Surreal Geometric Analysis

Roger Bishop Jones

Abstract

This document is an exploration into formalisation of geometric algebra and analysis usingsurreal numbers instead of real numbers.

Created 2006/12/04

Last Change Date: 2010/08/13 10:57:22

http://www.rbjones.com/rbjpub/pp/doc/t022.pdf

Id: t022.doc,v 1.8 2010/08/13 10:57:22 rbj Exp

c© Roger Bishop Jones; Licenced under Gnu LGPL

Contents

1 INTRODUCTION 41.1 Preliminary Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Motivations Connected with the Formalisation of Physics . . . . . . . . . . . . 41.1.2 Some Objections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.3 Other Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 GALACTIC SET THEORY 82.1 Relations and Operations over Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Proof Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 SURREAL NUMBERS 133.1 An Axiomatisation of the Surreal Numbers . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1 Primitive Types and Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.3 The Zero Axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.4 The Cut Axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.5 The Induction Axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.6 Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Constructing the Surreal Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.1 The Representation of Surreals . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.2 Defining The Constants used in The Axioms . . . . . . . . . . . . . . . . . . . 163.2.3 Defining the Arithmetic Operators . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 The Theory of Surreal Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 The Theory GST 204.1 Parents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Children . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.4 Aliases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.5 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.6 Fixity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.7 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.8 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.9 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 The Theory sra 235.1 Parents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3 Aliases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.4 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.5 Fixity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.6 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.7 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 The Theory src 246.1 Parents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.2 Children . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.3 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

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6.4 Fixity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.5 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.6 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7 The Theory sr 277.1 Parents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.2 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.3 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

8 INDEX 28

References

[1] R.D. Arthan and R.B. Jones. Some Differential Geometry. RBJones.com, 2010.http://www.rbjones.com/rbjpub/pp/doc/t003.pdf.

[2] Roger Bishop Jones. The Formalisation of Physics. RBJones.com, 2010.http://www.rbjones.com/rbjpub/pp/doc/t002.pdf.

3

1 INTRODUCTION

This document is a toe in the water to get a first idea of how hard it is to work with surreal numbers,with the objective ultimately of providing a formal basis for theoretical physics based on geometricalgebra.

The stages envisaged are:

1. The theory of surreals

2. Geometric algebra (GA(0, 8)) based on surreals.

3. Geometric analysis

4. General Relativity and other theoretical physics

We have here also, pro-tem, an axiomatic set theory in which to construct the surreals (but which Iintend to put to other uses as well).

1.1 Preliminary Discussion

My motivations for undertaking this exploration fall into two categories.

The most important of these, which may however be misconceived, are concerned with possibleproblems in the formalisation of Physics.

The less important motivation is simply curiosity, which arises I think for me primarily because ofthe relationship between the surreal numbers and the underlying set theory.

I will discuss these two kinds of motivation separately.

1.1.1 Motivations Connected with the Formalisation of Physics

I am interested in the formalisation of Physics:

• as an exercise in the scientific application of formal methods worthwhile in its own right

• as a way of approaching related philosophical problems (which I am inclined to class as meta-physics)

• as a part of that larger project in the automation of reason which we associate with Leibniz(his lingua characteristica and calculus ratiocinator).

There are some (rather slender) reasons which have encouraged me to consider whether the analysison which a formalisation of physics is based might best be a non-standard analysis, and some othertenuous reasons for considering that such a non-standard analysis might best be based on the surrealnumbers.

My provocations for considering non-standard analysis are as follows:

• In the course of my investigations to date (see [2]) I was offered by Rob Arthan some inputon differential geometry (see [1]), in which a centrepiece is the Frechet derivative. The Frechetgeneralises the derivative from ordinary analysis to a derivative over functions over vector

4

spaces. The formulation shown in [1] delivers for total function between two vector spaces aderivative at some point, if the function is differentiable at that point. In effect, we begin witha total function, but may well end up with a partial function (the derivative), and this rendersawkward the iteration of the function to obtain nth derivatives. Since the result of the firstapplication need not be a total function, we may not be able to apply the function again.

One way of resolving this problem is shown also in [1] which is to define the Frechet derivativeas delivering a total function of which the value at points where the argument function is notdifferentiable will be unknown. This can then be iterated.

Whether this is the right way to deal with iterated differentiation of functions which are noteverywhere differentiable unclear.

• One of the principle targets for the formalisation of physics envisaged is the theories of specialand general relativity, partly because of their relevance to the metaphysics of space and time.One of the interesting features of the general theory is the occurence of singularities (blackholes) under certain circumstances. Strictly speaking the occurence of a singularity is a break-down of the theory rather than a bona-fide solution to the differential equations. In informalmathematical treatments one can safely reason about these partial solutions to the equations,but in a formal treatment it will be necessary to spell out in detail what such a partial solutionis so that it can be reasoned about with complete formal rigour. This might turn out to beawkward and complicated.

The possibility that the use of non-standard analysis might admit full solutions to the equationsin these circumstances arises. If, in the interesting cases where the equations have partialsolutions under standard analysis the equations had full solutions under non-standard analysis,then this might provide a better route to formalisation of the theory.

Two questions then arise. First is this the case, and, if the answer to that question is not known,what is the best way to go about getting closer to an answer.

Up until recently I have not had positive views about non-standard analysis and my reasons for thiswere:

• I have not been well motivated by the usual grounds offered for adopting non-standard analysis.It is usually offered as permitting the rigorous use of infinitesimal in accounts of analysis, whichcorresponds to the best original formulations and is supposed to be more intelligible than thelater epsilon-delta formulations which were for about a century thought to be the only way tomake these matters fully rigorous.

For my part however, the epsilon-delta formulation has always seemed to me clearer, not leastbecause I could not begin to understand the treatment in terms of infinitesimals until I couldunderstand the number system in which infinitesimals feature.

• I have found it much more difficult to get an intuitive grasp of what the numbers used innon-standard analysis are. The fact that they are arrived at as a non-standard model of thefirst order axiom system definitely did not recommend it to me.

Conway’s surreal numbers seemed to me until recently disconnected from these issues. I thought thatto do analysis with them it was necessary to separate out a subset which corresponded to the numberseither of non-standard or of standard analysis (the hyper-reals or the reals), and this promised to bea difficult task. My interest in the surreal numbers had not been much exercised by the possibilityof using them for analysis.

However, in December 2006 I heard a talk by Philip Ehrlich in which he claimed that contrary towhat Conway had suggested it was not necessary to separate out a subdomain for non-standard

5

analysis, the surreal numbers as a whole provided a model for non-standard analysis. This made meimmediately consider the possibility of using the surreals as a base for a non-standard analysis to beused in the formalisation of physics.

The surreals can be constructed by two different methods both of which seemed to me to give clearintuitive semantic basis, and if non-standard analysis offered a smoother way to deal with singularitiesthen it was worth serious consideration.

This provoked the explorations in this document, which I undertook at first with a rather hazyindifference to the question whether non-standard analysis really would be any better for physicsthan standard analysis.

My present view is that formalisation is unlikely to contribute (in any sensible timescale) to thequestion whether non-standard analysis might be good for formalised physics, and that talking topeople who understand the physics and the mathematics better than I do would be more likely toclarify that point.

1.1.2 Some Objections

I briefly discuss some objections.

First I note two minor misundertandardings on my part which gave rise to easily dismissed objectionsto the idea that the surreals provide a model for non-standard analysis.

1. The first was that since the surreals include the ordinals, and addition on ordinals is non-commutative, the surreals cannot provide a field. This is a pretty dumb objection, because youonly have to look at the definition of addition on the surreals to see that it is commutative. Ofcourse that means that addition over the surreals does not agree with ordinal addition whenrestricted to the surreals which correspond to the ordinals, but there is nothing to stop theusual ordinal addition from being defined as a separate operation.

2. A second minor point to note is that the surreals are not complete in the same way as the reals.But non-standard analysis is based on a non-standard model of the field axioms, the hyperrealsare not equired to be complete. There is a different completeness axiom for the surreals, andit should be possible for formulate one which corresponds to completeness of the real subset fothe surreals (supposing as I am guessing at present that the reals are the finite surreals of ranknot greater than 2ω.

3. Finally, even in non-standard analysis you can’t divide by zero. Singularities may not go away.

It now appears to me the grounds I have for thinking that there might be solutions in non-standardanalysis corresponding to situations involving singularities in general relativity are very slender in-deed. I need to check out whether anyone has tried doing relativity with non-standard analysis.

Rob Arthan has expressed to me strong objections to the idea that surreals might be applicable inphysics on the grounds that the surreals go beyond his geometric intuitions. Though I can give somecredence to this objection, there are many aspects of modern physics which go beyond our intuitiveexpectations about the nature of space and time. My own intuitions object to the idea that theremight really be singularities in the physical universe, but the theories I seek to formalise embracethat possibility and must therefore depend on a conception of space and time which goes beyondmy intuitions. The possibilities which arise once gravitational fields are eliminated by incorporationinto the curvature of space time (for example, worm holes) seem to go beyond anything for whichwe have direct observational evidence.

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1.1.3 Other Motivations

My other motivation I declared as curiousity, but since it probably has the upper hand at present itmay be worth my while to tease that out a bit more.

A principal cause of the curiousity is the fact that the surreals expand to the size of the set theoreticuniverse in which they are formulated. There is therefore some special difficulty in an axiomaticformulation independent of set theory, and also the rather bizarre possibility that such a formulationmight itself technically suffice as a foundation for mathematics.

When doing mathematics in HOL (Higher Order Logic) an axiom of infinity is required. If you wantto do set theory in HOL (apart from the limited theory obtained by treated properties as typedsets) you either need to axiomatise the set theory, or else you need a stronger axiom of infinity.This latter option is philosophically and technically interesting (independently of considering surrealnumbers), but also interacts with the problem of formalising surreals in HOL. This is because thesurreals include the ordinals, and you have to find a way of saying how many ordinals you are goingto get (if this is not to be determined by doing a construction in a set theory and just picking upthe ordinals from the set theory).

So this question of how to formulate strong axioms of infinity outside the context of set theory (strongenough to assert that the cardinality of the indviduals is some large cardinal) and the question ofwhether and how such a strong axiom of infinity might provide a basis for a foundational theory(such as a set theory) is of interest. Bear in mind that I am working in the context of a HigherOrder Logic, so if standard semantics of set theory is factorised into two parts (think of iterativeconstruction of the cumulative heirarchy of sets), taking at each rank the set of all sets which canbe constructed from sets of lower rank, and then running through sufficienlty many ranks to getan adequate population, then the first part of the semantics (all sets of some rank) comes from thehigher order logic (second order would do), and the second part is what we need a strong axiom ofinfinity for. Implicit in this discussion is that you don’t have the option to go the whole hog and getALL the pure well-founded sets. You just make successively more outrageous speculations expressedin ever stronger axioms of infinity which place greater lower bound on how far you can go, it isactually incoherent to suppose that the process can be completed.

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2 GALACTIC SET THEORY

This is basically a higher order set theory with “universes” which I insist on calling “galaxies”. I’mgoing to follow an approach first adopted for the formalisation of the theory of PolySets in Isabell-HOL, in which I make maximal use of the set theoretic vocabulary already available in HOL bydefining maps from sets in galactic set theory and the sets (subsets of types) already available inHOL,

First we have a new theory.

SML

open theory "rbjmisc";

force new theory "GST";

new parent "fixp";

new parent "ordered sets";

set merge pcs r"hol1", "1savedthm cs D proof "s;

Now the new type and the primitive constant, membership:

SML

new typep"GST", 0 q;

new constp"Pg", p:GST Ñ GST Ñ BOOLqq;

declare infix p70 , "Pg"q;

The axioms of extensionality and well-foundedness can be stated immediately:

SML

val GST Ext ax � new axiompr"Ext"s,

p@x y x�y ô @z z Pg x ô z Pg yqq;

val GST wf ax � new axiompr"Wf"s,

p@P p@s p@t t Pg s ñ P tq ñ P sq ñ @x P xqq;

The final axiom states that every set is a member of a galaxy, and also asserts a more global versionof replacement. First the notion of Galaxy is defined. To do this I first introduce some mappingsbetween GST’s and set so GST’s.HOL Constant

Xg : GST Ñ GST SET

@s X g s � tx | x Pg su

HOL Constant

XXg : GST Ñ GST SET SET

@s XX g s � tx | Dy y Pg s ^ x � X g yu

HOL Constant

Dg : GST SET Ñ BOOL

@s Dg s ô Dt X g t � s

8

HOL Constant

Cg : GST SET Ñ GST

@s Dg s ñ X g pC g sq � s

We are now in a position to define galaxies.

HOL Constant

Gg : GST Ñ BOOL

@s Gg s ô @x x Pg s ñ

pDv v Pg s ^ X g v ��pXX g x qq

^ pDv v Pg s ^ XX g v � tz | z � X g xuq

^ p@r r P Functional ñ Dv v Pg x ^ X g v � r Image pX g x qq

Galaxies have the following properties:

G ñ H lemma �

$ @ x Gg x ñ Dg tu

The primarily ontological axiom is then:

SML

val G ax � new axiompr"G"s,

p@x Dg x Pg g ^ Gg gqq;

The ontological consequences of this need to be teased out.

D H lemma �

$ Dg tu

I might possibly also need a global replacement axiom.

2.1 Relations and Operations over Sets

SML

declare infix p50 , "�g"q;

declare infix p50 , "�g"q;

HOL Constant

$�g: GST Ñ GST Ñ BOOL

@s t s �g t ô @x x Pg s ñ x Pg t

HOL Constant

$�g: GST Ñ GST Ñ BOOL

@s t s �g t ô s �g t ^ s � t

9

HOL Constant�

g: GST Ñ GST

@s �

g s � C g p�pXX g sqq

We will need an ordered pair, for which the Wiener-Kuratowski version will do:

HOL Constant

Wkpg: GST Ñ GST Ñ GST

@l r Wkpg l r � C g tC gtlu; C gtl ;ruu

SML

declare alias p"ÞÑg", pWkpgqq;

declare infix p1100 , "ÞÑg"q;

2.2 Ordinals

Zero:HOL Constant

Hg: GST

Hg � C g tu

PH thm �

$ @ x x Pg Hg

C H lemma �

$ C g tu � Hg ^ C g tx |Fu � Hg

�gH lemma �

$ @ s Hg �g s

and the successor function:HOL Constant

succg: GST Ñ GST

@on succg on � C g tx | x � on _ x Pg onu

A property of sets is ordinal closed if it is true of the empty set and is closed under successor andlimit constructions.HOL Constant

Oclosed: pGST Ñ BOOLq Ñ BOOL

@ps Oclosed ps ô ps Hg

^ p@x ps x ñ ps psuccg x qq

^ p@ss p@x x Pg ss ñ ps x q ñ ps p�

g ssqq

10

HOL Constant

Ordinal: GST Ñ BOOL

@s Ordinal s ô @ps Oclosed ps ñ ps s

ord H lemma �

$ Ordinal Hg

HOL Constant

1g 2g: GST

1 g � succg Hg ^ 2 g � succg 1 g

HOL Constant

Transitiveg: GST Ñ BOOL

@s Transitiveg s ô @t t Pg s ñ t �g s

2.3 Relations and Functions

HOL Constant

Relation: GST Ñ BOOL

@s Relation s ô @x x Pg s ñ Dl r :GST x � l ÞÑg r

rel H lemma �

$ Relation Hg

HOL Constant

Function: GST Ñ BOOL

@s Function s ô Relation s ^ @x y z :GST x ÞÑg y Pg s ^ x ÞÑg z Pg s ñ y � z

func H lemma �

$ Function Hg

HOL Constant

domain: GST Ñ GST

@s domain s � C g tx | Dy x ÞÑg y Pg su

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dom H lemma �

$ domain Hg � Hg

HOL Constant

range: GST Ñ GST

@s range s � C g ty | Dx x ÞÑg y Pg su

ran H lemma �

$ range Hg � Hg

SML

declare infix p900 , "�g"q;

Cartesian product:

HOL Constant

$�g: GST Ñ GST Ñ GST

@s t s �g t � C g tp:GST | Dl r l Pg s ^ r Pg t ^ p � l ÞÑg ru

2.4 Proof Context

GST H clauses �

$ C g tu � Hg

^ C g tx |Fu � Hg

^ p@ s Hg �g sq

^ Ordinal Hg

^ Relation Hg

^ Function Hg

^ domain Hg � Hg

^ range Hg � Hg

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3 SURREAL NUMBERS

The plan of action here is as follows. First I speculate on an axiomatisation of the surreal numberssufficient for non-standard analysis (the surreals of rank up to 2*w would probably suffice). Thenthis axiomatisation is evaluated in two different ways. It is used for the development of parts of thetheory of surreals, heading as rapidly as is possible in the direction of analysis. It is also validatedby performing a construction which delivers the axioms.

Three theories are therefore produced. The first, sra, contains the axiomatisation and its develop-ment. The second, src, contains the construction preliminary to defining (in the third theory, srd) atype of surreals satisfying the axioms in the first theory.

If this were to go well, then the surreals would be used as a base for something like geometric algebraand analysis would then be developed in that context. If I can find a way of exploring these latter twooptions without completing the previous developments then I may do that. Unfortunately my mainmotivation for considering the surreal numbers is in the possibility that non-standard analysis wouldprovide a better base than standard analysis for reasoning formally about general relativity, and inparticular reasoning about “solutions” to the gravitational “field equations” in those circumstancesin which singularities arise (i.e. black holes). I really have no idea whether it will help. Even innon-standard analysis division by zero fails, so possibly the singularities remain.

From that point of view I should be looking at non-standard analysis not at the construction ofsurreal numbers.

3.1 An Axiomatisation of the Surreal Numbers

Since this document is exploratory in nature, and proofs take time, it is worthwhile to explore theaxiomatisation of surreal numbers independently of the construction. This may help in deciding howthe development of the theory on the basis of the construction should be done, or it may just be afaster way of deciding whether the entire enterprise is worth the candle.

I understand from Philip Ehrich that he has published an axiomatisation of the surreals independentof set theory in the Journal of Symbolic logic, but I don’t have ready access to it so I am importinginto this document an axiomatisation which I did a few years back.

For this purpose a new theory “sra” is introduced in a context which does not include an axiomaticset theory.

SML

open theory "rbjmisc";

force new theory "sra";

new parent "ordered sets";

set merge pcs r"hol1", "1savedthm cs D proof "s;

3.1.1 Primitive Types and Constants

SML

new type p"No", 0 q;

new const p"Hs", p:Noqq;

declare alias p"H", pHsqq;

new const p"IC", p:pNo Ñ BOOLq Ñ No Ñ Noqq;

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new const p" S", p:No Ñ No Ñ BOOLqq;

declare infix p240 , " S"q;

declare alias p" ", p$ Sqq;

declare infix p240 , "  "q;

3.1.2 Definitions

HOL Constant

rank : No Ñ No

@n rank n � IC pλx T q n

HOL Constant

$  : No Ñ No Ñ BOOL

@n m:No n    m � rank n  S rank m

3.1.3 The Zero Axiom

SML

new axiom pr"SZeroAx"s, p@x rank x � H ô x � Hq q;

3.1.4 The Cut Axiom

The following axiom states that, for:

1. any property p of surreals and

2. surreal n such that p is downward closed on the surreals of lower rank than n

there exists a surreal number (IC p n) such that:

• (IC p n) is in between the surreals of rank less than n with the property and those of rank lessthan n without the property, and

• where p and q define the same cut on the surreals of rank less than n then (IC p n) is the samesurreal as (IC q n).

SML

new axiom pr"SCutAx"s,

p@p: No Ñ BOOL; n: No

p@x y : No x    n ^ y    n ^ x   y ^ p y ñ p x q

ñ p@x : No x    n ñ pp x ô x   pIC p nqq ^ p p x ô pIC p nq   x qq

^ p@q : No Ñ BOOL p@x x    n ñ pp x ô q x qq ô pIC p nq � pIC q nqq

qq;

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3.1.5 The Induction Axiom

The following axiom states that for any property p of surreals, if p holds for a surreal n whenever itholds for all the surreals of lower rank than n, then it holds for all surreals.

This is the same as Conway’s induction axiom since the union of the two sides of the canonical cuts(”games”) on which this axiomatization is based is the set of all numbers of lower rank.

SML

new axiom pr"SIndAx"s, pWellFounded pUniverse, $  qqq;

3.1.6 Infinity

This is the ordinal version of my strong infinity for HOL axiom, asserted of the surreals rather thanthe individuals. I have a fairly low level of confidence in this as yet, and am no longer inclined tothis style of axiom. I don’t know whether it’s best to assert it of   , or of    restricted to ordinals.For the present I will just display rather than actually adopt the axiom and assert a much weakeraxiom which will probably suffice for the applications I have in mind (i.e. for analysis).

declare infix p240 , "e"q;

new axiom pr"SInfAx"s, p

@p Dq p    q

^ p@x x    q ñ

ppDy $e y    q

^ p@Z Du u    y

^ p@v v    x ñ pv e u ô Z vqq

q

q

^ p@G p@u u    x ñ pG uq    qq

ñ Dy y    q ^ @u u    x ñ pG uq    y

q

q

q

qq;

The weak axiom of infinity has some similarity with the galactic closure axiom for GST. Takingthe idea of a galaxy but limiting the closure properties to closure under replacement, we get thefollowing axiom. Think of it in terms of ordinals which we may think of according the the VonNeumann conception (though the model behind the surreals is not the same) in which each ordinalis the set of its predecessors (in the surreals it is more like an ordinal is the set of numbers of smallerrank). So we are asserting that for every surreal ordinal s there is a large ordinal g which is “closedunder replacement”, i.e. the image of an ordinal x under a function f is bounded in the galaxy if itis a subset of it.

15

SML

new axiom pr"WInfAx"s, p

@s Dg s    g

^ @x f

x    g

^ p@y y    x ñ f y    gq

ñ Dz z    g ^ @v v    x ñ f v    zqq;

3.2 Constructing the Surreal Numbers

SML

open theory "GST";

force new theory "src";

set merge pcs r"hol1", "1savedthm cs D proof ", "1GST1"s;

3.2.1 The Representation of Surreals

Its not clear what is the best way to do this, so I may end up trying more than one method. The firstmethod is to use transfinite binary expansions which I have done following the Wikpedia account(though not slavishly).

A surreal number is a function whose domain is an ordinal and whose range is a subset of 2g:

HOL Constant

Surreal rep: GST Ñ BOOL

@s Surreal rep s ô Function s ^ Ordinal pdomain sq ^ prange sq �g 2 g

Surreal exists thm �

$ D x Surreal rep x

Sometimes its handy to have this as a set:

HOL Constant

surreal reps: GST SET

surreal reps � ts | Surreal rep su

3.2.2 Defining The Constants used in The Axioms

We will need to define operations over the surreals using transfinite recursion, and for that purposea suitable well-founded relation is needed. The surreal representatives are partially ordered by theirlength, which is of course, their domain.

SML

declare infix p70 , "  srr"q;

16

HOL Constant

$  srr: GST Ñ GST Ñ BOOL

@x y x   srr y ô Surreal rep x ^ Surreal rep y ^ domain x �g domain y

SML

declare infix p70 , " srr"q;

The next key element of the construction is the usual linear ordering of the surreals.

HOL Constant

$ srr: GST Ñ GST Ñ BOOL

@s t s  srr t ô

s �g t ^ pdomain s ÞÑg 1 gq Pg t

_ t �g s ^ pdomain t ÞÑg Hgq Pg s

_ pDz z �g s ^ z �g t

^ pdomain z ÞÑg Hgq Pg s ^ pdomain z ÞÑg 1 gq Pg tq

In order to do this we need to define some auxiliary functions which translate between binary ex-pansions and cuts in the numbers.

First, from a number obtain a left and a right set of numbers:

HOL Constant

L: GST Ñ GST

@n L n � C gtm | m �g n ^ domain m ÞÑg 1 g Pg nu

HOL Constant

R: GST Ñ GST

@n R n � C g tm | m �g n ^ domain m ÞÑg Hg Pg nu

Each pair of sets of numbers L, R such that every element of L is less than every element of R,determines a unique number σpL,Rq which is the simplest number between the two sets.

This does not correspond precisely to the WIkpedia account, since in it σpL,Rq is supposed to bestrictly between L and R, and right now I can’t see how to do that in the case that one half containsa number which is an initial segment of a number in the other half.

HOL Constant

σ: GST � GST Ñ GST

@ls rs σpls, rsq � C g p�ti | Dle re le Pg ls ^ re Pg rs ^ i � pX g leq X pX g requq

17

3.2.3 Defining the Arithmetic Operators

These are the definitions of the arithmetic operations which go with the above definition of theconstruction of the surreals. They probably can and should be done in the new type of surreals afterit has been introduced, but I had already done them before that occurred to me, and will remain atleast until I have satisfied myself that definition the later is preferable.

SML

declare right infix p210 , "�srr"q;

declare right infix p210 , "�srr"q;

declare right infix p220 , "�srr"q;

The following definition is by transfinite recursion on the rank of the surreal numbers involved. Aproof script is required (not yet supplied) to show that the function is well-defined.

HOL Constant

$�srr: GST Ñ GST Ñ GST

@x y Surreal rep x ^ Surreal rep y ñ

x �srr y � σp

C gpta | Du a � u �srr y ^ u Pg Lpx qu Y ta | Dv a � x �srr v ^ v Pg Lpyquq,

C gpta | Du a � u �srr y ^ u Pg Rpx qu Y ta | Dv a � x �srr v ^ v Pg Rpyquqq

HOL Constant

$�srr: GST Ñ GST

@x Surreal rep x ñ

�srr x � C gtz | Du v z � u ÞÑg v ^ u ÞÑg pif v � Hg then 1 g else Hgq Pg xu

HOL Constant

$�srr: GST Ñ GST Ñ GST

@x y Surreal rep x ^ Surreal rep y ñ

x �srr y � x �srr p�srr yq

HOL Constant

$�srr: GST Ñ GST Ñ GST

@x y Surreal rep x ^ Surreal rep y ñ

x �srr y � σp

C gta | Du v a � u �srr y �srr x �srr v �srr u �srr v

^ pu Pg Lpx q ^ v Pg Lpyq _ u Pg Rpx q ^ v Pg Rpyqqu,

C gta | Du v a � u �srr y �srr x �srr v �srr u �srr v

^ pu Pg Lpx q ^ v Pg Rpyq _ u Pg Rpx q ^ v Pg Lpyqquq

18

3.3 The Theory of Surreal Numbers

SML

open theory "src";

force new theory "sr";

set merge pcs r"hol1", "1savedthm cs D proof ", "1GST1"s;

new type defn pr"sr"s, "sr", rs, Surreal exists thmq;

19

4 The Theory GST

4.1 Parents

ordered sets fixp rbjmisc

4.2 Children

poly�sets src

4.3 Constants

$Pg GST Ñ GST Ñ BOOLXg GST Ñ GST PXXg GST Ñ GST P PDg GST P Ñ BOOLCg GST P Ñ GSTGg GST Ñ BOOL$�g GST Ñ GST Ñ BOOL$�g GST Ñ GST Ñ BOOL�

g GST Ñ GSTWkpg GST Ñ GST Ñ GSTHg GSTsuccg GST Ñ GSTOclosed pGST Ñ BOOLq Ñ BOOLOrdinal GST Ñ BOOL2g GST1g GSTTransitiveg GST Ñ BOOLRelation GST Ñ BOOLFunction GST Ñ BOOLdomain GST Ñ GSTrange GST Ñ GST$�g GST Ñ GST Ñ GST

4.4 Aliases

ÞÑg Wkpg : GST Ñ GST Ñ GST

4.5 Types

GST

4.6 Fixity

Right Infix 50 :�g �g

Right Infix 70 :Pg

Right Infix 900 :�g

Right Infix 1100 :ÞÑg

20

4.7 Axioms

Ext $ @ x y x � y ô p@ z z Pg x ô z Pg yqWf $ @ P p@ s p@ t t Pg s ñ P tq ñ P sq ñ p@ x P x qG $ @ x D g x Pg g ^ Gg g

4.8 Definitions

Xg $ @ s X g s � tx |x Pg suXXg $ @ s XX g s � tx |D y y Pg s ^ x � X g yuDg $ @ s Dg s ô pD t X g t � sqCg $ @ s Dg s ñ X g pC g sq � sGg $ @ s

Gg sô p@ x x Pg s

ñ pD v v Pg s ^ X g v ��pXX g x qq

^ pD v v Pg s ^ XX g v � tz |z � X g xuq^ p@ r r P Functional

ñ pD v v Pg x ^ X g v � r Image X g x qqq

�g $ @ s t s �g t ô p@ x x Pg s ñ x Pg tq�g $ @ s t s �g t ô s �g t ^ s � t�

g $ @ s �

g s � C g p�pXX g sqq

Wkpg $ @ l r l ÞÑg r � C g tC g tlu; C g tl ; ruuHg $ Hg � C g tusuccg $ @ on succg on � C g tx |x � on _ x Pg onuOclosed $ @ ps

Oclosed psô ps Hg

^ p@ x ps x ñ ps psuccg x qq^ p@ ss p@ x x Pg ss ñ ps x q ñ ps p

�g ssqq

Ordinal $ @ s Ordinal s ô p@ ps Oclosed ps ñ ps sq1g

2g $ 1 g � succg Hg ^ 2 g � succg 1 g

Transitiveg $ @ s Transitiveg s ô p@ t t Pg s ñ t �g sqRelation $ @ s

Relation s ô p@ x x Pg s ñ pD l r x � l ÞÑg rqqFunction $ @ s

Function sô Relation s^ p@ x y z x ÞÑg y Pg s ^ x ÞÑg z Pg s ñ y � z q

domain $ @ s domain s � C g tx |D y x ÞÑg y Pg surange $ @ s range s � C g ty |D x x ÞÑg y Pg su�g $ @ s t

s �g t� C g tp|D l r l Pg s ^ r Pg t ^ p � l ÞÑg ru

21

4.9 Theorems

G ñ H lemma$ @ x Gg x ñ Dg tuD H lemma $ Dg tu PH thm $ @ x x Pg Hg

C H lemma $ C g tu � Hg ^ C g tx |Fu � Hg

�gH lemma $ @ s Hg �g sord H lemma $ Ordinal Hg

rel H lemma $ Relation Hg

func H lemma $ Function Hg

dom H lemma$ domain Hg � Hg

ran H lemma $ range Hg � Hg

GST H clauses$ C g tu � Hg

^ C g tx |Fu � Hg

^ p@ s Hg �g sq^ Ordinal Hg

^ Relation Hg

^ Function Hg

^ domain Hg � Hg

^ range Hg � Hg

22

5 The Theory sra

5.1 Parents

ordered sets rbjmisc

5.2 Constants

Hs NoIC pNo Ñ BOOLq Ñ No Ñ No$ S No Ñ No Ñ BOOLrank No Ñ No$   No Ñ No Ñ BOOL

5.3 Aliases

H Hs : No  $ S : No Ñ No Ñ BOOL

5.4 Types

No

5.5 Fixity

Right Infix 240 :    S

5.6 Axioms

SZeroAx $ @ x rank x � H ô x � HSCutAx $ @ p n

p@ x y x    n ^ y    n ^ x   y ^ p y ñ p x qñ p@ x x    n

ñ pp x ô x   IC p nq^ p p x ô IC p n   x qq

^ p@ q p@ x x    n ñ pp x ô q x qq

ô IC p n � IC q nqSIndAx $ WellFounded pUniverse, $  qWInfAx $ @ s

D g s    g

^ p@ x f x    g ^ p@ y y    x ñ f y    gq

ñ pD z z    g ^ p@ v v    x ñ f v    z qqq

5.7 Definitions

rank $ @ n rank n � IC pλ x T q n   $ @ n m n    m ô rank n   rank m

23

6 The Theory src

6.1 Parents

GST

6.2 Children

sr

6.3 Constants

Surreal rep GST Ñ BOOLsurreal reps GST P$  srr GST Ñ GST Ñ BOOL$ srr GST Ñ GST Ñ BOOLL GST Ñ GSTR GST Ñ GSTσ GST � GST Ñ GST$�srr GST Ñ GST Ñ GST�srr GST Ñ GST$�srr GST Ñ GST Ñ GST$�srr GST Ñ GST Ñ GST

6.4 Fixity

Right Infix 70 :  srr  srr

Right Infix 210 :�srr �srr

Right Infix 220 :�srr

6.5 Definitions

Surreal rep $ @ s Surreal rep s

ô Function s^ Ordinal pdomain sq^ range s �g 2 g

surreal reps $ surreal reps � ts|Surreal rep su  srr $ @ x y

x   srr yô Surreal rep x^ Surreal rep y^ domain x �g domain y

 srr $ @ s t s  srr t

ô s �g t ^ domain s ÞÑg 1 g Pg t_ t �g s ^ domain t ÞÑg Hg Pg s

24

_ pD z z �g s

^ z �g t^ domain z ÞÑg Hg Pg s^ domain z ÞÑg 1 g Pg tq

L $ @ n L n � C g tm|m �g n ^ domain m ÞÑg 1 g Pg nuR $ @ n R n � C g tm|m �g n ^ domain m ÞÑg Hg Pg nuσ $ @ ls rs

σ pls, rsq� C g

p�

ti|D le re le Pg ls

^ re Pg rs^ i � X g le X X g reuq

�srr $ ConstSpecpλ �srr

1

@ x y Surreal rep x ^ Surreal rep y

ñ �srr1 x y

� σpC g

pta|D u a � �srr

1 u y^ u Pg L xu

Y ta|D v a � �srr

1 x v^ v Pg L yuq,

C g

pta|D u a � �srr

1 u y^ u Pg R xu

Y ta|D v a � �srr

1 x v^ v Pg R yuqqq

$�srr

�srr $ ConstSpecpλ �srr

1

@ x Surreal rep x

ñ �srr1 x

� C g

tz|D u v z � u ÞÑg v

^ u

25

ÞÑg pif v � Hg

then 1 g

else HgqPg xuq

�srr

�srr $ ConstSpecpλ �srr

1

@ x y Surreal rep x ^ Surreal rep y

ñ �srr1 x y � x �srr �srr yq

$�srr

�srr $ ConstSpecpλ �srr

1

@ x y Surreal rep x ^ Surreal rep y

ñ �srr1 x y

� σpC g

ta|D u v a

� �srr1 u y

�srr �srr1 x v

�srr �srr1 u v

^ pu Pg L x^ v Pg L y

_ u Pg R x^ v Pg R yqu,

C g

ta|D u v a

� �srr1 u y

�srr �srr1 x v

�srr �srr1 u v

^ pu Pg L x^ v Pg R y

_ u Pg R x^ v Pg L yquqq

$�srr

6.6 Theorems

Surreal exists thm$ D x Surreal rep x

26

7 The Theory sr

7.1 Parents

src

7.2 Types

sr

7.3 Definitions

sr $ D f TypeDefn Surreal rep f

27

8 INDEX

�srr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18, 24, 26�srr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18, 24, 25�srr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18, 24, 26  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23   srr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17, 24  S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23  srr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17, 24�

g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10, 20, 21H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Hg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10, 20, 21Hs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23D H lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Dg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8, 20, 21P g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 P H thm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22ÞÑ g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17, 24, 25� srr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18, 24, 25� g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9, 20, 21� g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9, 20, 21� gH lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22�g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12, 20, 211g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11, 20, 212g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11, 20, 21

C H lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Cg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9, 20, 21

dom H lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11, 20, 21

Ext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8, 21

func H lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11, 20, 21

G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21G ñ H lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22GST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20GST H clauses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22GST Ext ax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8GST wf ax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Gg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9, 20, 21

IC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17, 24, 25

No . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Oclosed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10, 20, 21ord H lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Ordinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11, 20, 21

R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17, 24, 25ran H lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12, 20, 21rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23rel H lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11, 20, 21

SCutAx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23SIndAx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23sr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27succg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10, 20, 21Surreal exists thm . . . . . . . . . . . . . . . . . . . . . . . . . . 26Surreal rep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16, 24surreal reps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16, 24SZeroAx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Transitiveg . . . . . . . . . . . . . . . . . . . . . . . . . . . 11, 20, 21

Wf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8, 21WInfAx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Wkpg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10, 20, 21

XXg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8, 20, 21Xg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8, 20, 21

28