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 v¨alordningsprincipen ¨arekvivalenta ¨arett vanligt p˚ast˚aenden¨arde naturliga talen studeras. Utg˚angspunktenf¨ordetta examensarbete ¨arartikeln The equivalence of the multiplication, pigeonhole, induction, and well ordering principles, av Swanson och Hansen. D¨arp˚ast˚asatt multiplikationsprincipen , l˚adprincipen, induktionsprincipen och v¨alordningsprincipen ¨arekvivalenta. Detta bevisas g¨allaf¨orde naturliga talen i Zermelo-Fraenkel-Skolem m¨angdl¨ara,d¨aralla dessa principer ¨arsatser. Detta examensarbete inneh˚alleren presentation av generella axiomatiska system, och den logiska strukturen i s˚adanasystem. Syftet med denna presentation ¨aratt redog¨oraf¨orvad det inneb¨arf¨orprinciper, som ¨arsatser i ett axiomatiskt system, att vara ekvivalenta, samt att se p˚askillnaden mellan denna typ av ekvivalens och ekvivalens relativt till ett axiomatiskt system d¨arprinciperna inte ¨arsatser. De naturliga talen, Peano axiomen och Zermelo-Fraenkel-Skolem m¨angdl¨arapresenteras som ett underlag f¨orforts¨attningenp˚aexamensarbetet. Huvudsyftet med detta examensarbete ¨aratt motbevisa ekvivalens mellan in- duktionsprincipen och v¨alordninsprincipen relativt till ett axiomatiskt system d¨ar dessa principer inte ¨arsatser. D¨armedmotbevisas ¨aven ekvivalens mellan alla de fyra principerna i artikeln av Swanson och Hansen relativt ett s˚adant system. Detta utf¨orsgenom att konstruera ett axiomatiskt system d¨arvarken induktionsprincipen eller v¨alordningsprincipen ¨arsatser, och d¨areftervisa att induktionsprincipen och v¨alordninsprincipen inte ¨arekvivalenta 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, in- duction, 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 nat- ural 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 multiplica- tion, 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 2 S, and (b) n + 1 2 S whenever n 2 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 prin- ciple 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.