor is the successor of the set A = {∅, {∅}}. S (A) is by deﬁnition {∅, {∅}} ∪ {{∅, {∅}}} = {∅, {∅}, {∅, {∅}}}.

Axiom 6 (Axiom of Inﬁnity). There exists a set containing the empty set ∅ and containing the successor of each of its elements.

The Axiom of Infinity assures the existence of an infinite set, namely the set ω = {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}, ...}. This kind of set constructed and assured by the Axiom of Infinity is called an inductive set, and ω is the smallest inductive set, i.e. ω is a subset of every inductive set.

Axiom 7 (Axiom of Choice). If A is a set, all of whose elements are non- empty sets and no two of which have any elements in common, then there is a set B which has precisely one element in common with each element of A.

The Axiom of Choice gives another way to construct new sets.

4.1 The natural numbers defined in ZFS set theory In ZFS theory everything at disposal is sets. But the natural numbers can be deﬁned in ZFS set theory. This is possible by the following two deﬁnitions:

22 4.1 The natural numbers deﬁned in ZFS set theory

Deﬁnition 4.2. 0 := ∅.

Deﬁnition 4.3. n = S(n − 1) := S∗(n − 1) = n − 1 ∪ {n − 1} = {0, 1, 2, ..., n − 1}, where n is an arbitrary natural number. This is one way to deﬁne the natural numbers and it is called the von Neumann construction of the natural numbers, after the Hungar- ian mathematician John von Neumann[19].

This deﬁnition gives that the natural number one in ZFS set theory is deﬁned as

∗ Deﬁnition 4.4. 1 := S (0) = ∅ ∪ {∅} = {∅} = {0}, and two as

∗ Deﬁnition 4.5. 2 := S (1) = {∅} ∪ {{∅}} = {∅, {∅}} = {0, 1}, three as

Definition 4.6. 3 := S∗(2) = 2 ∪ {2} = {0, 1, 2}, and so on. The natural numbers in ZFS set theory, as defined above, are exactly the members of the inductive set ω, thus the set of all natural numbers, N, is defined as ω.

A way to know that this definition of the natural numbers is acceptable, that it really is the natural numbers that is defined, is to show that the natural numbers, defined in ZFS set theory, fulfils the Peano axioms. In ZFS set theory P1 and P2 becomes;

Peano axiom 1 (P1). ∅ is a member of ω; ∅ ∈ ω.

Peano axiom 2 (P2). The successor of an element in ω is a member of ω; If x ∈ ω, then S∗(x) ∈ ω.

Both P1 and P2 are automatically guaranteed by the Axiom of Inﬁnity. Axiom P3 becomes:

∗ Peano axiom 3 (P3). ∅ is not the successor of anything; S (x) 6= ∅ for all x in ω.

∗ The proof of why this is true is trivial; S (x) always contains x, by deﬁnition, since ∅ is empty it can not be the successor of anything. P4 becomes:

23 4.1 The natural numbers deﬁned in ZFS set theory

Peano axiom 4 (P4). Two sets of which the successors are equal are themselves equal; If x and y are in ω, and if S∗(x) = S∗(y), then x = y.

This can also be proven in ZFS, but is a little more intricate. Halmos gives a proof of this [6, pp. 46-47]. The induction principle, P5, becomes;

Peano axiom 5 (P5). If S ⊂ ω is such that (i) ∅ ∈ S (ii) S∗(x) ∈ S whenever x ∈ S then S = ω.

This is guaranteed by the Axiom of Inﬁnity and the Axiom of Extension which gives that the inductive set ω is unique.

This shows that the Peano axioms can be derived from ZFS set theory, which implies that ZFS set theory contains all properties of Peano arithmetic plus additional ones. This means that the ZFS set theory is a stronger theory, i.e. has more properties, then Peano arithmetic.

24 5 What is order?

The axioms of ZFS set theory generate a system where a set is totally determined by its elements, and where the order of the elements is insigniﬁcant. In fact the axioms give no information about what order of elements is. So then what is a least element? To understand the well-ordering principle, where this is a central concept, the concept of order will be introduced in this section.

To make the distinction between sets with the same elements, but with diﬀerent order of the elements, the concept of a relation must be deﬁned.

Deﬁnition 5.1. R is a relation on the set A if for every z ∈ R there exists x, y ∈ A so that z = (x, y), where (x, y) is an ordered pair. If (x, y) ∈ R it is denoted xRy, where an ordered pair is deﬁned as:

Deﬁnition 5.2. (a, b) = {{a}, {a, b}}

If (x, y) and (y, x) are just ordinary sets, then they are the exact same set, i.e (x, y) = (y, x). But definition 5.2 says that if (x, y) and (y, x) are ordered pairs they are different sets since the set {{x}, {x, y}} differs from {{y}, {y, x}}.

Relations on a set can have different properties, thus be different kinds of relations. The definition of three such properties are:

Deﬁnition 5.3. A relation R on a set A is called reﬂexive if xRx for every x in A.

Deﬁnition 5.4. A relation R on a set A is called transitive if xRy and yRz imply that xRz for every x, y, z ∈ A.

Deﬁnition 5.5. A relation R on a set A is called antisymmetric if xRy and yRx imply that x = y for all x, y ∈ A.

With these relation properties a type of order can be deﬁned on a set. This order is called a partial order.

Deﬁnition 5.6. A relation R that is reﬂexive, transitive and antisymmetric on a set A, is called a partial order in A.

With one additional property another relation called a total order can be deﬁned;

Deﬁnition 5.7. Let R be a partial order on A, if for every x, y ∈ A either xRy or yRx, then R is called a total order in A.

Now the concept of an order in a set makes sense, but what about an ordered set? An ordered set is a set A with some order R in it. A way to specify this is to, in some exact way, combine a relation with a set, this can be done by the ordered pair (A, R) where R is a relation on the set A. If R is a partial order in A, by deﬁnition 5.6, then (A, R) is

25 called a partially ordered set. If R is a total order in A, by deﬁnition 5.7, then (A, R) is called a totally ordered set.

One important example of a totally ordered set is the ordered pair (N, ≤), i.e. the set of all natural numbers with the relation ≤ on it. The relation ≤ is reﬂexive on N since a ≤ a for all a ∈ N, it is transitive in N since if a ≤ b and b ≤ c is true, then a ≤ c for all a, b, c, ∈ N and it is antisymmetric in N since if a ≤ b and b ≤ a then a = b for all a, b ∈ N. For the natural numbers it is also true that for all a, b ∈ N either a ≤ b or b ≤ a is true, thus it is a total order [6, pp. 54-55]. The logical connective ”or” includes the possibility that both a ≤ b and b ≤ a are true, and in that case a = b.

Since the natural numbers are totally ordered with ≤ in it, the set has the property that given any two distinct elements in the set, one of them must be smaller than the other.

Another type of order is called a well-ordering:

Deﬁnition 5.8. A well-ordering on a set A is a total order R on A so that for any non-empty subset of A, the subset contains a least element. An ordered set (A, R) where R is a well-ordering is called a well-ordered set.

Theorem 5.9. The set of natural numbers, N, with the relation ≤ on it is a well-ordered set, i.e. the ordered pair (N, ≤) is a well-ordered set.

Proof. We must show that every non-empty subset of N with the relation ≤ on it has a least element. Let S ⊂ N, where S is non-empty. First let us establish the fact that if S has a least element, then it is unique; Assume that both n1 and n2 are least elements of S, then the following would apply; both n1 ≤ n2 and n2 ≤ n1 which imply that n1 = n2. Now assume that S does not have a least element. We construct the set A with all natural numbers that are not in S, i.e

A = {n ∈ N : n∈ / S}.

Let’s prove by induction that A = N; Base step: The least of all natural numbers is 0, thus 0 can not be in S, because that would make it the least element of S. This implies 0 ∈ A. Induction step: Assume 0, 1, 2, ..., k ∈ A. If k + 1 ∈ S, then for k + 1 not to be the least element of S some element ki must be such that ki ∈ S and ki < k + 1. But there is no element, ki such that k < ki and ki < k + 1, since N is totally ordered, and all 0, 1, 2, ..., k is in A. This shows that k + 1 would be a least element in S and can therefore not be in S, thus k + 1 ∈ A.

Then by the induction principle A = N. This implies that S must be empty, which

26 is a contradiction. Thus S has a least element, and by deﬁnition 5.8 the natural numbers, N, with the relation ≤ on it is a well-ordered set.

Theorem 5.9 says that the set of natural numbers, with the relation ≤ on it, is well- ordered. Because of this the successor of some natural number will always be bigger than the number itself. The second Peano axiom, P2, gives that whenever x is in N then S(x) is in N, this makes it impossible to ﬁnd a biggest natural number. So there exists inﬁnitely many natural numbers.

27 6 An axiomatic system, P , weak enough for the principles in question not to be theorems of the system

The aim of this section is to refute the equivalence between the multiplication, pigeonhole, well-ordering and induction principles relative to an axiomatic system where the principles are not theorems. This will be done by focusing on the refutation of the equivalence between the well-ordering principle and the induction principle. To obtain this a theory, P, where neither the induction principle nor the well-ordering principle is a theorem, will be constructed. Relative to this theory it will be determined if the two principles are equivalent or not in the way described in deﬁnition 2.7. This means adding the induction principle to the axioms of P resulting in a new theory, P1, and determining if the well-ordering principle is a theorem of P1, and then adding the well-ordering principle to P resulting in a new theory, P2, and determining if the induction principle is a theorem of P2.

For the well-ordering principle to be a theorem of P1 there has to exist a proof of the well-ordering principle in P1 in complete logical terms, and the same applies for the induction principle to be a theorem of P2. Theorem 2.8 gives that ”every theorem of a theory is a tautology relative to that same theory”, and because Modus Ponens is logically equivalent to Modus Tollens this is exactly the same as ”if a wff is not a tautology relative to some theory that wff is not a theorem of the same theory”. This means that to refute the equivalence between the induction principle and the well-ordering principle relative to P it suffices to show that either the well-ordering principle is not a tautology relative to P1 or the induction principle is not a tautology relative to P2.

The Peano axioms give the foundation of pure arithmetic, but gives little foundation when it comes to set theory. To make sense of the well-ordering principle within this theory some additional properties of sets, such as being able to construct subsets, are needed. Looking at the induction principle as stated in the article by Swanson and Hansen, see MI, it looks like some set theory is needed to make sense of the induction principle as well. But the induction principle can be formulated in another way not using set theory:

The induction principle. Let S(x) be a statement involving n. If (i) S(0) holds, and (ii) for every k ≥ 0, if S(k) holds then S(k + 1) holds, then for every n ≥ 0, the statement S(n) holds.

The formulation of the induction principle in the article by Swanson and Hansen is really just a special case of the induction principle given above, where the condition S(x) is x ∈ S. This explains why the Peano axioms, as stated in section 3, suﬃces when founding the natural numbers, i.e. that no additional set theory is needed in that system.

28 Because the well-ordering principle requires set theoretic properties a theory, called P, containing the Peano axioms, except the induction axiom, and some axioms from set theory is needed to refute the equivalence between the induction principle and the well-ordering principle. P has to be constructed with care because if too many of the set theoretic axioms are used there is a risk of adding so many properties to the system that the induction principle and the well-ordering principle become theorems of the system.

The ﬁrst thing to establish in this new axiomatic system is its primitive notions. The combination of the primitive notions from arithmetic and set theory have to be combined, thus the primitive notions of the system are: set, element, natural number, 0 and successor function. Let the ﬁrst four Peano axioms be axioms of the system:

Axiom 1 (P1). 0 is a member of ω; 0 ∈ ω.

Axiom 2 (P2). The successor of an element in ω is a member of ω; If x ∈ ω, then S(x) ∈ ω,

Axiom 3 (P3). 0 is not the successor of anything; S(x) 6= 0 for all x in ω.

Axiom 4 (P4). Two numbers of which the successors are equal are themselves equal; If x and y are in ω, and if S(x) = S(y), then x = y.

Since the concept of a set is one of the primitive notions of the system the set ω is at disposal as an existing set. But as already mentioned; it must be possible to form a subset in this new theory. To add the property of making subsets the Axiom of Speciﬁcation is added to the system.

Axiom 5 (Axiom of Speciﬁcation). To every set A and to every condition C(x) there corresponds a set B whose elements are exactly those elements x of A for which C(x) holds.

29 6.1 P with the induction principle

The next axiom added is the Axiom of Extension. This is a property that is needed to be able to determine what is meant by the equality of sets.

Axiom 6 ( Axiom of Extension). Two sets, A and B, are equal if and only if they have the same elements. We write A = B.

In this constructed system the induction principle is not a theorem. This can be concluded because the weakest known set theory where the Peano axioms are theorems is called general set theory, and consists of the Axiom of Speciﬁcation, the Axiom of Extension and one additional axiom called the Axiom of Adjunction[1, pp. 196]. Where the axiom of Adjunction is as follows:

Axiom (Axiom of Adjunction). For any two sets, A and B, there is a set W = A ∪ {B} given by adjoining the set B to the elements of A.

This means that without the Axiom of Adjunction the Peano axioms are not theorems of the system, thus the induction principle is not a theorem of the system. Since the induction principle is needed to prove the well-ordering principle, see the proof of theorem 5.9, the well-ordering principle is also not a theorem of this system.

So the theory P is now an axiomatic system in which neither the induction principle nor the well-ordering principle is a theorem.

6.1 The axiomatic system P with the induction principle added as an axiom Now the induction principle will be added to the theory P to form a new theory called P1. Thus the theory P1 consists of the following axioms:

Axiom 1 (P1). 0 is a member of ω; 0 ∈ ω.

Axiom 2 (P2). The successor of an element in ω is a member of ω; If x ∈ ω, then S(x) ∈ ω,

Axiom 3 (P3). 0 is not the successor of anything; S(x) 6= 0 for all x in ω.

30 6.1 P with the induction principle

Axiom 4 (P4). Two numbers of which the successors are equal are themselves equal; If x and y are in ω, and if x+ = y+, then x = y.

Axiom 5 (The induction principle). If S ⊂ ω is such that (i) 0 ∈ S (ii) S(x) ∈ S whenever x ∈ S then S = ω.

Axiom 6 (Axiom of Speciﬁcation). To every set A and to every condition C(x) there corresponds a set B whose elements are exactly those elements x of A for which C(x) holds.

Axiom 7 ( Axiom of Extension). Two sets, A and B, are equal if and only if they have the same elements. We write A = B.

Now let the model of this new theory be the natural numbers. This is done with the same definition as in section 3. By defining the natural number 0 as 0, defining the successor function S(x) as: Definition 6.1. S(x) := x + 1, and defining the natural number n + 1 as: Definition 6.2. n + 1 := S(n), where n is a natural number.

This makes the natural numbers N fulﬁl all Peano axioms, and in addition this theory gives the possibility of taking subsets of the set of natural numbers N and comparing such subsets.

Theorem 5.9 gives that the natural numbers are well-ordered, thus the well-ordering principle is true in this model. In section 3 it was mentioned that every additional model of the Peano arithmetic is isomorphic to the natural numbers, thus the natural number is really the only model of the Peano axioms. This fact also applies to the theory P1 that consists of the Peano axioms and two additional axioms, since adding more properties to a system does not generate new models of the system. Since the natural numbers then are the only model of the theory P1, and the well-ordering principle is true in this model the well-ordering principle is a tautology relative to P1. Looking at this theory one can therefore not conclude that the induction principle and the well-ordering principle are not equivalent relative to P.

31 6.2 P with the well-ordering principle

6.2 The axiomatic system P with the well-ordering principle added as an axiom To further evaluate the equivalence between the induction principle and the well-ordering principle relative to P the axiomatic system, P2, which consists of the axioms of the theory P with the well-ordering principle added to these axioms, must be considered. The axioms of P2 is then the following:

Axiom 1 (P1). 0 is a member of ω; 0 ∈ ω.

Axiom 2 (P2). The successor of an element in ω is a member of ω; If x ∈ ω, then S(x) ∈ ω,

Axiom 3 (P3). 0 is not the successor of anything; S(x) 6= 0 for all x in ω.

Axiom 4 (P4). Two numbers of which the successors are equal are themselves equal; If x and y are in ω, and if x+ = y+, then x = y.

Axiom 5 (The well-ordering principle). Every non-empty subset of ω has a least element, thus ω is well-ordered.

Axiom 6 (Axiom of Speciﬁcation). To every set A and to every condition C(x) there corresponds a set B whose elements are exactly those elements x of A for which C(x) holds.

Axiom 7 ( Axiom of Extension). Two sets, A and B, are equal if and only if they have the same elements. We write A = B.

To refute the equivalence between the well-ordering principle and the induction principle relative to P it must be shown that there exists a model of P2 where the induction principle is not true, thus that the induction principle is not a theorem of P2.

First let the natural numbers, N, be the model of the system. This is a valid model

32 6.3 Ordinal numbers

of P2 since it was a valid model of P1 and the natural numbers are well-ordered. So far, when just looking at the natural numbers as a model of P2, it might be possible for the induction principle and the well-ordering principle to be equivalent relative to P, since the induction principle is true for the natural numbers.

But now a diﬀerent kind of numbers than the natural will be presented for the purpose of considering these numbers as a model of the theory P2.

6.3 Ordinal numbers

The Peano axioms founds the natural numbers N, which contains 0 and contains the successor of x whenever the set contains x. The ZFS set theory also assures the natural numbers since the Peano arithmetic is containd in it. In ZFS set theory the natural numbers are deﬁned with the successor function S∗(x) = x ∪ {x}. But is it also possible to take the successor of the set of all natural numbers. Lets call the set of all natural numbers ω, and let the successor function be as in deﬁnition 4.1. The successor of ω then becomes S∗(ω) := ω ∪{ω} = {0, 1, 2, ..., ω}, this set is called ω +1. This procedure can continue by taking the successor of ω+1: ω+2 = S∗(ω+1) := ω+1∪{ω+1} = {0, 1, 2, ..., ω, ω+1}; and so on until this is done the same number of times as there are natural numbers, which is called countably many times [5, pp. 55]. Then the resulting number is ω+ω. One can take the successor of this number again and get S(ω+ω) = ω+ω+1, and repeat the procedure.

To make a more rigorous definition of all ordinal numbers the concept of an initial segment must be defined. Definition 6.3. For a well ordered set (A, ≤) and t ∈ A, the initial segment of (A, ≤) determined by t is defined as seg(t)= x ∈ A : x < t. Now the ordinal numbers can be defined as: Definition 6.4. An ordinal number, a, is a well ordered set (A, ≤) such that a := seg(a). This definition of the ordinal numbers gives for example that ω + 3 = {0, 1, 2, 3, ..., ω, ω + 1, ω + 2}. This is called the von Neumann ordinals, and is the same type of construction as the the von Neumann construction of the natural numbers, see section 4. The von Neumann construction of the natural numbers is really just a subset of the ordinal numbers.

Since the set of these new ordinal numbers 1, 2, 3, ..., ω, ω + 1, ω + 2, ..., ω + ω, ω + ω + 1, ... is well ordered, choosing any two distinct elements from this set one of them will be smaller than the other. But something that distinguishes this new type of numbers from the natural numbers is that not every number has an immediate predecessor. Where a predecessor, and an immediate predecessor is deﬁned as:

Deﬁnition 6.5. Let (A, ≤) be a set with an order in it, and let x ∈ A. Then all xi ∈ A where xi ≤ x is a predecessor of x. y is called an immediate predecessor of x if S(y) = x.

33 6.4 The refutation of equivalence

A consequence of deﬁnition 6.5 is that if y is the immediate predecessor of x there is no element strictly between the two [6, pp. 74-75].

Theorem 6.6. The ordinal number ω has no immediate predecessor.

Proof. Assume that ω do have an immediate predecessor, and let x be it. Then x is a natural number because of the construction of ω. There is always a natural number that is bigger than x because there are inﬁnitely many natural numbers. Every natural number is smaller than ω. This implies that here is some number between x and ω which is a contradiction to the assumption that x is the immediate predecessor of ω. Thus ω has no immediate predecessor.

This sort of ordinal number, with no immediate predecessor, is called a limit ordinal. The ordinal number ω is also the first infinite ordinal number. All the natural numbers are finite numbers. All ordinal numbers besides the naturals, i.e. the infinite ones, are called transfinite numbers [6, pp. 97].

In the theory P2 this von Neumann definition of the ordinal numbers can not be applied since the tool of taking the union is unavailable in this theory. But a subset of the ordinal numbers can be defined in another way which is valid in P2. This is because only the properties available in P2 are used in the following definition of the ordinals up to ω + ω: Definition 6.7. Let the natural number 0 be an ordinal number.

Deﬁnition 6.8. Let the set of all natural numbers, denoted ω, be an ordinal number.

Definition 6.9. Let all S(n) be an ordinal number whenever n is an ordinal number, where the successor function S(x) is defined as S(x) := n+1. Definition 6.9 generates the ordinal number that is the successor of 0, namely 0 + 1 = 1, and generates the ordinals 2, 3, 4, 5, .... This definition also generates the ordinal number that is the successor of ω, namely ω+1, and generates the ordinals ω+2, ω+3, ω+4, .... The definition 6.7, 6.8 and 6.9 makes it possible to use the ordinals up to ω +ω, i.e. the set {0, 1, 2, ..., ω, ω +1, ω +2, ...}, in the theory P2.

6.4 The refutation of equivalence between the induction principle and the well-ordering principle relative to a theory where non of the principles are theorems Is the set of ordinal numbers up to ω + ω, i.e. {0, 1, 2, ..., ω, ω + 1, ω + 2, ...} a model of the theory P2? For it to be a model it must fulfil all the axioms of P2. The set fulfils P1 since the natural numbers is a subset of ω + ω and 0 ∈ N. It fulfils P2 because of the way the ordinal numbers are constructed, see subsection 6.3. P3 and P4 are also fulfilled by construction. The set of all ordinals up to ω + ω is a subset of for example the ordinal number ω + ω + 3 which is a well-ordered set, both according to definition 6.4, thus ω + ω

34 6.4 The refutation of equivalence must be a well-ordered set, so it fulﬁls The well-ordering principle. The two set theoretic axioms are also fulﬁlled by the ordinal numbers up to ω + ω. This means that the set ω + ω is a model of P2.

The following must apply for the induction principle to be true for the ordinal numbers up to ω + ω:

The induction principle. If S is a subset of the ordinal numbers up to ω + ω such that: (a) 1 ∈ S, and (b) n + 1 ∈ S whenever n ∈ S, then S is the set of all ordinal numbers up to ω + ω.

Lets assume that the induction principle does apply in this model. And then assume that the subset S of the ordinals up to ω + ω contains ω. n + 1 is by deﬁnition the successor of n, and every element, n + 1, in S ensured by (b), has an immediate predecessor, namely n. But in subsection 6.3 it was proven that ω has no immediate predecessor. The consequence of this is that ω can not be in S. This contradicts the assumption that ω is in S, thus S can not be the set of all ordinal numbers up to ω + ω.

This means that the induction principle is not true in this model of P2, thus the induction principle is not a theorem of P2. Figure 3 shows an overview of the information found in this section.

35 6.4 The refutation of equivalence

Theory: Model: Principles:

P 1

- P1 The Well-ordering - P2 The natural numbers principle - P3 - P4 - The Induction principle - Axiom of Specification - Axiom of Extension

P 2 The induction principle The natural numbers - P1 - P2 - P3 - P4 The ordinal numbers The induction - The Well-ordering principle up to ω+ω principle - Axiom of Specification - Axiom of Extension

Figure 3: A schematic diagram of the information found in this section. The axioms of theory P1 and P2 respectively is shown, and the applicable models for each of them. An applicable model for the theory P1 is the natural numbers, and it is shown that the well-ordering principle is true in this system. For the theory P2 both the natural numbers and the ordinal numbers up to ω + ω are applicable models. With the natural numbers as a model of theory P2 it is shown that the induction principle is true, but with the ordinal numbers up to ω + ω as a model of P2 it is shown that the induction principle is not true.

For the induction principle and the well-ordering principle to be equivalent relative to P the induction principle must be a theorem of the theory P2. The overview in ﬁgure 3 shows that this is not the case because the induction principle is not true when using the ordinal numbers up to ω + ω as a model of P2. This leads to the important conclusion that the well-ordering principle and the induction principle are not equivalent relative to an axiomatic system where non of the principles are theorems. This conclusion implies of course that the four principles: induction, well-ordering, multiplication and pigeonhole can not all be equivalent relative to an axiomatic system where non of the principles are theorems when two of them are not. This means that the claim that because of logical equivalence between the multiplication, pigeonhole, induction, and well-ordering principles relative to ZFS set theory, the same principles are equivalent to each other relative to a system where the principles are not theorems, is indeed refuted.

36 7 Discussion

The refutation of the equivalence between the four principles evaluated in this thesis con- sisted of two separate parts. The ﬁrst part was to show that all theorems of an axiomatic system are logically equivalent to each other relative to that system, see subsection 2.3. With this in mind, Swanson and Hansen are not wrong in their main statement; that the four principles are equivalent when working in the ZFS set theory. Since the ZFS set theory is not complete, there is no guarantee that logical equivalence between theorems in this theory means that there exists a proof between them. So the only real result of Swanson’s and Hanson’s article would be, assuming that their proof (MP) ⇒ (PH) ⇒ (WO) ⇒ (MI) ⇒ (MP) is correct, that they have shown that there exists a proof between these four theorems in ZFS set theory.

If this was the only statement made in their article, their only mistake would have been to publish a quite inconsequential article. But this sentence suggests that they are not aware of this: It is interesting to note that these two basic concepts(the MP and PH) are really more basic than they might at ﬁrst appear. They are, in fact, equivalent to each other and they are a part of the axiomatic structure of the natural numbers in the sense that they are equivalent to the following principles(the WO and MI). [15] They seem to imply that this logical equivalence relative the ZFS set theory, that is the result of their article, automatically leads to equivalence in another system. This other system they call ”the axiomatic structure of the natural numbers”, and it is indicated that the well-ordering principle and the induction principle are part of the axioms of this system. Because of this indication that the induction principle is a part of this axiomatic structure they mention, it is assumed in this thesis that they refer to the Peano axioms. In this system all of the principles are not theorems and one can no longer guarantee logical equivalence between the principles relative such a system. If this is the conclusion that is indicated; that because principles that are equivalent relative one axiomatic system, they must also be equivalent relative another axiomatic system, Swanson and Hanson show a lack of understanding of the concept of equivalence and of the subject of the structure of axiomatic systems.

The main purpose of the thesis was to refute the equivalence between these four principles relative to a system where the principles are not theorems, this was done by refuting the equivalence between the induction principle and the well-ordering principle relative to such a system. Swanson and Hansen seem to imply that both the induction principle and the well-ordering principle are part of the ”axiomatic structure of the natural numbers”, thus are equivalent relative to a weak system where none of the principles are theorems. This assertion is actually more usual that one might think. There seems to be a common misconception of the concept of the equivalence between these two principles, which is probably why Swanson and Hansen also have this misconception relating this

37 matter. In their article they actually just refer to some other source when reporting the ”Well-ordering implies mathematical induction” part of their proof with the comment: ”This well known result may be found in Long [3,pp.20-21] and many other sources.”.

Nievergelt asserts that these two principles are equivalent with the additional condition ”on the set N”[13, pp.250]. Vaderlind also claim this equivalence of the well-ordering principle and the induction principle with just the natural numbers as a model [17, pp. 118-119]. These assumptions probably rely on the fact that by using the natural numbers as a model of P1 the well-ordering principle is true, and if using the natural numbers as a model of P2 the induction principle is true. But this is not the case with the ordinal numbers up to ω + ω as a model of P2. With the right understanding of equivalence relative an axiomatic system this implies that the two principles are not equivalent relative to P because the induction principle does not apply in every model of P2. This relationship between the induction principle and the well-ordering principle when using the natural numbers as a model do not imply equivalence between them, but gives information about the properties of the natural numbers. The fact that this misconception has become quite common is probably just spreading it even more.

The existence of a model of the theory P2 where the induction principles is not true gives the consequence that the induction principle is not a theorem of P2, thus there exists no proof of the induction principle in P2 in completely logical terms. This can be compared to the equivalence between the parallel axiom and Playfair’s axiom relative to Euclidean geometry, see Theorem 2.10.

I think that a lack of knowledge on the subject of logic might be a contributing factor in the spread of the misconception that the induction principle and the well-ordering principle are equivalent. Another contributing factor could be that it is so common to be working in an axiomatic system, proving theorems and doing mathematics in that axiomatic system without keeping in mind the foundation and structure of the system.

The refutation of the equivalence between the induction principle and the well-ordering principle relative to an axiomatic system where the two principles are not theorems has been done. But this does not eliminate the possibility of relative equivalence between pairs of the remaining principles, for example between the multiplication principle and the well-ordering principle. It does not eliminate the possibility of relative equivalence between three of the principles either, as long as not two of them are the induction principle and the well-ordering principle. This could be a reason for continued work on this subject. These possibilities can all be considered in the same way as the induction principle and the well-ordering principle were considered in section 6 of this thesis; by creating an axiomatic system where the principles in question are not theorems, choosing one of the principles to be an axiom of the system, and ﬁnding out if the other principle is a theorem of the system, and vice versa.

38 A weakness of this thesis is the lack of proofs showing that the pigeonhole principle and the multiplication principle really are theorems of ZFS set theory. It proved diﬃcult to ﬁnd sources that showed this. One proof of the pigeonhole principle in logical terms is attempted by Megill [10], but my grasp of logic is not good enough to insure that the proof is correct.

39 References

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