Nonstandard Analysis

Armin Straub Technische Universität Darmstadt

Dec 10, 2007

Contents

1 Introduction...... 2 1.1 Approaches to Getting Inﬁnitesimals ...... 2 1.2 The Use of Inﬁnitesimals ...... 5

2 Diving In...... 6 2.1 Limits and Ultraﬁlters ...... 6 2.2 A First Nonstandard Peak ...... 9 2.3 An Axiomatic Approach ...... 11 2.4 Basic Usage ...... 12 2.5 The Source of Nonstandard Strength ...... 16 2.6 Stronger Nonstandard Models ...... 17 2.7 Studying Z in its Own Right ...... 19

3 Conclusions...... 20

References...... 21

Abstract

These notes complement a seminar talk I gave as a student at TU Darm- stadt as part of the StuVo (Studentische Vortragsreihe), Dec 10, 2007. While this text is meant to be informal in nature, any corrections as well as suggestions are very welcome. We discuss some ways to add inﬁnitesimals to our usual numbers, get acquainted with ultrafilters and how they can be used to construct nonstandard extensions, and then provide an axiomatic framework for nonstandard analysis. As basic working examples we present nonstandard characterizations of continuity and uniform continuity. We close with a short external look at the nonstandard integers and point out connections withp-adic integers.

1 2 Nonstandard Analysis

1 Introduction

1.1 Approaches to Getting Inﬁnitesimals

One aspect of nonstandard analysis is that it introduces a new kind of numbers, namely infinitely large numbers (and hence infinitesimally small ones as well). This is something that may be philosophically challenging to a Platonist. It's not just that we are talking about infinity but its quality here is of a particular kind. When we have a never-ending process, like counting the natural numbers, we talk of potential infinity. Thiscounting never ends can be formalized as

x N y y>x; 8 2 9 and is usually philosophically well accepted by most people. By contrast, the actual inﬁnity of nonstandard numbers corresponds to the statement that

y x N y>x; 9 8 2 and usually troubles people more easily. On the other hand, mathematicians like Euler, Leibniz or Newton had a very good conceptual understanding of infinitesimals. It wasn't before 1961, however, that they were rigorously introduced into mathematics as formal objects by Robinson, see also [Rob67]. But Robinson's approach does more than just introduce infinitesimals and therefore justifying the non-Archimedean approach prevailing before the"- infection by Weierstrass around 1850. It enriches all infinite structures as well and turns out to be very suitable for applications infields like topology or probability theory where a metric concept is absent in thefirst place. In 1934 Skolem pointed out that the natural numbers can't be characterized as afirst-order theory. The unusual structures satisfying the same properties as the natural numbers are therefore called nonstandard models of arithmetic. Since we learned a lot about (in)completeness in the previous talk we start by showing how logic leads to nonstandard models. In the sequel, however, we will try to be moreconstructive.

Example 1.1. Recall Gödel's incompleteness theorem. In one guise it states that every consistentﬁrst-order theory that includes arithmetic cannot prove its own consistency. Starting with the usual axioms of Peano arithmetic1, PA for short, we can add its inconsistency Con(PA) as an axiom (we're very : informal here). If PA is consistent then so is PA Con(PA). Obviously, a ^: model for the latter can't be our usual (or standard) natural numbers. Hence we must have obtained a nonstandard model of the natural numbers. But well, that was kind of magic. What do we know about this model after all?

1. Note that in PA the second-order induction axiom is replaced by aﬁrst-order induction schema which consists of one axiom for each of the countably many formulas in PA. Armin Straub 3

Example 1.2. There is another common way to get nonstandard models with the help of logic and the so-called compactness theorem. Gödel's compactness theorem roughly states that a set of axioms has a model if and only if every ﬁnite subset of these axioms has a model. Let's start with PA arithmetic again and consider the theory obtained by adding an elementc to the signature of PA as well as the following axioms

c >1; c >1+1;:::

Everyﬁnite subset of these axioms clearly has a model, namely the natural numbersN with a big elementc N exhibited. The compactness theorem 2 then asserts that there also exists a model satisfying all the above axioms. Forgetting the special role ofc, this gives a nonstandard model for PA. While this was still rather magical, we at least have an inﬁnitely large elementc at hand.

Remark 1.3. We cannot resist to demonstrate at least one cool and inspiring implication based on the compactness theorem. Namely, if a statementS (first-order, of course) is true for everyfield of characteristic0 then it is true for everyfield of characteristicp as long asp is big enough. Just note that the assumption implies that

S;field axioms; 1+1=/0; 1+1+1=/0;::: : is not satisfiable. By the compactness theorem afinite subset of these statements isn't satisfiable already.

Remark 1.4. By the way, ZF proves that Gödel's compactness theorem as well as Gödel's completeness theorem are equivalent to the ultraﬁlter theorem. So in the case that you despise our later use of the ultraﬁlter theorem to construct nonstandard models, you should feel uncomfortable with this part of mathematics as well.

On the other hand, it is easy to introduce inﬁnitely large objects that serve some speciﬁc but limited purpose. We present some examples in the hope that they will increase our appreciation of the nonstandard world that Robinson invited us to.

Example 1.5. For example, see [O'D05], we can extend the ordered abelian groupZ of integers toZ 2 by adding a second coordinate and using lexico- graphic ordering. Then elements with a nonzero second component could be called inﬁnite since each of them is indeed larger than any copy(n;0) of a usual integern Z. While we are able this way to preserve the group structure and 2 the ordering ofZ, chances are that these inﬁnite numbers won't make us too happy. They can't even be multiplied. 4 Nonstandard Analysis

Example 1.6. We can extend the previous example to allow for multiplication. Note that the previous construction could have been obtained in the following way: Starting withZ and a symbolc we interpretc as being larger than any integer. Then wefill in all the additional stuff which is needed to 2 get addition to work like2c+1. What we end up with isZ+Zc = Z , just as in the previous example. Since we also want to have multiplication, we again fill in all the additional stuff needed for multiplication, for instancec 2 + 1. It won't come as a surprise that we get the polynomial ringZ[c]. But, oh well, true happiness is hard to achieve, and this world of infinite numbers still isn't good enough for us. We can't even talk about2 c. As before we could keep going in the same fashion but the resulting structure will always be too small (even if we kept doing that infinitely often, see [Kos96] where the inherent and provable complexity oftrue infinitesimals is discussed).

At this point we can already feel that there is going to be some trouble with the nonstandard worlds of integers. Namely, they are going to be very large. In fact, every nonstandard model of the integers that obeys the transfer principle is going to be uncountably large. That's why use of the axiom of choice or a slightly weaker axiom will be needed to shed some light.

Example 1.7. Another way to get inﬁnitesimals into play is to consider the ring of dual numbers

R,R["]/(" 2); that is we introduce a new quantity" that squares to0. While such a property of a number might seem strange atfirst, this is actually found in computers when they do numerical calculations withfinite precision. Because of thefinite precision there is a smallest positive number which hence has to square to0. We can then write

(x+") 2 =x 2 + 2x":

Similarly, for any polynomial we getf(x+") =f(x)+:::" and we can deﬁne ::: as the derivative off(x). To handle a broader class of functions, including the trigonometric ones, in this way we can postulate that

exp(") = 1 +":

Then also cos(") = 1 and sin(") = 0, so that

cos(x+") = cos(x)cos(") sin(x)sin(") = cos(x) sin(x)" ¡ ¡ Armin Straub 5 whence the derivative of cos(x) is sin(x). This approach has actually been ¡ used for example in a freshmen calculus class as described in [Hoo07]. It can be adapted to handle higher order derivatives or multiple variables. An appli- cation to elliptic curves can be found in [Bel07].

1.2 The Use of Inﬁnitesimals

There are several reasons why one would wish to have inﬁnitesimal numbers at hand together with a rigorous foundation. We only state a few.

Resurrection of the Good Old Days. With inﬁnitesimal numbers the arguments of mathematicians like Leibniz, Newton or Euler canﬁnally be made rigorous.

Better Understanding of the Standard World. In the words of Kossak [Kos96],we can compute anything we want without irrationals, but we could never understand geometry and calculus without them. In the same spirit nonstandard thinking can illuminate our understanding of the standard world.

Improving the Teaching of Mathematics. Indicated by the fact that mathematicians like Euler used infinitesimals in their reasoning (suc- cessfully despite the lack of afirm foundation), it is often argued that a nonstandard approach to teaching calculus may be easier to understand for students. At any rate, there are excellent books, for instance [Kei00], that demonstrate that calculus can be taught using infinitesimals instead of sequences or"- concepts. As an informal example, continuity ofx x 2 is derived as 7! (x+d) 2 =x 2 + 2dx+d 2 x 2: Enrichment of the Mathematical Environment. Nonstandard objects are worth considering in their own right. We'll for example take a closer look at the hyperintegers Z at the end of this text. For another example consider R R. It forms a model for plane Euclidean geometry. But when we restrict tofinite points in both coordinates we get a plane, called the Dehn plane, that violates the parallel postulate (for instance a line through (0; 1) with infinitesimal slope won't meet thex-axis).

Automatic"- Management. As illustrated in [ Tao07] the use of nonstandard analysis can make arguments more succinct and understand- able. This is in particular true for arguments that make heavy use

of interdependent" i's and i's. 6 Nonstandard Analysis

2 Diving In

2.1 Limits and Ultraﬁlters

This section is heavily based on [Tao07]. Given a nice sequence(x n) we can talk about its limit limx n (nice meaning convergent here). This limit can be interpreted, as Tao does, as the outcome of a voting system. Every natural numbern votes for the candidatex n, and the voting system then chooses limx n to be the winner. The limit has several properties, including: (a) lim is an algebra homomorphism, that is

lima n +b n = lima n + limb n;

limca n =c lima n;

lima n bn = lima n limb n:

(b) lim is bounded as follows

infx n 6 limx n 6 supx n:

(c) lim is non-principal, that is limx n =limy n whenever the sequences(x n) and(y n) diﬀer atﬁnitely many indices only. (d) lim is shift-invariant, that is for all shiftsh N we have 2 limx n+h = limx n:

Remark 2.1. Notice that the shift-invariance of lim actually implies that lim is non-principal. However, boundedness, and the non-principality alone suffice to characterize lim . To see this, take a sequence(x n) that converges tox . 1 For any" wefindh N such that after setting thefirsth values of(x n) tox 2 1 we have

x "< infx n 6 supx n < x +" 1 ¡ 1 whence the only candidate for limx n isx . 1 Can we give a definition of a limit that satisfies the above conditions and that applies to more than just the usually convergent sequences? More specifically, can we define such a limit for merely bounded sequences?

Example 2.2. Consider the sequence(x n) = (0;1;0;1;0;1;:::). It is certainly bounded, and by the algebra homomorphism property of lim , we have limx n = 2 (limx n) . Hence the only candidates for limits are0 and1. On the other hand, using the algebra homomorphism property and the shift-invariance we get the contradiction limx = limx = lim1 x = 1 limx n n+1 ¡ n ¡ n Armin Straub 7

which would imply that limx n = 1/2.

So we see that we have to abandon either the algebra homomorphism property or the shift-invariance of lim . We dismiss shift-invariance. In the voting interpretation this means that we give up the fairness of the election (which means arbitrary choices in the voting system which, as Tao points out, can be taken as aﬁrst hint that the Axiom of Choice might get involved). Suppose we found such an extension of lim to bounded sequences. Let's consider what happens when we apply lim to boolean sequences only, that is to sequences that take only values0 and1. By the algebra homomorphism property (as before) weﬁnd that for a boolean sequence(x n) the only candidates for limx n are0 and1. Such a sequence can be represented by its indicator set A= n N:x = 1 , and we can consider the collectionF of those indicator f 2 n g setsA such that limx n = 1 (those voters that can decide an election by voting in unison). Using the homomorphism property as well as boundedness and non-principality we easily derive the following properties ofF: (a)F is an upper set, that is

A F;B A= B F: 2 ) 2 (b)F is closed underﬁnite intersections,

A;B F= A B F: 2 ) \ 2 (c)F is dichotomous, that is forA N exactly one of the following holds A F A c F: 2 _ 2 (d)F is non-principal meaning that changing a setA byfinitely many elements doesn't affect whetherA F. 2 Thefirst two conditions state thatF is afilter, the third adds that it is in fact an ultrafilter and the last one exclude the principalfilters (which are just all the sets containing a particular index, and which correspond to a dictator in the voting interpretation).

On the other hand, given an ultrafilterF on a setI we can defineF -limx n for a sequence(x ) taking values in 0;1 by setting n f g F -limx = 1 n I:x = 1 F; n () f 2 n g 2 andF -limx n = 0 otherwise. ThenF -lim satisfies the properties discussed above. Whenever we have a propertyp(n), likex n = 1 here, we say thatp(n) holds w.r.t.F if n I:p(n) F . The properties of an ultrafilter translate into: f 2 g2 (a) (Modus-Ponens) Ifp(n) q(n) then ) p(n) w.r.t.F= q(n) w.r.t.F: ) 8 Nonstandard Analysis

(b) The closure ofF underﬁnite intersections implies that

p(n) w.r.t.F;q(n) w.r.t.F= p(n) q(n) w.r.t.F: ) ^ (c) (Law of the Excluded Middle) Exactly one of the following holds,

p(n) w.r.t.F p(n) w.r.t.F: _:

Now, given any bounded sequence(x n) we can deﬁneF -limx n as the unique x such that

x x <" w.r.t.F j n ¡ j for all">0. But wait! We actually obtained an ultraﬁlterF only under the assumption that we found an extension of lim to bounded sequences taking values in 0; f 1 . So do ultraﬁlters exist after all? g

Theorem 2.3. (UFT) Everyﬁlter is contained in an ultraﬁlter.

Proof. Well, this is independent from ZF but it can be proved very straight- forward using Zorn's Lemma (which is equivalent to the Axiom of Choice). Note, however, that the ultraﬁlter theorem is strictly weaker than both.

Tofind a non-principal ultrafilter we can just start with thefilter of cofinite sets. We now present two statements which assert that actually there are incredibly many ultrafilters on an infinite set (on afinite setX there are only the principal ultrafilters and hence only X many). j j

X Theorem 2.4. An inﬁnite setX allows2 2j j many (non-principal) ultraﬁlters.

Proof. More is not possible any way. The hard part is to construct a setS X of2 j j subsets ofX such that any Boolean combination

s s ::: s =/ ; s S s c S 1 \ 2 \ \ n ; i 2 _ i 2 is nonempty. Then any subsetT S gives rise to aﬁlter basis T s c:s S T [f 2 n g

X which in turn define distinct ultrafilters. Of course, there are2 S = 22j j many such subsetsT . Finally, note that are only exactly X many principal ultra- j j filters onX. Armin Straub 9

Corollary 2.5. There are non-isomorphic non-principal ultrafilters on any infinite setX (two ultrafilters being isomorphic if one arises from the other by a bijection onX).

X 2 X Proof. There are no more than2 j j many bijections onX but2 j j many non- principal ultraﬁlters.

2.2 A First Nonstandard Peak

For theﬁeld extension fromQ toR what you do is consider all sequences QN, take the ring of Cauchy sequences, and factor by the maximal ideal a: f lima=0 . Now, what do we get if we start withR, consider the ring of all g sequencesR N, and factor by a maximal ideal? Ok, of course we get aﬁeld but what else can we say. Is it for example going to be ordered again? And what do maximal ideals ofR N look like?

Theorem 2.6. Ultraﬁlters onI are in a1-1 correspondence with maximal ideals ofR I via

F (x ) R I:x = 0 w.r.t.F : 7! f n 2 n g

N WheneverF is a principal ultrafilter onN thenR /F = R and we don't win anything. That's why we need a non-principal ultrafilterF . Those exist by the ultrafilter theorem though we'll never be able to describe one explicitly. N LetF be an ultrafilter, and set R=R /F . Elements of R are equivalence classes of sequences where two sequences(x n) and(y n) are identified whenever xn =y n w.r.t.F . We denote the equivalence class of a sequence(x n) by[x n]. R is embedded in R by identifying a real numberx with the sequence that is constantlyx. We write[x n]6[y n] whenever

xn 6y n w.r.t.F which gives a well-deﬁned binary relation on R.

Corollary 2.7. R is a totally orderedﬁeld.

N Proof. By the preceding theorem, R is the quotient of the ringR by a maximal ideal and hence afield. Alternatively, we can check the axioms for a field, use their validity inR and then use the logic w.r.t.F to deduce them for R. We only prove the existence of inverses. Given a sequence(x n) such that 1 [xn]=/0, we define(y n) byy n =x n¡ ifx n =/0 andy n =0 otherwise. Then for alln

x =/0 = x y = 1: n ) n n 10 Nonstandard Analysis

Thus the logic w.r.t.F implies

[x ] =/0 = x =/0 w.r.t.F n ) n = x y = 1 w.r.t.F ) n n = [x ][y ] = 1 ) n n whence[y n] is the inverse of[x n]. That6 is a total order is again just checking the axioms for a total order. For instance, given sequences(x n) and(y n) we know that for eachn eitherx n < yn, xn =y n, orx n > yn. Thus it follows that we have exactly one of the cases

xn < yn w.r.t.F; x n =y n w.r.t.F; x n > yn w.r.t.F; that is exactly one of[x n]<[y n],[x n] = [yn],[x n]>[y n]. Compatibility with theﬁeld operations follows from the logic w.r.t.F as well. Let for example(x n);(y n);(z n) be sequences. SinceR is an orderedﬁeld we have for eachn that x y z 0 = x z y z : n 6 n ^ n > ) n n 6 n n Thus we employ the logic w.r.t.F to get

[x ] [y ] [z ] 0 = x y w.r.t.F z 0 w.r.t.F n 6 n ^ n > ) n 6 n ^ n > = x y z 0 w.r.t.F ) n 6 n ^ n > = x z y z w.r.t.F ) n n 6 n n = [x ][z ] [y ][z ]: ) n n 6 n n Basically what we observe is that statements inR pass over to statements in R.

Our initial goal was to extend the real numbers so that we have inﬁnitesimals at hand. Did we achieve this goal? Well, consider the sequence(x n) = (1;1/2; 1/3;:::). IfF is a non-principal ultraﬁlter then

( " R )[x ]<" 8 2 + n because onlyfinitely many of thex n are larger than anyfixed". Hence[x n] is an infinitesimal. So we succeeded in extending the real numbers as an orderedfield to the nonstandardfield R which contains infinitesimal and infinite numbers. We could go on from here, for instance lift functionsR R to functions R R, ! ! make explicit what kind of statements overR transfer to statements over R and lots more. This ultrapower construction is described in detail in [LR94]. Here we'll switch to an axiomatic approach now that hopefully some credibility in the possibility of such a construction is won. Armin Straub 11

2.3 An Axiomatic Approach Starting with our usual mathematical world, we want to construct a nonstandard extension of this world, in which it is more fun to fool around. Since we know about the sneaky trap of playing with the set of all sets, we cave in and reduce our usual mathematical world to be based on the set of allreasonable sets (in a sense that most mathematicians stay reasonable most of the time). More details of the axiomatic approach exposited here and how to work with it can be found in [ST99].

Deﬁnition 2.8. (Superstructure) LetS be a set, and deﬁneV 0(S)=S and inductivelyV (S)=V (S) (V (S)). The superstructureV(S) ofS is n+1 n [P n

V(S)= Vn(S): n N [2 We usually chooseS=R. The superstructureV(R) entails lots of the mathematical world that we use to work in (recall that tuples, relations, functions and pretty much anything else can be encoded as a set). Our goal is to extend this superstructure to some strictly larger one, namelyV( R) such that true statements inV(R) transfer to true statements inV( R). R can be con- structed as in the preceding section, and some details need to be considered to define how to embed the whole ofV(R). We have to be a bit careful about what statements we can transfer. These are formulas in which quantification is always bounded, that is quantification appears in the forms( x A) and 8 2 ( x A) only. The forms( x) or( x) are said to be unbounded and are 9 2 8 9 not allowed. Such formulas are called bounded quantifier formulas. We write

(x1; :::; xn) to denote a formula with free variablesx 1; :::; xn.

Deﬁnition 2.9. (Nonstandard Model) LetS be an inﬁnite set. A map

:V(S) V(S );A A ! 7! is a nonstandard extension if it satisﬁes the following three axioms.

Extension Principle.s =s for alls S. 2

Transfer Principle. For every bounded quantiﬁer formula(x 1; :::; xn) and allA ;:::;A V(S), 1 n 2 (A ; :::; A ) is true in (V(S)) 1 n L (A ; :::; A ) is true in (V(S )): () 1 n L

Saturation Principle. The extension is nontrivial, that isS )S.

To get a feeling how to work in the setting of a nonstandard extension we prove some basic properties. 12 Nonstandard Analysis

Proposition 2.10. (Basic Properties of ) Let be a nonstandard extension.

(a) is injective.

(b) preserves the usual boolean operations, i.e. for setsA;B V(S) 2

(A B)=A B ; (A B)=A B ; (A B)=A B : [ [ \ \ n n

(c) a ; :::; a = a ; :::; a for alla V(S). f 1 ng f 1 ng i 2 (d) Iff:A B is a function between setsA;B V(S) thenf is a function ! 2 A B . !

Proof.

(a) LetA =B forA;B V(S). By transferA=B. 2 (b) We prove the claim for unions. The other statements can be handled the same way. LetC=A B. Choosen such thatA;B;C V (S). [ n Then the following is true in (V(S)) L ( x V (S))(x C (x A x B)) 8 2 n 2 , 2 _ 2 and transfer yields

( x V (S))(x C (x A x B )) 8 2 n 2 , 2 _ 2

in (V (S)). Since (also by transfer)A ;B ;C V (S), the latter L n amounts to sayingC =A B . [ (c) LetA= a ; :::; a . Then f 1 ng ( a A)a=a ::: a=a 8 2 1 _ _ n is true. By transfer

( a A )a=a ::: a=a 8 2 1 _ _ n

is true as well which shows that a ; :::; a =A = a ; :::; a . f 1 ng f 1 ng (d) Since we didn't even discuss how to encode a function as a set we won't provide a proof but hope for the reader's belief (or willingness to work it out as an exercise).

2.4 Basic Usage

As for the explicit construction based on ultraﬁlters we have the following.

Theorem 2.11. R is a totally orderedﬁeld which properly extendsR. Armin Straub 13

Proof. We have to check lots of properties of R but all of them follow from the respective property ofR. For instance the commutativity of addition in R formalized as

( x; y R)x+ y=y+ x 8 2 follows from transfer of

( x; y R)x+y=y+ x: 8 2

Since R was build to contain inﬁnitely large numbers, we don't expect it to be Archimedean. But why doesn't this property follow by transfer as well? Because the statement

( x R)( n N)n > x 8 2 9 2 transfers to

( x R)( n N)n > x 8 2 9 2 which just asserts that R ishyper-Archimedean.

Since it shouldn't lead to any confusion we'll just write+ instead of+ , and similarly for other symbols like<, and so on. We introduce the following jj intuitive language.

Definition 2.12. Letx;y R. 2 x is calledfinite if x n for somen N, and infinite otherwise. j j6 2 x is called infinitesimal if 1/x is infinite. x andy are said to be infinitesimally close, denoted byx y, ifx y ¡ is infinitesimal.

We just note that intuitively true statements likefinite+finite=finite are just as easily derived from these definitions. But one question lingers: Do there exist infinite numbers at all, or are we talking in a sophisticated way about empty attributes here?

Theorem 2.13. N N =/ and everyh N N is inﬁnite. ¡ ; 2 ¡

Proof. Leth N N. To prove thath is inﬁnite we have to show that 2 ¡ h > n for alln N. We proveh>n inductively. Clearly,h>1 since 2 1 N. Suppose thath>n. By transfer,h n+1 (consider the formula 2 > ( m N)m > n m n+1 and recall thatn =n). Butn+1 N and hence 8 2 ) > 2 h > n+1. 14 Nonstandard Analysis

We now prove that N N =/ . Assume otherwise that N=N. Since the ¡ ; rationals are dense inR we have the following true statement

r s 1 ( x R)( n N)( r;s;t N) x ¡ < 8 2 8 2 9 2 ¡ t n

which under the assumption N=N transfers to

r s 1 ( x R)( n N)( r;s;t N) x ¡ < : 8 2 8 2 9 2 ¡ t n

Letx R. We thusﬁnd a sequencex Q such that x x <1/n. This 2 k 2 j ¡ nj makes(x n) a Cauchy sequence which hence converges to somex R. Note 1 2 that

x x 6 x x n + x n x 0 asn : j ¡ 1j j ¡ j j ¡ 1j ! !1

In particular, x x <1/n for alln N= N. The transfer of j ¡ 1j 2 ( x;y R)(( n N) x y <1/n) = x=y 8 2 8 2 j ¡ j ) now yields thatx=x R. But then R R which is a contradiction. 1 2 This language is very nice for deﬁning notions as limits, accumulation points, continuity or uniform continuity.

Theorem 2.14. Let(x ) R be a sequence andx R. The following are n 2 equivalent.

limx =x if and only ifx x for allh N N. n h 2 ¡ x is an accumulation point of(x ) if and only ifx x for some n h h N N. 2 ¡

Proof. We only prove thefirst statement. Supposefirst that limx n =x, and leth N N. Then for any" R wefindn such that 2 ¡ 2 + 0 ( n N)n>n = x x < ": 8 2 0 ) j n ¡ j This transfers to

( n N)n>n = x x < ": 8 2 0 ) j n ¡ j

Sinceh>n and" was arbitrary this implies that1/ x x >n for anyn N, 0 j h ¡ j 2 and hencex x as claimed. h Now suppose thatx x for allh N N. Let" R . Then h 2 n 2 +

( n N)( n N)n>n = x x <" 9 0 2 8 2 0 ) j n ¡ j Armin Straub 15 holds, and hence by transfer

( n N)( n N)n>n = x x <" 9 0 2 8 2 0 ) j n ¡ j which proves that limx n =x.

Theorem 2.15. LetD R,x D, andf:D R be a function. The following 0 2 ! are equivalent. f is continuous inx . 0 ( x D )x x = f (x) f(x ). 8 2 0 ) 0

Proof. The proof is along the lines of the previous one. We give it anyway. First, letf be continuous inx . Let" R . Weﬁnd R such that 0 2 + 2 + ( x D) x x <= f(x) f(x ) < ": 8 2 j ¡ 0j ) j ¡ 0 j This transfers to

( x D ) x x <= f (x) f(x ) < ": 8 2 j ¡ 0j ) j ¡ 0 j

Letx D such thatx x . The condition x x < is then trivially satisﬁed 2 0 j ¡ 0j for any choice of". We thus have that f (x) f(x ) <" for all" R , and j ¡ 0 j 2 + hencef (x) f(x ). 0 Now suppose thatf (x) f(x ) for allx D such thatx x . Let" R . Then 0 2 0 2 +

( R )( x D ) x x <= f (x) f(x ) <" 9 2 + 8 2 j ¡ 0j ) j ¡ 0 j holds, and hence by transfer

( R )( x D) x x <= f(x) f(x ) <" 9 2 + 8 2 j ¡ 0j ) j ¡ 0 j which proves thatf is continuous atx 0.

Theorem 2.16. LetD R, andf:D R be a function. The following are ! equivalent. f is uniformly continuous. ( x;y D )x y= f (x) f (y). 8 2 )

Proof. The proof is almost exactly the previous one. Again, we give it anyway. First, letf be uniformly continuous. Let" R . Weﬁnd R such that 2 + 2 + ( x;y D) x y <= f(x) f(y) < ": 8 2 j ¡ j ) j ¡ j 16 Nonstandard Analysis

This transfers to

( x;y D ) x y <= f (x) f (y) < ": 8 2 j ¡ j ) j ¡ j

Letx;y D such thatx y. The condition x y < is then trivially satisﬁed 2 j ¡ j for any choice of". We thus have that f (x) f (y) <" for all" R , and j ¡ j 2 + hencef (x) f (y). Now suppose thatf (x) f (y) for allx;y D such thatx y. Let" R . 2 2 + Then

( R )( x;y D ) x y <= f (x) f (y) <" 9 2 + 8 2 j ¡ j ) j ¡ j holds, and hence by transfer

( R )( x; y D) x y <= f(x) f(y) <" 9 2 + 8 2 j ¡ j ) j ¡ j which proves thatf is uniformly continuous.

All these examples show that using nonstandard language we can deﬁne standard notions not only in an intuitive way but even in a more succinct form that allows to focus on the essentials.

2.5 The Source of Nonstandard Strength

Let's take another look at the hypernatural numbers N. We proved before that N N =/ and everyh N N is infinite. In particular, for infiniteh ¡ ; 2 ¡ alsoh 1 N N is infinite. Therefore, N N has no minimal element! ¡ 2 ¡ ¡ What is going on here? Any subset ofN has a minimal element. Shouldn't this by transfer be true for N as well? To understand this, look what happens when we transfer statements fromR to R. For example

( m A):::( n B)::: 8 2 9 2 transfers to

( m A ):::( n B )::: 8 2 9 2

Objects of the formA are called standard, and we observe that after transfer we talk about elements of standard objectsA only. Those are called internal. Since A = A standard objects are internal. f g f g

Remark 2.17. It's not hard to see that the setV int(S ) of internal objects is

Vint(S ) = Vn(S): n N [2 Armin Straub 17

Objects that are not internal are called external. Vaguely, external objects are those that exist in the nonstandard model but which can not be accessed by transfer. N N is an example of an external object. While this might ¡ seem like an obstruction it actually is source for the strength of working with nonstandard models. What would we win when it behaved exactly the same as our standard world? No, it's great that it doesn't, and essentially what we gain is the possibility to talk about an object being standard, that is of the form A, or not. In ZF we only have the binary relation but in the nonstandard 2 model we also have this unary relation st that expresses whether an object is standard.

Remark 2.18. In fact, nonstandard analysis can be build axiomatically around this unary relation st and some axioms how to work with it. This approach was ﬁrst used by Nelson who introduced his internal set theory IST as an extension of ZF. See for instance [Nel87] where he uses IST to develop probability theory.

2.6 Stronger Nonstandard Models

The saturation property which asserts thatS )S can be reformulated in the following equivalent way, see [LR94]. For any countable family of standard sets

(A n)n I that has theﬁnite intersection property, 2

A =/ : n ; n I \2

Example 2.19. Contrast this with the situation in the standard world. For instance the family(N 1; :::; n ) n N clearly has theﬁnite intersection ¡f g 2 property but still has empty intersection. SetB N 1; :::; n . Observe n , ¡f g that

( n; m N)n>m= B B : 8 2 ) n m

Leth N N. By transfer,B B for anyn N. Hence, 2 ¡ h n 2

B B : h n n N \2 Also by transferB =/ . h ; For a countable family of standard sets that has thefinite intersection property we can do thefirst-entry decomposition to obtain a decreasing sequence of sets. By the same argument as in this example we thenfind that the intersection of the nonstandard sets is nonempty.

Especially in branches like topology or probability theory, we would like to have a stronger version of the above saturation property. 18 Nonstandard Analysis

Deﬁnition 2.20. Let be an inﬁnite cardinal. A nonstandard extension

:V(S) V(S );A A ! 7! is said to be-saturated if the following holds.

-Saturation Principle. For any family of internal sets(A n)n I of car- 2 dinality cardI6 that has theﬁnite intersection property,

A =/ : n ; n I \2

We wish to present an easy example that makes use of a-saturated nonstandard extension. First recall the Riesz' representation theorem in the following guise.

Theorem 2.21. Letf:X R be a continuous linear functional on a Hilbert ! spaceX with inner product( ; ). Then there existsy X such that 2 f(x)=(y;x) for allx X. 2

What if we drop the condition thatf is continuous and that we're working in a Hilbert space? Suppose that we have a-saturated nonstandard model where X . We then have the following extension of the representation theorem, >j j see [LR94].

Theorem 2.22. Letf:X R be a linear functional on a pre-Hilbert spaceX ! with inner product( ; ). Then there existsy X such that 2

f(x)= (y; x) for allx X. 2

Proof. The sets

C y X:f(x)=(y;x) x ,f 2 g wherex X are easily checked to have thefinite intersection property by 2 restricting tofinite dimensions where Riesz' representation theorem trivially applies. Thus wefind

y C ; 2 x x X \2 Armin Straub 19

or equivalentlyy X such that 2

( x X)f (x) = (y; x): 8 2

Sincef (x)=f(x) the claim follows.

2.7 Studying Z in its Own Right The purpose of this section is at least two-fold. Firstly, to demonstrate that nonstandard objects are interesting objects in their own right. And secondly, to (hopefully) wet the readers appetite to play withp-adic numbers. n Letp beaprime. Thep-adic integersZ p are the inverse limit of the ringsZ/p . From a practical point of view they can be regarded as sequences(a n)n N such 2 thata Z/p n anda a (modp n) wheneverm>n. Of course, addition and n 2 m n multiplication are then given pointwisely.

N Remark 2.23. As a set we can also think ofZ p as(Z/p) but then addition and multiplication are no longer just the pointwise ones. At any rate, we see thatZ p is uncountable.

The integersZ canonically embed intoZ . A numberz Z corresponds to the p 2 sequence of its remainders modulop n, that is

n Z Z p; z (zmodp )n N: ! 7! 2

Example 2.24. The number 23 corresponds to (1;3;7;7; 23; 23;:::) inZ 2.

Remark 2.25. The invertible elements inZ p are exactly those sequences not starting with0. For this reason, thefield of fractionsQ p ofZ p consists of the elements that can be written asp ¡ z for somez Z .Q is afield extension 2 p p ofQ which can't be turned into an orderedfield.

Remark 2.26. One importance of consideringQ p is rooted in number theory. Certain equations (eg. quadratic forms) satisfy the Hasse principle, that is they have solutions overQ if and only if they have solutions overR and all theﬁeldsQ p (this is called a local-global principle because a solution inQ is global in the sense thatQ embeds in theQ p andR). Here as well as in other occasions thep-adicﬁelds naturally occur alongsideR.

For everyn N we have the projectionsZ Z/n and hence also the induced 2 ! projections Z Z/n. By the universal property ofZ p as the inverse limit n ! of theZ/p we have unique morphismsZ Z and Z Z which commute ! p ! p with the canonical projections ontoZ/p n. These morphisms are

n Z Z p; z (zmodp )n N; ! 7! 2 20 Nonstandard Analysis and

n Z Z p; z (z modp )n N: ! 7! 2

While we saw that the embedding ofZ intoZ p is injective but deﬁnitely not surjective, the canonical morphism Z Z is no longer injective but surjective. ! p We thus get thatZ p is a quotient of the hyperintegers. Using some analog of the Chinese remainder theorem, we actually have that

^ Z , Zp p Y is a quotient as well, see [Fes06]. The kernel are those elements of Z that are 0 modulo anyp . Equivalently, they are multiples of any natural number, namely

n Z n Z \2 which is a torsion-free divisible abelian group and hence isomorphic toQ () (because it can be viewed as a vector space overQ). In fact,

^ () Z = Z Q for some. We thusﬁnd that, in particular, the endomorphism ring of Z is noncommutative which is a drastic diﬀerence to the standard case where the endomorphisms ofZ are isomorphic toZ again.

3 Conclusions

Given a single non-principal ultrafilter (the existence statement is called the weak ultrafilter theorem) we are able to construct a nonstandard extension as defined by the above axioms. To further construct-saturated nonstandard extensions we can use ultrafilters with special properties which do exist if we assume the axiom of choice. On the other hand, we could ask if the existence of nonstandard extensions really requires ultrafilters. Maybe there is another ultrafilter-free way to construct nonstandard extensions. Well, there is not. Recall our initial discussion of extending lim to bounded sequences. We con- cluded that such an extension would give rise to a non-principal ultrafilter. But given a nonstandard extension and some infiniteh N we can define 2 for a bounded real sequence(x n)

h-limx n , st(xh) Armin Straub 21 where st(x) denotes the unique real numberx R such thatx x . It's easy to 0 2 0 check that the so-defined h-lim is an extension of lim in the above sense. Thus we canonically get a non-principal ultrafilter from any nonstandard extension. On the other hand, if we are willing to give up things like commutativity and invertibility we can do without ultrafilters. Basically, we can replace the non- principal ultrafilter onN in our construction by thefilter of cofinite sets. A few more details can be found in [Con07] and [Tao07].

References

[Bel07] Juliana V. Belding. A weil pairing on thep-torsion of ordinary elliptic curves over the dual numbers ofk. arXiv:math/0703906, 2007. [Con07] Alain Connes. Noncommutative geometry: Infinitesimal variables. Published online at http://noncommutativegeometry.blogspot.com/2007/06/ infinitesimal-variables.html, June 2007. [Fes06] Ivan Fesenko. Several nonstandard remarks. American Mathematical Society Trans- lations, 217(2), 2006. [Hoo07] Matt Hoorn. Non-nonstandard calculus, I. Published online at http:// cornellmath.wordpress.com/2007/08/28/non-nonstandard-calculus-i/, Aug. 2007. [Kei00] H. Jerome Keisler. Elementary Calculus: An Approach Using Infinitesimals. Pub- lished online at https://people.math.wisc.edu/~keisler/calc.html, 2nd edition, 2000. [Kos96] Roman Kossak. What are infinitesimals and why they cannot be seen. The Amer- ican Mathematical Monthly, 103(10):846853, 1996. [LR94] Dieter Landers and Lothar Rogge. Nichtstandard Analysis. Springer, 1994. [Nel87] Edward Nelson. Radically Elementary Probability Theory. Annals of Mathematics Studies. Princeton University Press, 1987. [O'D05] Richard O'Donovan. An additive group containing infinities. Seminarberichte aus dem Fachbereich Mathematik, Band 76, FernUniversität in Hagen, 2005. [Rob67] Abraham Robinson. Nonstandard arithmetic. Bull. Amer. Math. Soc., 73(6):818843, 1967. [ST99] Sergio Salbany and Todor Todorov. Nonstandard analysis in point-set topology. Preprint ESI 666, 1999. [Tao07] Terence Tao. Ultrafilters, nonstandard analysis, and epsilon management. Pub- lished online at http://terrytao.wordpress.com/2007/06/25/ultrafilters- nonstandard-analysis-and-epsilon-management/, June 2007.