real closed field

Real closed eldFrom Wikipedia, the free encyclopedia

Contents

1 Hyperreal number 11.1 The transfer principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Use in analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Calculus with algebraic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4.1 From Leibniz to Robinson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4.2 The ultrapower construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4.3 An intuitive approach to the ultrapower construction . . . . . . . . . . . . . . . . . . . . . 4

1.5 Properties of innitesimal and innite numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Hyperreal elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Non-standard analysis 82.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Basic denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1 Historical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 Pedagogical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.3 Technical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Approaches to non-standard analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Robinsons book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Invariant subspace problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.7 Other applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.7.1 Applications to calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.8 Critique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.9 Logical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.10 Internal sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.11 First consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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ii CONTENTS

2.12 -saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.13 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.14 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.15 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.16 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.17 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Real closed eld 173.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Model theory: decidability and quantier elimination . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Order properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 The generalized continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Examples of real closed elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Real closed ring 214.1 Examples of real closed rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 The real closure of a commutative ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.4 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.5 Model theoretic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.6 Comparison with characterizations of real closed elds . . . . . . . . . . . . . . . . . . . . . . . . 234.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Standard part function 245.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.2 Not internal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.3.1 Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3.2 Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3.3 Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3.4 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6 Superreal number 276.1 Formal Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.3 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

CONTENTS iii

7 Surreal number 287.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.2 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7.2.1 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.2.2 Numeric forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.2.3 Equivalence classes of numeric forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.2.4 Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.2.5 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

7.3 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.3.1 Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.3.2 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.3.3 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.3.4 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.3.5 Arithmetic closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7.4 To Innity ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.4.1 Contents of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7.5 "... And Beyond. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.6 Powers of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.7 Surcomplex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.8 Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.9 Application to combinatorial game theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.10 Alternative realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7.10.1 Sign expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.10.2 Axiomatic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.10.3 Hahn series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

7.11 Relation to hyperreals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.12 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.14 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.15 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.16 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7.16.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.16.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.16.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Chapter 1

Hyperreal number

"*R redirects here. For R*, see R* (disambiguation).

The system of hyperreal numbers is a way of treating innite and innitesimal quantities. The hyperreals, or non-standard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of theform

1 + 1 + + 1:

Such a number is innite, and its reciprocal is innitesimal. The term hyper-real was introduced by Edwin Hewittin 1948.[1]

The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibnizs heuristic Law of Continuity. Thetransfer principle states that true rst order statements about R are also valid in *R. For example, the commutativelaw of addition, x + y = y + x, holds for the hyperreals just as it does for the reals; since R is a real closed eld, so is*R. Since sinn = 0 for all integers n, one also has sinH = 0 for all hyperintegers H. The transfer principle forultrapowers is a consequence of o' theorem of 1955.Concerns about the soundness of arguments involving innitesimals date back to ancient Greek mathematics, withArchimedes replacing such proofs with ones using other techniques such as the method of exhaustion.[2] In the 1960s,Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were. This put to restthe fear that any proof involving innitesimals might be unsound, provided that they were manipulated according tothe logical rules which Robinson delineated.The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called non-standard analysis. One immediate application is the denition of the basic concepts of analysis such as derivative andintegral in a direct fashion, without passing via logical complications of multiple quantiers. Thus, the derivative off(x) becomes f 0(x) = st

f(x+x)f(x)

x

for an innitesimal x , where st() denotes the standard part function,

which rounds o each nite hyperreal to the nearest real. Similarly, the integral is dened as the standard part ofa suitable innite sum.

1.1 The transfer principleMain article: Transfer principle

The idea of the hyperreal system is to extend the real numbers R to form a system *R that includes innitesimal andinnite numbers, but without changing any of the elementary axioms of algebra. Any statement of the form forany number x... that is true for the reals is also true for the hyperreals. For example, the axiom that states for anynumber x, x + 0 = x" still applies. The same is true for quantication over several numbers, e.g., for any numbers xand y, xy = yx. This ability to carry over statements from the reals to the hyperreals is called the transfer principle.However, statements of the form for any set of numbers S ... may not carry over. The only properties that dier

1

2 CHAPTER 1. HYPERREAL NUMBER

between the reals and the hyperreals are those that rely on quantication over sets, or other higher-level structuressuch as functions and relations, which are typically constructed out of sets. Each real set, function, and relation has itsnatural hyperreal extension, satisfying the same rst-order properties. The kinds of logical sentences that obey thisrestriction on quantication are referred to as statements in rst-order logic.The transfer principle, however, doesn't mean that R and *R have identical behavior. For instance, in *R there existsan element such that

1 < !; 1 + 1 < !; 1 + 1 + 1 < !; 1 + 1 + 1 + 1 < !; : : : :

but there is no such number inR. (In other words, *R is not Archimedean.) This is possible because the nonexistenceof cannot be expressed as a rst order statement.

1.2 Use in analysis

1.2.1 Calculus with algebraic functionsInformal notations for non-real quantities have historically appeared in calculus in two contexts: as innitesimals likedx and as the symbol , used, for example, in limits of integration of improper integrals.As an example of the transfer principle, the statement that for any nonzero number x, 2x x, is true for the realnumbers, and it is in the form required by the transfer principle, so it is also true for the hyperreal numbers. Thisshows that it is not possible to use a generic symbol such as for all the innite quantities in the hyperreal system;innite quantities dier in magnitude from other innite quantities, and innitesimals from other innitesimals.Similarly, the casual use of 1/0 = is invalid, since the transfer principle applies to the statement that division byzero is undened. The rigorous counterpart of such a calculation would be that if is a non-zero innitesimal, then1/ is innite.For any nite hyperreal number x, its standard part, st x, is dened as the unique real number that diers from it onlyinnitesimally. The derivative of a function y(x) is dened not as dy/dx but as the standard part of dy/dx.For example, to nd the derivative f (x) of the function f(x) = x2, let dx be a non-zero innitesimal. Then,

The use of the standard part in the denition of the derivative is a rigorous alternative to the traditional practice ofneglecting the square of an innitesimal quantity. After the third line of the dierentiation above, the typical methodfrom Newton through the 19th century would have been simply to discard the dx2 term. In the hyperreal system, dx2 0, since dx is nonzero, and the transfer principle can be applied to the statement that the square of any nonzeronumber is nonzero. However, the quantity dx2 is innitesimally small compared to dx; that is, the hyperreal systemcontains a hierarchy of innitesimal quantities.

1.2.2 IntegrationOne way of dening a denite integral in the hyperreal system is as the standard part of an innite sum on a hypernitelattice dened as a, a + dx, a + 2dx, ... a + ndx, where dx is innitesimal, n is an innite hypernatural, and the lowerand upper bounds of integration are a and b = a + n dx.[3]

1.3 PropertiesThe hyperreals *R form an ordered eld containing the reals R as a subeld. Unlike the reals, the hyperreals do notform a standard metric space, but by virtue of their order they carry an order topology.The use of the denite article the in the phrase the hyperreal numbers is somewhat misleading in that there is not aunique ordered eld that is referred to in most treatments. However, a 2003 paper by Vladimir Kanovei and Shelah[4]

1.4. DEVELOPMENT 3

shows that there is a denable, countably saturated (meaning -saturated, but not of course countable) elementaryextension of the reals, which therefore has a good claim to the title of the hyperreal numbers. Furthermore, the eldobtained by the ultrapower construction from the space of all real sequences, is unique up to isomorphism if oneassumes the continuum hypothesis.The condition of being a hyperreal eld is a stronger one than that of being a real closed eld strictly containing R. Itis also stronger than that of being a superreal eld in the sense of Dales and Woodin.[5]

1.4 DevelopmentThe hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence ofthe axiomatic approach is to assert (1) the existence of at least one innitesimal number, and (2) the validity ofthe transfer principle. In the following subsection we give a detailed outline of a more constructive approach. Thismethod allows one to construct the hyperreals if given a set-theoretic object called an ultralter, but the ultralteritself cannot be explicitly constructed.

1.4.1 From Leibniz to Robinson

When Newton and (more explicitly) Leibniz introduced dierentials, they used innitesimals and these were stillregarded as useful by later mathematicians such as Euler and Cauchy. Nonetheless these concepts were from thebeginning seen as suspect, notably byGeorge Berkeley. Berkeleys criticism centered on a perceived shift in hypothesisin the denition of the derivative in terms of innitesimals (or uxions), where dx is assumed to be nonzero at thebeginning of the calculation, and to vanish at its conclusion (see Ghosts of departed quantities for details). When inthe 1800s calculus was put on a rm footing through the development of the (, )-denition of limit by Bolzano,Cauchy, Weierstrass, and others, innitesimals were largely abandoned, though research in non-Archimedean eldscontinued (Ehrlich 2006).However, in the 1960s Abraham Robinson showed how innitely large and innitesimal numbers can be rigorouslydened and used to develop the eld of non-standard analysis.[6] Robinson developed his theory nonconstructively,using model theory; however it is possible to proceed using only algebra and topology, and proving the transferprinciple as a consequence of the denitions. In other words hyperreal numbers per se, aside from their use innonstandard analysis, have no necessary relationship tomodel theory or rst order logic, although they were discoveredby the application of model theoretic techniques from logic. Hyper-real elds were in fact originally introduced byHewitt (1948) by purely algebraic techniques, using an ultrapower construction.

1.4.2 The ultrapower construction

We are going to construct a hyperreal eld via sequences of reals.[7] In fact we can add and multiply sequencescomponentwise; for example:

(a0; a1; a2; : : :) + (b0; b1; b2; : : :) = (a0 + b0; a1 + b1; a2 + b2; : : :)

and analogously for multiplication. This turns the set of such sequences into a commutative ring, which is in fact areal algebra A. We have a natural embedding of R in A by identifying the real number r with the sequence (r, r, r, ...)and this identication preserves the corresponding algebraic operations of the reals. The intuitive motivation is, forexample, to represent an innitesimal number using a sequence that approaches zero. The inverse of such a sequencewould represent an innite number. As we will see below, the diculties arise because of the need to dene rulesfor comparing such sequences in a manner that, although inevitably somewhat arbitrary, must be self-consistent andwell dened. For example, we may have two sequences that dier in their rst n members, but are equal after that;such sequences should clearly be considered as representing the same hyperreal number. Similarly, most sequencesoscillate randomly forever, and we must nd some way of taking such a sequence and interpreting it as, say, 7 + ,where is a certain innitesimal number.Comparing sequences is thus a delicate matter. We could, for example, try to dene a relation between sequences ina componentwise fashion:


(a0; a1; a2; : : :) (b0; b1; b2; : : :) () a0 b0 ^ a1 b1 ^ a2 b2 : : :but here we run into trouble, since some entries of the rst sequence may be bigger than the corresponding entries ofthe second sequence, and some others may be smaller. It follows that the relation dened in this way is only a partialorder. To get around this, we have to specify which positions matter. Since there are innitely many indices, wedon't want nite sets of indices to matter. A consistent choice of index sets that matter is given by any free ultralterU on the natural numbers; these can be characterized as ultralters which do not contain any nite sets. (The goodnews is that Zorns lemma guarantees the existence of many such U; the bad news is that they cannot be explicitlyconstructed.) We think of U as singling out those sets of indices that matter": We write (a0, a1, a2, ...) (b0, b1,b2, ...) if and only if the set of natural numbers { n : an bn } is in U.This is a total preorder and it turns into a total order if we agree not to distinguish between two sequences a andb if ab and ba. With this identication, the ordered eld *R of hyperreals is constructed. From an algebraicpoint of view, U allows us to dene a corresponding maximal ideal I in the commutative ring A (namely, the set ofthe sequences that vanish in some element of U), and then to dene *R as A/I; as the quotient of a commutativering by a maximal ideal, *R is a eld. This is also notated A/U, directly in terms of the free ultralter U; the twoare equivalent. The maximality of I follows from the possibility of, given a sequence a, constructing a sequence binverting the non-null elements of a and not altering its null entries. If the set on which a vanishes is not in U, theproduct ab is identied with the number 1, and any ideal containing 1 must be A. In the resulting eld, these a and bare inverses.The eld A/U is an ultrapower of R. Since this eld contains R it has cardinality at least that of the continuum. SinceA has cardinality

(2@0)@0 = 2@20 = 2@0 ;

it is also no larger than 2@0 , and hence has the same cardinality as R.One question we might ask is whether, if we had chosen a dierent free ultralter V, the quotient eld A/U wouldbe isomorphic as an ordered eld to A/V. This question turns out to be equivalent to the continuum hypothesis; inZFC with the continuum hypothesis we can prove this eld is unique up to order isomorphism, and in ZFC with thenegation of continuum hypothesis we can prove that there are non-order-isomorphic pairs of elds which are bothcountably indexed ultrapowers of the reals.For more information about this method of construction, see ultraproduct.

1.4.3 An intuitive approach to the ultrapower constructionThe following is an intuitive way of understanding the hyperreal numbers. The approach taken here is very close tothe one in the book by Goldblatt.[8] Recall that the sequences converging to zero are sometimes called innitely small.These are almost the innitesimals in a sense; the true innitesimals include certain classes of sequences that containa sequence converging to zero.Let us see where these classes come from. Consider rst the sequences of real numbers. They form a ring, that is, onecan multiply, add and subtract them, but not always divide by a non-zero element. The real numbers are consideredas the constant sequences, the sequence is zero if it is identically zero, that is, an = 0 for all n.In our ring of sequences one can get ab = 0 with neither a = 0 nor b = 0. Thus, if for two sequences a; b one has ab= 0, at least one of them should be declared zero. Surprisingly enough, there is a consistent way to do it. As a result,the equivalence classes of sequences that dier by some sequence declared zero will form a eld which is called ahyperreal eld. It will contain the innitesimals in addition to the ordinary real numbers, as well as innitely largenumbers (the reciprocals of innitesimals, including those represented by sequences diverging to innity). Also everyhyperreal which is not innitely large will be innitely close to an ordinary real, in other words, it will be the sum ofan ordinary real and an innitesimal.This construction is parallel to the construction of the reals from the rationals given by Cantor. He started with thering of the Cauchy sequences of rationals and declared all the sequences that converge to zero to be zero. The resultis the reals. To continue the construction of hyperreals, let us consider the zero sets of our sequences, that is, thez(a) = fi : ai = 0g , that is, z(a) is the set of indexes i for which ai = 0 . It is clear that if ab = 0 ,then the union of z(a) and z(b) is N (the set of all natural numbers), so:

1.5. PROPERTIES OF INFINITESIMAL AND INFINITE NUMBERS 5

1. One of the sequences that vanish on two complementary sets should be declared zero

2. If a is declared zero, ab should be declared zero too, no matter what b is.

3. If both a and b are declared zero, then a2 + b2 should also be declared zero.

Now the idea is to single out a bunch U of subsets X of N and to declare that a = 0 if and only if z(a) belongsto U. From the above conditions one can see that:

1. From two complementary sets one belongs to U

2. Any set containing a set that belongs to U, also belongs to U.

3. An intersection of any two sets belonging to U belongs to U.

4. Finally, we do not want an empty set to belong toU because then everything becomes zero, as every set containsan empty set.

Any family of sets that satises (24) is called a lter (an example: the complements to the nite sets, it is calledthe Frchet lter and it is used in the usual limit theory). If (1) also holds, U is called an ultralter (because you canadd no more sets to it without breaking it). The only explicitly known example of an ultralter is the family of setscontaining a given element (in our case, say, the number 10). Such ultralters are called trivial, and if we use it in ourconstruction, we come back to the ordinary real numbers. Any ultralter containing a nite set is trivial. It is knownthat any lter can be extended to an ultralter, but the proof uses the axiom of choice. The existence of a nontrivialultralter (the ultralter lemma) can be added as an extra axiom, as it is weaker than the axiom of choice.Now if we take a nontrivial ultralter (which is an extension of the Frchet lter) and do our construction, we get thehyperreal numbers as a result.If f is a real function of a real variable x then f naturally extends to a hyperreal function of a hyperrealvariable by composition:

f(fxng) = ff(xn)g

where f: : : g means the equivalence class of the sequence : : : relative to our ultralter, two sequences being in thesame class if and only if the zero set of their dierence belongs to our ultralter.All the arithmetical expressions and formulas make sense for hyperreals and hold true if they are true for the ordinaryreals. One can prove that any nite (that is, such that jxj < a for some ordinary real a ) hyperreal x will beof the form y + d where y is an ordinary (called standard) real and d is an innitesimal.Now one can see that f is continuous means that f(a) f(x) is innitely small whenever x a is, and fis dierentiable means that

(f(x) f(a))/(x a) f 0(a)

is innitely small whenever x a is. Remarkably, if one allows a to be hyperreal, the derivative will be auto-matically continuous (because, f being dierentiable at x ,

f 0(x) (f(x) f(a))/(x a) = f 0(x) (f(a) f(x))/(a x)

is innitely small when x a is, therefore f 0(x) f 0(a) is also innitely small when x a is).

1.5 Properties of innitesimal and innite numbersThe nite elements F of *R form a local ring, and in fact a valuation ring, with the unique maximal ideal S beingthe innitesimals; the quotient F/S is isomorphic to the reals. Hence we have a homomorphic mapping, st(x), from


F to R whose kernel consists of the innitesimals and which sends every element x of F to a unique real numberwhose dierence from x is in S; which is to say, is innitesimal. Put another way, every nite nonstandard realnumber is very close to a unique real number, in the sense that if x is a nite nonstandard real, then there existsone and only one real number st(x) such that x st(x) is innitesimal. This number st(x) is called the standard partof x, conceptually the same as x to the nearest real number. This operation is an order-preserving homomorphismand hence is well-behaved both algebraically and order theoretically. It is order-preserving though not isotonic; i.e.x y implies st(x) st(y) , but x < y does not imply st(x) < st(y) .

We have, if both x and y are nite,

st(x+ y) = st(x) + st(y)

st(xy) = st(x) st(y)

If x is nite and not innitesimal.

st(1/x) = 1/ st(x)

x is real if and only if

st(x) = x

The map st is continuous with respect to the order topology on the nite hyperreals; in fact it is locally constant.

1.6 Hyperreal eldsSupposeX is a Tychono space, also called a T. space, and C(X) is the algebra of continuous real-valued functions onX. Suppose M is a maximal ideal in C(X). Then the factor algebra A = C(X)/M is a totally ordered eld F containingthe reals. If F strictly contains R then M is called a hyperreal ideal (terminology due to Hewitt (1948)) and F ahyperreal eld. Note that no assumption is being made that the cardinality of F is greater than R; it can in fact havethe same cardinality.An important special case is where the topology on X is the discrete topology; in this case X can be identied with acardinal number and C(X) with the real algebra R of functions from to R. The hyperreal elds we obtain in thiscase are called ultrapowers of R and are identical to the ultrapowers constructed via free ultralters in model theory.

1.7 See also Hyperinteger Real closed elds Non-standard calculus Constructive non-standard analysis Inuence of non-standard analysis Surreal number

1.8. REFERENCES 7

1.8 References[1] Hewitt (1948), p. 74, as reported in Keisler (1994)

[2] Ball, p. 31

[3] Keisler

[4] Kanovei, Vladimir; Shelah, Saharon (2004), A denable nonstandard model of the reals (PDF), Journal of SymbolicLogic 69: 159164, doi:10.2178/jsl/1080938834

[5] Woodin, W. H.; Dales, H. G. (1996), Super-real elds: totally ordered elds with additional structure, Oxford: ClarendonPress, ISBN 978-0-19-853991-9

[6] Robinson, Abraham (1996), Non-standard analysis, Princeton University Press, ISBN 978-0-691-04490-3. The classicintroduction to nonstandard analysis.

[7] Loeb, Peter A. (2000), An introduction to nonstandard analysis, Nonstandard analysis for the working mathematician,Math. Appl. 510, Dordrecht: Kluwer Acad. Publ., pp. 195

[8] Goldblatt, Robert (1998), Lectures on the hyperreals: an introduction to nonstandard analysis, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98464-3

1.9 Further reading Ball, W.W. Rouse (1960), A Short Account of the History of Mathematics (4th ed. [Reprint. Original publica-tion: London: Macmillan & Co., 1908] ed.), New York: Dover Publications, pp. 5062, ISBN 0-486-20630-0

Hatcher, William S. (1982) Calculus is Algebra, American Mathematical Monthly 89: 362370. Hewitt, Edwin (1948) Rings of real-valued continuous functions. I. Trans. Amer. Math. Soc. 64, 4599. Jerison, Meyer; Gillman, Leonard (1976), Rings of continuous functions, Berlin, New York: Springer-Verlag,ISBN 978-0-387-90198-5

Keisler, H. Jerome (1994) The hyperreal line. Real numbers, generalizations of the reals, and theories ofcontinua, 207237, Synthese Lib., 242, Kluwer Acad. Publ., Dordrecht.

Kleinberg, Eugene M.; Henle, James M. (2003), Innitesimal Calculus, New York: Dover Publications, ISBN978-0-486-42886-4

1.10 External links Crowell, Calculus. A text using innitesimals. Hermoso, Nonstandard Analysis and the Hyperreals. A gentle introduction. Keisler, Elementary Calculus: An Approach Using Innitesimals. Includes an axiomatic treatment of the hy-perreals, and is freely available under a Creative Commons license

Stroyan, A Brief Introduction to Innitesimal Calculus Lecture 1 Lecture 2 Lecture 3

Chapter 2

Non-standard analysis

The history of calculus is fraught with philosophical debates about the meaning and logical validity of uxions orinnitesimal numbers. The standard way to resolve these debates is to dene the operations of calculus using epsilondelta procedures rather than innitesimals. Non-standard analysis[1][2][3] instead reformulates the calculus using alogically rigorous notion of innitesimal numbers.Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson.[4][5] He wrote:

[...] the idea of innitely small or innitesimal quantities seems to appeal naturally to our intuition.At any rate, the use of innitesimals was widespread during the formative stages of the Dierentialand Integral Calculus. As for the objection [...] that the distance between two distinct real numberscannot be innitely small, Gottfried Wilhelm Leibniz argued that the theory of innitesimals implies theintroduction of ideal numbers which might be innitely small or innitely large compared with the realnumbers but which were to possess the same properties as the latter

Robinson argued that this law of continuity of Leibnizs is a precursor of the transfer principle. Robinson continued:

However, neither he nor his disciples and successors were able to give a rational development leadingup to a system of this sort. As a result, the theory of innitesimals gradually fell into disrepute and wasreplaced eventually by the classical theory of limits.[6]

Robinson continues:

It is shown in this book that Leibnizs ideas can be fully vindicated and that they lead to a novel andfruitful approach to classical Analysis and tomany other branches ofmathematics. The key to ourmethodis provided by the detailed analysis of the relation between mathematical languages and mathematicalstructures which lies at the bottom of contemporary model theory.

In 1973, intuitionist Arend Heyting praised non-standard analysis as a standard model of important mathematicalresearch.[7]

2.1 IntroductionA non-zero element of an ordered eld F is innitesimal if and only if its absolute value is smaller than any elementof F of the form 1n , for n , a standard natural number. Ordered elds that have innitesimal elements are also callednon-Archimedean. More generally, non-standard analysis is any form of mathematics that relies on non-standardmodels and the transfer principle. A eld which satises the transfer principle for real numbers is a hyperreal eld,and non-standard real analysis uses these elds as non-standard models of the real numbers.Robinsons original approach was based on these non-standard models of the eld of real numbers. His classicfoundational book on the subject Non-standard Analysis was published in 1966 and is still in print.[8] On page 88,Robinson writes:

8

2.1. INTRODUCTION 9

Gottfried Wilhelm Leibniz argued that idealized numbers containing innitesimals be introduced.

The existence of non-standard models of arithmetic was discovered by Thoralf Skolem (1934).Skolems method foreshadows the ultrapower construction [...]

Several technical issues must be addressed to develop a calculus of innitesimals. For example, it is not enough toconstruct an ordered eld with innitesimals. See the article on hyperreal numbers for a discussion of some of therelevant ideas.

10 CHAPTER 2. NON-STANDARD ANALYSIS

2.2 Basic denitionsIn this section we outline one of the simplest approaches to dening a hyperreal eld R . Let R be the eld of realnumbers, and let N be the semiring of natural numbers. Denote by RN the space of sequences of real numbers. Aeld R is dened as a suitable quotient of RN , as follows. Take a nonprincipal ultralter F P (N) . In particular,F contains the Frchet lter. Consider a pair of sequences

u = (un); v = (vn) 2 RN

We say that u and v are equivalent if they coincide on a set of indices which is a member of the ultralter, or informulas:

fn 2 N : un = vng 2 FThe quotient ofRN by the resulting equivalence relation is a hyperreal eld R , a situation summarized by the formulaR = RN/F .

2.3 MotivationThere are at least three reasons to consider non-standard analysis: historical, pedagogical, and technical.

2.3.1 HistoricalMuch of the earliest development of the innitesimal calculus by Newton and Leibniz was formulated using ex-pressions such as innitesimal number and vanishing quantity. As noted in the article on hyperreal numbers, theseformulations were widely criticized by George Berkeley and others. It was a challenge to develop a consistent theoryof analysis using innitesimals and the rst person to do this in a satisfactory way was Abraham Robinson.[6]

In 1958 Curt Schmieden and Detlef Laugwitz published an Article Eine Erweiterung der Innitesimalrechnung[9] -An Extension of Innitesimal Calculus, which proposed a construction of a ring containing innitesimals. The ringwas constructed from sequences of real numbers. Two sequences were considered equivalent if they diered only ina nite number of elements. Arithmetic operations were dened elementwise. However, the ring constructed in thisway contains zero divisors and thus cannot be a eld.

2.3.2 PedagogicalH. Jerome Keisler, David Tall, and other educators maintain that the use of innitesimals is more intuitive and moreeasily grasped by students than the so-called epsilon-delta approach to analytic concepts.[10] This approach cansometimes provide easier proofs of results than the corresponding epsilon-delta formulation of the proof. Much ofthe simplication comes from applying very easy rules of nonstandard arithmetic, viz:

innitesimal bounded = innitesimal

innitesimal + innitesimal = innitesimal

together with the transfer principle mentioned below.Another pedagogical application of non-standard analysis is Edward Nelson's treatment of the theory of stochasticprocesses.[11]

2.3.3 TechnicalSome recent work has been done in analysis using concepts from non-standard analysis, particularly in investigatinglimiting processes of statistics andmathematical physics. Sergio Albeverio et al.[12] discuss some of these applications.

2.4. APPROACHES TO NON-STANDARD ANALYSIS 11

2.4 Approaches to non-standard analysisThere are two very dierent approaches to non-standard analysis: the semantic or model-theoretic approach and thesyntactic approach. Both these approaches apply to other areas of mathematics beyond analysis, including numbertheory, algebra and topology.Robinsons original formulation of non-standard analysis falls into the category of the semantic approach. As de-veloped by him in his papers, it is based on studying models (in particular saturated models) of a theory. SinceRobinsons work rst appeared, a simpler semantic approach (due to Elias Zakon) has been developed using purelyset-theoretic objects called superstructures. In this approach a model of a theory is replaced by an object called asuperstructure V(S) over a set S. Starting from a superstructure V(S) one constructs another object *V(S) using theultrapower construction together with a mapping V(S) *V(S) which satises the transfer principle. The map *relates formal properties of V(S) and *V(S). Moreover it is possible to consider a simpler form of saturation calledcountable saturation. This simplied approach is also more suitable for use by mathematicians who are not specialistsin model theory or logic.The syntactic approach requiresmuch less logic andmodel theory to understand and use. This approachwas developedin the mid-1970s by the mathematician Edward Nelson. Nelson introduced an entirely axiomatic formulation of non-standard analysis that he called Internal Set Theory (IST).[13] IST is an extension of Zermelo-Fraenkel set theory (ZF)in that alongside the basic binary membership relation , it introduces a new unary predicate standard which can beapplied to elements of the mathematical universe together with some axioms for reasoning with this new predicate.Syntactic non-standard analysis requires a great deal of care in applying the principle of set formation (formally knownas the axiom of comprehension) which mathematicians usually take for granted. As Nelson points out, a commonfallacy in reasoning in IST is that of illegal set formation. For instance, there is no set in IST whose elements areprecisely the standard integers (here standard is understood in the sense of the new predicate). To avoid illegal setformation, one must only use predicates of ZFC to dene subsets.[13]

Another example of the syntactic approach is the Alternative Set Theory[14] introduced by Vopnka, trying to ndset-theory axioms more compatible with the non-standard analysis than the axioms of ZF.

2.5 Robinsons bookAbraham Robinsons book Non-standard analysis was published in 1966. Some of the topics developed in the bookwere already present in his 1961 article by the same title (Robinson 1961). In addition to containing the rst fulltreatment of non-standard analysis, the book contains a detailed historical section where Robinson challenges someof the received opinions on the history of mathematics based on the pre-NSA perception of innitesimals as incon-sistent entities. Thus, Robinson challenges the idea that Augustin-Louis Cauchy's sum theorem in Cours d'Analyseconcerning the convergence of a series of continuous functions was incorrect, and proposes an innitesimal-basedinterpretation of its hypothesis that results in a correct theorem.

2.6 Invariant subspace problemAbraham Robinson and Allen Bernstein proved that every polynomially compact linear operator on a Hilbert spacehas an invariant subspace.[15]

Given an operator T on Hilbert space H, consider the orbit of a point v in H under the iterates of T. Applying Gram-Schmidt one obtains an orthonormal basis (ei) for H. Let (Hi) be the corresponding nested sequence of coordinatesubspaces of H. The matrix ai,j expressing T with respect to (ei) is almost upper triangular, in the sense that thecoecients ai,i are the only nonzero sub-diagonal coecients. Bernstein and Robinson show that if T is polynomiallycompact, then there is a hypernite index w such that the matrix coecient aw,w is innitesimal. Next, considerthe subspace Hw of *H. If y in Hw has nite norm, then T(y) is innitely close to Hw.Now let Tw be the operator Pw T acting on Hw, where Pw is the orthogonal projection to Hw. Denote by q thepolynomial such that q(T) is compact. The subspace Hw is internal of hypernite dimension. By transferring uppertriangularisation of operators of nite-dimensional complex vector space, there is an internal orthonormal Hilbertspace basis (ek) for Hw where k runs from 1 to w, such that each of the corresponding k-dimensional subspacesEk is T-invariant. Denote by k the projection to the subspace Ek. For a nonzero vector x of nite norm in H,


one can assume that q(T)(x) is nonzero, or |q(T)(x)| > 1 to x ideas. Since q(T) is a compact operator, (q(Tw))(x)is innitely close to q(T)(x) and therefore one has also |q(Tw)(x)| > 1. Now let j be the greatest index such thatjq(Tw) (j(x)) j < 12 . Then the space of all standard elements innitely close to Ej is the desired invariant subspace.Upon reading a preprint of the Bernstein-Robinson paper, Paul Halmos reinterpreted their proof using standardtechniques.[16] Both papers appeared back-to-back in the same issue of the Pacic Journal of Mathematics. Some ofthe ideas used in Halmos proof reappeared many years later in Halmos own work on quasi-triangular operators.

2.7 Other applicationsOther results were received along the line of reinterpreting or reproving previously known results. Of particularinterest is Kamaes proof[17] of the individual ergodic theorem or van denDries andWilkies treatment[18] ofGromovstheorem on groups of polynomial growth. NSA was used by Larry Manevitz and Shmuel Weinberger to prove a resultin algebraic topology.[19]

The real contributions of non-standard analysis lie however in the concepts and theorems that utilizes the new extendedlanguage of non-standard set theory. Among the list of new applications in mathematics there are new approaches toprobability [11] hydrodynamics,[20] measure theory,[21] nonsmooth and harmonic analysis,[22] etc.There are also applications of non-standard analysis to the theory of stochastic processes, particularly constructionsof Brownian motion as random walks. Albeverio et-al[12] have an excellent introduction to this area of research.

2.7.1 Applications to calculusAs an application to mathematical education, H. Jerome Keisler wrote Elementary Calculus: An Innitesimal Ap-proach.[10] Covering non-standard calculus, it develops dierential and integral calculus using the hyperreal num-bers, which include innitesimal elements. These applications of non-standard analysis depend on the existence ofthe standard part of a nite hyperreal r. The standard part of r, denoted st(r), is a standard real number innitelyclose to r. One of the visualization devices Keisler uses is that of an imaginary innite-magnication microscope todistinguish points innitely close together. Keislers book is now out of print, but is freely available from his website;see references below.

2.8 CritiqueMain article: Criticism of non-standard analysis

Despite the elegance and appeal of some aspects of non-standard analysis, criticisms have been voiced, as well, suchas those by E. Bishop, A. Connes, and P. Halmos, as documented at Criticism of non-standard analysis.

2.9 Logical frameworkGiven any set S, the superstructure over a set S is the set V(S) dened by the conditions

V0(S) = S;

Vn+1(S) = Vn(S) [ 2Vn(S);V (S) =

[n2N

Vn(S):

Thus the superstructure over S is obtained by starting from S and iterating the operation of adjoining the power setof S and taking the union of the resulting sequence. The superstructure over the real numbers includes a wealth ofmathematical structures: For instance, it contains isomorphic copies of all separable metric spaces and metrizabletopological vector spaces. Virtually all of mathematics that interests an analyst goes on within V(R).

2.10. INTERNAL SETS 13

Theworking view of nonstandard analysis is a set *R and amapping * : V(R)V(*R) which satises some additionalproperties. To formulate these principles we rst state some denitions.A formula has bounded quantication if and only if the only quantiers which occur in the formula have rangerestricted over sets, that is are all of the form:

8x 2 A;(x; 1; : : : ; n)9x 2 A;(x; 1; : : : ; n)For example, the formula

8x 2 A; 9y 2 2B ; x 2 yhas bounded quantication, the universally quantied variable x ranges over A, the existentially quantied variable yranges over the powerset of B. On the other hand,

8x 2 A; 9y; x 2 ydoes not have bounded quantication because the quantication of y is unrestricted.

2.10 Internal setsA set x is internal if and only if x is an element of *A for some element A of V(R). *A itself is internal if A belongsto V(R).We now formulate the basic logical framework of nonstandard analysis:

Extension principle: The mapping * is the identity on R. Transfer principle: For any formula P(x1, ..., xn) with bounded quantication and with free variables x1, ..., xn,and for any elements A1, ..., An of V(R), the following equivalence holds:

P (A1; : : : ; An) () P (A1; : : : ; An)

Countable saturation: If {Ak}k N is a decreasing sequence of nonempty internal sets, with k ranging overthe natural numbers, then

\k

Ak 6= ;

One can show using ultraproducts that such a map * exists. Elements of V(R) are called standard. Elements of *Rare called hyperreal numbers.

2.11 First consequencesThe symbol *N denotes the nonstandard natural numbers. By the extension principle, this is a superset of N. The set*N N is nonempty. To see this, apply countable saturation to the sequence of internal sets

An = fk 2 N : k ng


The sequence {An}n N has a nonempty intersection, proving the result.We begin with some denitions: Hyperreals r, s are innitely close if and only if

r = s () 8 2 R+; jr sj

A hyperreal r is innitesimal if and only if it is innitely close to 0. For example, if n is a hyperinteger, i.e. an elementof *N N, then 1/n is an innitesimal. A hyperreal r is limited (or nite) if and only if its absolute value is dominatedby (less than) a standard integer. The limited hyperreals form a subring of *R containing the reals. In this ring, theinnitesimal hyperreals are an ideal.The set of limited hyperreals or the set of innitesimal hyperreals are external subsets of V(*R); what this means inpractice is that bounded quantication, where the bound is an internal set, never ranges over these sets.Example: The plane (x, y) with x and y ranging over *R is internal, and is a model of plane Euclidean geometry.The plane with x and y restricted to limited values (analogous to the Dehn plane) is external, and in this limited planethe parallel postulate is violated. For example, any line passing through the point (0, 1) on the y-axis and havinginnitesimal slope is parallel to the x-axis.Theorem. For any limited hyperreal r there is a unique standard real denoted st(r) innitely close to r. The mappingst is a ring homomorphism from the ring of limited hyperreals to R.The mapping st is also external.One way of thinking of the standard part of a hyperreal, is in terms of Dedekind cuts; any limited hyperreal s denesa cut by considering the pair of sets (L, U) where U is the set of standard rationals a less than s and U is the set ofstandard rationals b greater than s. The real number corresponding to (L, U) can be seen to satisfy the condition ofbeing the standard part of s.One intuitive characterization of continuity is as follows:Theorem. A real-valued function f on the interval [a, b] is continuous if and only if for every hyperreal x in theinterval *[a, b], we have: *f(x) *f(st(x)).(see microcontinuity for more details). Similarly,Theorem. A real-valued function f is dierentiable at the real value x if and only if for every innitesimal hyperrealnumber h, the value

f 0(x) = stf(x+ h) f(x)

h

exists and is independent of h. In this case f(x) is a real number and is the derivative of f at x.

2.12 -saturation

It is possible to improve the saturation by allowing collections of higher cardinality to be intersected. A model is-saturated if whenever fAigi2I is a collection of internal sets with the nite intersection property and jIj ,

\i2I

Ai 6= ;

This is useful, for instance, in a topological space X, where we may want |2X |-saturation to ensure the intersection ofa standard neighborhood base is nonempty.[23]

For any cardinal , a -saturated extension can be constructed.[24]

2.13. SEE ALSO 15

2.13 See also

2.14 Further reading E. E. Rosinger, [math/0407178]. Short introduction to Nonstandard Analysis. arxiv.org.

2.15 References[1] Nonstandard Analysis in Practice. Edited by Francine Diener, Marc Diener. Springer, 1995.[2] Nonstandard Analysis, Axiomatically. By V. Vladimir Grigorevich Kanovei, Michael Reeken. Springer, 2004.[3] Nonstandard Analysis for the Working Mathematician. Edited by Peter A. Loeb, Manfred P. H. Wol. Springer, 2000.[4] Non-standard Analysis. By Abraham Robinson. Princeton University Press, 1974.[5] Abraham Robinson and Nonstandard Analysis: History, Philosophy, and Foundations of Mathematics. By Joseph W.

Dauben. www.mcps.umn.edu.[6] Robinson, A.: Non-standard analysis. North-Holland Publishing Co., Amsterdam 1966.[7] Heijting, A. (1973) Address to Professor A. Robinson. At the occasion of the Brouwer memorial lecture given by Prof.

A.Robinson on the 26th April 1973. Nieuw Arch. Wisk. (3) 21, pp. 134137.[8] Robinson, Abraham (1996). Non-standard analysis (Revised edition ed.). Princeton University Press. ISBN 0-691-04490-

2.[9] Curt Schmieden and Detlef Laugwitz: Eine Erweiterung der Innitesimalrechnung, Mathematische Zeitschrift 69 (1958),

1-39[10] H. Jerome Keisler, Elementary Calculus: An Innitesimal Approach. First edition 1976; 2nd edition 1986: full text of 2nd

edition[11] Edward Nelson: Radically Elementary Probability Theory, Princeton University Press, 1987, full text[12] Sergio Albeverio, Jans Erik Fenstad, Raphael Hegh-Krohn, Tom Lindstrm: Nonstandard Methods in Stochastic Analysis

and Mathematical Physics, Academic Press 1986.[13] Edward Nelson: Internal Set Theory: A New Approach to Nonstandard Analysis, Bulletin of the American Mathematical

Society, Vol. 83, Number 6, November 1977. A chapter on Internal Set Theory is available at http://www.math.princeton.edu/~{}nelson/books/1.pdf

[14] Vopnka, P. Mathematics in the Alternative Set Theory. Teubner, Leipzig, 1979.[15] Allen Bernstein and Abraham Robinson, Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos, Pacic

Journal of Mathematics 16:3 (1966) 421-431[16] P. Halmos, Invariant subspaces for Polynomially Compact Operators, Pacic Journal of Mathematics, 16:3 (1966) 433-437.[17] T. Kamae: A simple proof of the ergodic theorem using nonstandard analysis, Israel Journal ofMathematics vol. 42, Number

4, 1982.[18] L. van den Dries and A. J. Wilkie: Gromovs Theorem on Groups of Polynomial Growth and Elementary Logic, Journal of

Algebra, Vol 89, 1984.[19] Manevitz, Larry M.; Weinberger, Shmuel: Discrete circle actions: a note using non-standard analysis. Israel J. Math. 94

(1996), 147-155.[20] Capinski M., Cutland N. J. Nonstandard Methods for Stochastic Fluid Mechanics. Singapore etc., World Scientic Publish-

ers (1995)[21] Cutland N. Loeb Measures in Practice: Recent Advances. Berlin etc.: Springer (2001)[22] Gordon E. I., Kutateladze S. S., and Kusraev A. G. Innitesimal Analysis Dordrecht, Kluwer Academic Publishers (2002)[23] Salbany, S.; Todorov, T. Nonstandard Analysis in Point-Set Topology. Erwing Schrodinger Institute for Mathematical

Physics.[24] Chang, C. C.; Keisler, H. J. Model theory. Third edition. Studies in Logic and the Foundations of Mathematics, 73.

North-Holland Publishing Co., Amsterdam, 1990. xvi+650 pp. ISBN 0-444-88054-2


2.16 Bibliography

2.17 External links The Ghosts of Departed Quantities by Lindsay Keegan.

Chapter 3

Real closed eld

ArtinSchreier theorem redirects here. For the branch of Galois theory, see ArtinSchreier theory.

In mathematics, a real closed eld is a eld F that has the same rst-order properties as the eld of real numbers.Some examples are the eld of real numbers, the eld of real algebraic numbers, and the eld of hyperreal numbers.

3.1 DenitionsA real closed eld is a eld F in which any of the following equivalent conditions are true:

1. F is elementarily equivalent to the real numbers. In other words it has the same rst-order properties as thereals: any sentence in the rst-order language of elds is true in F if and only if it is true in the reals. (Thechoice of signature is not signicant.)

2. There is a total order on F making it an ordered eld such that, in this ordering, every positive element of Fhas a square root in F and any polynomial of odd degree with coecients in F has at least one root in F.

3. F is a formally real eld such that every polynomial of odd degree with coecients in F has at least one rootin F, and for every element a of F there is b in F such that a = b2 or a = b2.

4. F is not algebraically closed but its algebraic closure is a nite extension.5. F is not algebraically closed but the eld extension F (p1) is algebraically closed.6. There is an ordering on F which does not extend to an ordering on any proper algebraic extension of F.7. F is a formally real eld such that no proper algebraic extension of F is formally real. (In other words, the eld

is maximal in an algebraic closure with respect to the property of being formally real.)8. There is an ordering on F making it an ordered eld such that, in this ordering, the intermediate value theorem

holds for all polynomials over F with degree 0.9. F is a real closed ring.

If F is an ordered eld, the ArtinSchreier theorem states that F has an algebraic extension, called the real closureK of F, such that K is a real closed eld whose ordering is an extension of the given ordering on F, and is unique upto a unique isomorphism of elds identical on F[1] (note that every ring homomorphism between real closed eldsautomatically is order preserving, because x y if and only if z y = x + z2). For example, the real closure of theordered eld of rational numbers is the eld Ralg of real algebraic numbers. The theorem is named for Emil Artinand Otto Schreier, who proved it in 1926.If (F,P) is an ordered eld, and E is a Galois extension of F, then by Zorns Lemma there is a maximal ordered eldextension (M,Q) with M a subeld of E containing F and the order on M extending P. M, together with its orderingQ, is called the relative real closure of (F,P) in E. We call (F,P) real closed relative to E if M is just F. When Eis the algebraic closure of F the relative real closure of F in E is actually the real closure of F described earlier.[2]

17

18 CHAPTER 3. REAL CLOSED FIELD

If F is a eld (no ordering compatible with the eld operations is assumed, nor is assumed that F is orderable) thenF still has a real closure, which may not be a eld anymore, but just a real closed ring. For example, the real closureof the eld Q(

p2) is the ring Ralg Ralg (the two copies correspond to the two orderings of Q(

p2) ). On the other

hand, if Q(p2) is considered as an ordered subeld of R , its real closure is again the eld Ralg .

3.2 Model theory: decidability and quantier eliminationAlthough the theory of real closed elds was rstly developed by algebraists, it has received considerable attentionfrom Model Theory. By adding to the ordered eld axioms

an axiom asserting that every positive number has a square root, and an axiom scheme asserting that all polynomials of odd degree have at least one root

one obtains a rst-order theory. Alfred Tarski (1951) proved that the theory of real closed elds in the rst orderlanguage of partially ordered rings (consisting of the binary predicate symbols "=" and "", the operations of addition,subtraction and multiplication and the constant symbols 0,1) admits elimination of quantiers. The most importantmodel theoretic consequences hereof: The theory of real closed elds is complete, o-minimal and decidable.Decidability means that there exists at least one decision procedure, i.e., a well-dened algorithm for determiningwhether a sentence in the rst order language of real closed elds is true. Euclidean geometry (without the ability tomeasure angles) is also a model of the real eld axioms, and thus is also decidable.The decision procedures are not necessarily practical. The algorithmic complexities of all known decision proceduresfor real closed elds are very high, so that practical execution times can be prohibitively high except for very simpleproblems.The algorithm Tarski proposed for quantier elimination has NONELEMENTARY complexity, meaning that notower 22

n

can bound the execution time of the algorithm if n is the size of the problem. Davenport and Heintz(1988) proved that quantier elimination is in fact (at least) doubly exponential: there exists a family of formulaswith n quantiers, of length O(n) and constant degree such that any quantier-free formula equivalent to mustinvolve polynomials of degree 22(n) and length 22(n) , using the asymptotic notation. Ben-Or, Kozen, andReif (1986) proved that the theory of real closed elds is decidable in exponential space, and therefore in doublyexponential time.Basu and Roy (1996) proved that there exists a well-behaved algorithm to decide the truth of a formula x1,,xP1(x1,,x)0P(x1,,x)0 where is or =, with complexity in arithmetic operations sk+1dO(k). In fact,the existential theory of the reals can be decided in PSPACE.Adding additional functions symbols, for example, the sine or the exponential function, can change the decidabilityof the theory.Yet another important model-theoretic property of real closed elds is that they are weakly o-minimal structures.Conversely, it is known that any weakly o-minimal ordered eld must be real closed.[3]

3.3 Order propertiesA crucially important property of the real numbers is that it is an Archimedean eld, meaning it has the Archimedeanproperty that for any real number, there is an integer larger than it in absolute value. An equivalent statement is thatfor any real number, there are integers both larger and smaller. Such real closed elds that are not Archimedean, arenon-Archimedean ordered elds. For example, any eld of hyperreal numbers is real closed and non-Archimedean.The Archimedean property is related to the concept of conality. A set X contained in an ordered set F is conal in Fif for every y in F there is an x in X such that y < x. In other words, X is an unbounded sequence in F. The conalityof F is the size of the smallest conal set, which is to say, the size of the smallest cardinality giving an unboundedsequence. For example natural numbers are conal in the reals, and the conality of the reals is therefore @0 .We have therefore the following invariants dening the nature of a real closed eld F:

The cardinality of F.

3.4. THE GENERALIZED CONTINUUM HYPOTHESIS 19

The conality of F.

To this we may add

The weight of F, which is the minimum size of a dense subset of F.

These three cardinal numbers tell us much about the order properties of any real closed eld, though it may be dicultto discover what they are, especially if we are not willing to invoke generalized continuum hypothesis. There are alsoparticular properties which may or may not hold:

A eld F is complete if there is no ordered eld K properly containing F such that F is dense in K. If theconality of F is , this is equivalent to saying Cauchy sequences indexed by are convergent in F.

An ordered eld F has the eta set property , for the ordinal number , if for any two subsets L and U of Fof cardinality less than @ such that every element of L is less than every element of U, there is an element xin F with x larger than every element of L and smaller than every element of U. This is closely related to themodel-theoretic property of being a saturated model; any two real closed elds are if and only if they are@ -saturated, and moreover two real closed elds both of cardinality @ are order isomorphic.

3.4 The generalized continuum hypothesisThe characteristics of real closed elds become much simpler if we are willing to assume the generalized continuumhypothesis. If the continuum hypothesis holds, all real closed elds with cardinality the continuum and having the 1property are order isomorphic. This unique eld can be dened by means of an ultrapower, as RN/M , whereM isa maximal ideal not leading to a eld order-isomorphic toR . This is the most commonly used hyperreal number eldin non-standard analysis, and its uniqueness is equivalent to the continuum hypothesis. (Even without the continuumhypothesis we have that if the cardinality of the continuum is @ then we have a unique eld of size .)Moreover, we do not need ultrapowers to construct , we can do so much more constructively as the subeld of serieswith a countable number of nonzero terms of the eld R((G)) of formal power series on a totally ordered abeliandivisible group G that is an 1 group of cardinality @1 (Alling 1962). however is not a complete eld; if we take its completion, we end up with a eld of larger cardinality. has thecardinality of the continuum which by hypothesis is @1 , has cardinality @2 , and contains as a dense subeld.It is not an ultrapower but it is a hyperreal eld, and hence a suitable eld for the usages of nonstandard analysis. Itcan be seen to be the higher-dimensional analogue of the real numbers; with cardinality @2 instead of @1 , conality@1 instead of @0 , and weight @1 instead of @0 , and with the 1 property in place of the 0 property (which merelymeans between any two real numbers we can nd another).

3.5 Examples of real closed elds the real algebraic numbers

the computable numbers

the denable numbers

the real numbers

superreal numbers

hyperreal numbers

the Puiseux series with real coecients

20 CHAPTER 3. REAL CLOSED FIELD

3.6 Notes[1] Rajwade (1993) pp. 222223

[2] Efrat (2006) p. 177

[3] D. Macpherson et. al, (1998)

3.7 References Alling, Norman L. (1962), On the existence of real-closed elds that are -sets of power ., Trans. Amer.Math. Soc. 103: 341352, doi:10.1090/S0002-9947-1962-0146089-X, MR 0146089

Basu, Saugata, Richard Pollack, and Marie-Franoise Roy (2003) Algorithms in real algebraic geometry inAlgorithms and computation in mathematics. Springer. ISBN 3-540-33098-4 (online version)

Michael Ben-Or, Dexter Kozen, and John Reif, The complexity of elementary algebra and geometry, Journalof Computer and Systems Sciences 32 (1986), no. 2, pp. 251264.

Caviness, B F, and Jeremy R. Johnson, eds. (1998) Quantier elimination and cylindrical algebraic decompo-sition. Springer. ISBN 3-211-82794-3

Chen Chung Chang and Howard Jerome Keisler (1989) Model Theory. North-Holland. Dales, H. G., and W. Hugh Woodin (1996) Super-Real Fields. Oxford Univ. Press. Davenport, James H.; Heintz, Joos (1988). Real quantier elimination is doubly exponential. J. Symb.Comput. 5 (1-2): 2935. Zbl 0663.03015.

Efrat, Ido (2006). Valuations, orderings, and Milnor K-theory. Mathematical Surveys and Monographs 124.Providence, RI: American Mathematical Society. ISBN 0-8218-4041-X. Zbl 1103.12002.

Macpherson, D., Marker, D. and Steinhorn, C., Weakly o-minimal structures and real closed elds, Trans. ofthe American Math. Soc., Vol. 352, No. 12, 1998.

Mishra, Bhubaneswar (1997) "Computational Real Algebraic Geometry," in Handbook of Discrete and Com-putational Geometry. CRC Press. 2004 edition, p. 743. ISBN 1-58488-301-4

Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series 171. Cambridge Univer-sity Press. ISBN 0-521-42668-5. Zbl 0785.11022.

Alfred Tarski (1951) A Decision Method for Elementary Algebra and Geometry. Univ. of California Press. Erds, P.; Gillman, L.; Henriksen, M. (1955), An isomorphism theorem for real-closed elds, Ann. of Math.(2) 61: 542554, MR 0069161

3.8 External links Real Algebraic and Analytic Geometry Preprint Server Model Theory preprint server

Chapter 4

Real closed ring

In mathematics, a real closed ring is a commutative ring A that is a subring of a product of real closed elds, whichis closed under continuous semi-algebraic functions dened over the integers.

4.1 Examples of real closed ringsSince the rigorous denition of a real closed ring is of technical nature it is convenient to see a list of prominentexamples rst. The following rings are all real closed rings:

real closed elds. These are exactly the real closed rings that are elds.

the ring of all real valued continuous functions on a completely regular space X. Also, the ring of all boundedreal valued continuous functions on X is real closed.

convex subrings of real closed elds. These are precisely those real closed rings which are also valuation ringsand were initially studied by Cherlin and Dickmann (they used the term 'real closed ring' for what is now called'real closed valuation ring').

the ring A of all continuous semi-algebraic functions on a semi-algebraic set of a real closed eld (with valuesin that eld). Also, the subring of all bounded (in any sense) functions in A is real closed.

(generalizing the previous example) the ring of all (bounded) continuous denable functions on a denable setS of an arbitrary rst-order expansionM of a real closed eld (with values inM). Also, the ring of all (bounded)denable functions S !M is real closed.

Real closed rings are precisely the rings of global sections of ane real closed spaces (a generalization ofsemialgebraic spaces) and in this context they were invented by Niels Schwartz in the early 1980s.

4.2 DenitionA real closed ring is a reduced, commutative unital ring A which has the following properties:

1. The set of squares of A is the set of nonnegative elements of a partial order on A and (A,) is an f-ring.

2. Convexity condition: For all a,b from A, if 0ab then b|a2.

3. For every prime ideal p of A, the residue class ring A/p is integrally closed and its eld of fractions is a realclosed eld.

The link to the denition at the beginning of this article is given in the section on algebraic properties below.

21

22 CHAPTER 4. REAL CLOSED RING

4.3 The real closure of a commutative ringEvery commutative unital ring R has a so-called real closure rcl(R) and this is unique up to a unique ring homomor-phism over R. This means that rcl(R) is a real closed ring and there is a (not necessarily injective) ring homomorphismr : R ! rcl(R) such that for every ring homomorphism f : R ! A to some other real closed ring A, there is aunique ring homomorphism g : rcl(R)! A with f = g r .For example the real closure of the polynomial ring R[T1; :::; Tn] is the ring of continuous semi-algebraic functionsRn ! R .Note that an arbitrary ring R is semi-real (i.e. 1 is not a sum of squares in R) if and only if the real closure of R isnot the null ring.Note also that the real closure of an ordered eld is in general not the real closure of the underlying eld. Forexample, the real closure of the ordered subeldQ(

p2) ofR is the eldRalg of real algebraic numbers, whereas the

real closure of the eldQ(p2) is the ring RalgRalg (corresponding to the two orders ofQ(

p2) ). More generally

the real closure of a eld F is a certain subdirect product of the real closures of the ordered elds (F,P), where P runsthrough the orderings of F.

4.4 Algebraic properties The category RCR of real closed rings which has real closed rings as objects and ring homomorphisms as mapshas the following properties:

1. Arbitrary products, direct limits and inverse limits (in the category of commutative unital rings) of real closedrings are again real closed. The bre sum of two real closed rings B,C over some real closed ring A exists inRCR and is the real closure of the tensor product of B and C over A.

2. RCR has arbitrary limits and co-limits.

3. RCR is a variety in the sense of universal algebra (but not a subvariety of commutative rings).

For a real closed ringA, the natural homomorphism ofA to the product of all its residue elds is an isomorphismonto a subring of this product that is closed under continuous semi-algebraic functions dened over the integers.Conversely, every subring of a product of real closed elds with this property is real closed.

If I is a radical ideal of a real closed ring A, then also the residue class ring A/I is real closed. If I and J areradical ideals of a real closed ring then the sum I + J is again a radical ideal.

All classical localizations S1A of a real closed ring A are real closed. The epimorphic hull and the completering of quotients of a real closed ring are again real closed.

The (real) holomorphy ring H(A) of a real closed ring A is again real closed. By denition, H(A) consists ofall elements f in A with the property N f N for some natural number N. Applied to the examples above,this means that the rings of bounded (semi-algebraic/denable) continuous functions are all real closed.

The support map from the real spectrum of a real closed ring to its Zariski spectrum, which sends an orderingP to its support P \ P is a homeomorphism. In particular, the Zariski spectrum of every real closed ring Ais a root system (in the sense of graph theory) and therefore A is also a Gel'fand ring (i.e. every prime ideal ofA is contained in a unique maximal ideal of A). The comparison of the Zariski spectrum of A with the Zariskispectrum of H(A) leads to a homeomorphism between the maximal spectra of these rings, generalizing theGel'fand-Kolmogorov theorem for rings of real valued continuous functions.

The natural map r from an arbitrary ring R to its real closure rcl(R) as explained above, induces a homeomor-phism from the real spectrum of rcl(R) to the real spectrum of R.

Summarising and signicantly strengthening the previous two properties, the following is true: The naturalmap r from an arbitrary ring R to its real closure rcl(R) induces an identication of the ane scheme of rcl(R)with the ane real closed space of R.

4.5. MODEL THEORETIC PROPERTIES 23

4.5 Model theoretic propertiesThe class of real closed rings is rst-order axiomatizable and undecidable. The class of all real closed valuation ringsis decidable (by Cherlin-Dickmann) and the class of all real closed elds is decidable (by Tarski). After naming adenable radical relation, real closed rings have a model companion, namely von Neumann regular real closed rings.

4.6 Comparison with characterizations of real closed eldsThere are many dierent characterizations of real closed elds. For example in terms of maximality (with respect toalgebraic extensions): a real closed eld is a maximally orderable eld; or, a real closed eld (together with its uniqueordering) is a maximally ordered eld. Another characterization says that the intermediate value theorem holds forall polynomials in one variable over the (ordered) eld. In the case of commutative rings, all these properties can be(and are) analyzed in the literature. They all lead to dierent classes of rings which are unfortunately also called 'realclosed' (because a certain characterization of real closed elds has been extended to rings). None of them lead tothe class of real closed rings and none of them allow a satisfactory notion of a closure operation. A central point inthe denition of real closed rings is the globalisation of the notion of a real closed eld to rings when these rings arerepresented as rings of functions on some space (typically, the real spectrum of the ring).

4.7 References Cherlin, Gregory. Rings of continuous functions: decision problems Model theory of algebra and arithmetic(Proc. Conf., Karpacz, 1979), pp. 4491, Lecture Notes in Math., 834, Springer, Berlin, 1980.

Cherlin, Gregory(1-RTG2); Dickmann, Max A. Real closed rings. II. Model theory. Ann. Pure Appl. Logic25 (1983), no. 3, 213231.

A. Prestel, N. Schwartz. Model theory of real closed rings. Valuation theory and its applications, Vol. I(Saskatoon, SK, 1999), 261290, Fields Inst. Commun., 32, Amer. Math. Soc., Providence, RI, 2002.

Schwartz, Niels. The basic theory of real closed spaces. Memoirs of the American Mathematical Society 1989(ISBN 0821824600 )

Schwartz, Niels; Madden, James J. Semi-algebraic function rings and reectors of partially ordered rings.Lecture Notes in Mathematics, 1712. Springer-Verlag, Berlin, 1999

Schwartz, Niels. Real closed rings. Algebra and order (Luminy-Marseille, 1984), 175194, Res. Exp. Math.,14, Heldermann, Berlin, 1986

Schwartz, Niels. Rings of continuous functions as real closed rings. Ordered algebraic structures (Curaao,1995), 277313, Kluwer Acad. Publ., Dordrecht, 1997.

Tressl, Marcus. Super real closed rings. Fundamenta Mathematicae 194 (2007), no. 2, 121177.

Chapter 5

Standard part function

In non-standard analysis, the standard part function is a function from the limited (nite) hyperreal numbers tothe real numbers. Briey, the standard part function rounds o a nite hyperreal to the nearest real. It associatesto every such hyperreal x , the unique real x0 innitely close to it, i.e. x x0 is innitesimal. As such, it is amathematical implementation of the historical concept of adequality introduced by Pierre de Fermat,[1] as well asLeibniz's Transcendental law of homogeneity.The standard part function was rst dened by Abraham Robinson who used the notation x for the standard part ofa hyperreal x (see Robinson 1974). This concept plays a key role in dening the concepts of the calculus, such ascontinuity, the derivative, and the integral, in non-standard analysis. The latter theory is a rigorous formalisation ofcalculations with innitesimals. The standard part of x is sometimes referred to as its shadow.

5.1 Denition

The standard part function rounds o a nite hyperreal to the nearest real number. The innitesimal microscope is used to viewan innitesimal neighborhood of a standard real.

24

5.2. NOT INTERNAL 25

Nonstandard analysis deals primarily with the pairR R , where the hyperreals R are an ordered eld extension ofthe realsR , and contain innitesimals, in addition to the reals. In the hyperreal line every real number has a collectionof numbers (called a monad, or halo) of hyperreals innitely close to it. The standard part function associates to anite hyperreal x, the unique standard real number x0 which is innitely close to it. The relationship is expressedsymbolically by writing

st(x) = x0:

The standard part of any innitesimal is 0. Thus if N is an innite hypernatural, then 1/N is innitesimal, and st(1/N)= 0.If a hyperreal u is represented by a Cauchy sequence hun : n 2 Ni in the ultrapower construction, then

st(u) = limn!1un:

More generally, each nite u 2 R denes a Dedekind cut on the subset R R (via the total order on R ) and thecorresponding real number is the standard part of u.

5.2 Not internalThe standard part function st is not dened by an internal set. There are several ways of explaining this. Perhaps thesimplest is that its domain L, which is the collection of limited (i.e. nite) hyperreals, is not an internal set. Namely,since L is bounded (by any innite hypernatural, for instance), L would have to have a least upper bound if L wereinternal, but L doesn't have a least upper bound. Alternatively, the range of st is R R which is not internal; infact every internal set in R which is a subset of R is necessarily nite, see (Goldblatt, 1998).

5.3 ApplicationsAll the traditional notions of calculus are expressed in terms of the standard part function, as follows.

5.3.1 Derivative

The standard part function is used to dene the derivative of a function f. If f is a real function, and h is innitesimal,and if f(x) exists, then

f 0(x) = stf(x+ h) f(x)

h

:

Alternatively, if y = f(x) , one takes an innitesimal increment x , and computes the corresponding y =f(x+x) f(x) . One forms the ratio yx . The derivative is then dened as the standard part of the ratio:

dy

dx= st

y

x

5.3.2 Integral

Given a function f on [a; b] , one denes the integralR baf(x)dx as the standard part of an innite Riemann sum

S(f; a; b;x) when the value of x is taken to be innitesimal, exploiting a hypernite partition of the interval[a,b].

26 CHAPTER 5. STANDARD PART FUNCTION

5.3.3 LimitGiven a sequence (un) , its limit is dened by limn!1 un = st(uH) where H 2 N n N is an innite index. Herethe limit is said to exist if the standard part is the same regardless of the innite index chosen.

5.3.4 ContinuityA real function f is continuous at a real point x if and only if the composition st f is constant on the halo of x . Seemicrocontinuity for more details.

5.4 See also Adequality Non-standard calculus

5.5 Notes[1] Karin Usadi Katz and Mikhail G. Katz (2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Math-

ematics and its Historiography. Foundations of Science. doi:10.1007/s10699-011-9223-1 See arxiv. The authors refer tothe Fermat-Robinson standard part.

5.6 References H. Jerome Keisler. Elementary Calculus: An Innitesimal Approach. First edition 1976; 2nd edition 1986.(This book is now out of print. The publisher has reverted the copyright to the author, who has made availablethe 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~{}keisler/calc.html.)

Goldblatt, Robert. Lectures on the hyperreals. An introduction to nonstandard analysis. Graduate Texts inMathematics, 188. Springer-Verlag, New York, 1998.

Abraham Robinson. Non-standard analysis. Reprint of the second (1974) edition. With a foreword byWilhelmus A. J. Luxemburg. Princeton Landmarks in Mathematics. Princeton University Press, Princeton,NJ, 1996. xx+293 pp. ISBN 0-691-04490-2

Chapter 6

Superreal number

In abstract algebra, the superreal numbers are a class of extensions of the real numbers, introduced by H. GarthDales and W. Hugh Woodin as a generalization of the hyperreal numbers and primarily of interest in non-standardanalysis, model theory, and the study of Banach algebras. The eld of superreals is itself a subeld of the surrealnumbers.Dales andWoodins superreals are distinct from the super-real numbers of David O. Tall, which are lexicographicallyordered fractions of formal power series over the reals.[1]

6.1 Formal DenitionSuppose X is a Tychono space, also called a T. space, and C(X) is the algebra of continuous real-valued functionson X. Suppose P is a prime ideal in C(X). Then the factor algebra A = C(X)/P is by denition an integral domainwhich is a real algebra and which can be seen to be totally ordered. The eld of fractions F of A is a superreal eldif F strictly contains the real numbers R , so that F is not order isomorphic to R .If the prime ideal P is a maximal ideal, then F is a eld of hyperreal numbers (Robinsons hyperreals being a veryspecial case).

6.2 References[1] David Tall, Looking at graphs through innitesimal microscopes, windows and telescopes, Mathematical Gazette, 64 22

49, reprint at http://www.warwick.ac.uk/staff/David.Tall/downloads.html

6.3 Bibliography Dales, H. Garth; Woodin, W. Hugh (1996), Super-real elds, LondonMathematical SocietyMonographs. NewSeries 14, The Clarendon Press Oxford University Press, ISBN 978-0-19-853991-9, MR 1420859

L. Gillman and M. Jerison: Rings of Continuous Functions, Van Nostrand, 1960.

27

Chapter 7

Surreal number

In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well asinnite and innitesimal numbers, respectively larger or smaller in absolute value than any positive real number. Thesurreals share many properties with the reals, including a total order and the usual arithmetic operations (addition,subtraction, multiplication, and division); as such, they form an ordered eld. (Strictly speaking, the surreals are nota set, but a proper class.[1]) If formulated in Von NeumannBernaysGdel set theory, the surreal numbers are thelargest possible ordered eld; all other ordered elds, such as the rationals, the reals, the rational functions, the Levi-Civita eld, the superreal numbers, and the hyperreal numbers, can be realized as subelds of the surreals.[2] It hasalso been shown (in Von NeumannBernaysGdel set theory) that the maximal class hyperreal eld is isomorphic tothe maximal class surreal eld; in theories without the axiom of global choice, this need not be the case, and in suchtheories it is not necessarily true that the surreals are the largest ordered eld. The surreals also contain all transniteordinal numbers; the arithmetic on them is given by the natural operations.In 1907 Hahn introduced Hahn series as a generalization of formal power series, and Hausdor introduced certainordered sets called -sets for ordinals and asked if it was possible to nd a compatible ordered group or eldstructure. In 1962 Alling used a modied form of Hahn series to construct such ordered elds associated to certainordinals , and taking to be the class of all ordinals in his construction gives a class that is an ordered eld isomor-phic to the surreal numbers.[3] Research on the go endgame by John Horton Conway led to a simpler denition andconstruction of the surreal numbers.[4] Conways construction was introduced in Donald Knuth's 1974 book SurrealNumbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. In his book, which takesthe form of a dialogue, Knuth coined the term surreal numbers for what Conway had called simply numbers. Conwaylater adopted Knuths term, and used surreals for analyzing games in his 1976 book On Numbers and Games.

7.1 OverviewThe surreal numbers are constructed in stages, along with an ordering such that for any two surreal numbers a andb either a b or b a. (Both may hold, in which case a and b are equivalent and denote the same number.) Numbersare formed by pairing subsets of numbers already constructed: given subsets L and R of numbers such that all themembers of L are strictly less than all the members of R, then the pair { L | R } represents a number intermediate invalue between all the members of L and all the members of R.Dierent subsets may end up dening the same number: { L | R } and { L | R } may dene the same number evenif L L and R R. (A similar phenomenon occurs when rational numbers are dened as quotients of integers:1/2 and 2/4 are dierent representations of the same rational number.) So strictly speaking, the surreal numbers areequivalence classes of representations of form { L | R } that designate the same number.In the rst stage of construction, there are no previously existing numbers so the only representation must use theempty set: { | }. This representation, where L and R are both empty, is called 0. Subsequent stages yield forms like:

{ 0 | } = 1

{ 1 | } = 2

{ 2 | } = 3

28

7.1. OVERVIEW 29

The surreal number tree visualization.

and

{ | 0 } = 1

{ | 1 } = 2

{ | 2 } = 3

The integers are thus contained within the surreal numbers. Similarly, representations arise like:

30 CHAPTER 7. SURREAL NUMBER

{ 0 | 1 } = 1/2

{ 0 | 1/2 } = 1/4

{ 1/2 | 1 } = 3/4

so that the dyadic rationals (rational numbers whose denominators are powers of 2) are contained within the surrealnumbers.After an innite number of stages, innite subsets become available, so that any real number a can be represented by{ La | Ra }, where La is the set of all dyadic rationals less than a and Ra is the set of all dyadic rationals greater thana (reminiscent of a Dedekind cut). Thus the real numbers are also embedded within the surreals.But there are also representations like

{ 0, 1, 2, 3, | } =

{ 0 | 1, 1/2, 1/4, 1/8, } =

where is a transnite number greater than all integers and is an innitesimal greater than 0 but less than any positivereal number. Moreover, the standard arithmetic operations (addition, subtraction, multiplication, and division) canbe extended to these non-real numbers in a manner that turns the collection of surreal numbers into an ordered eld,so that one can talk about 2 or 1 and so forth.

7.2 ConstructionSurreal numbers are constructed inductively as equivalence classes of pairs of sets of surreal

real closed field

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examples of real closed

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