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Introduction to Nonstandard Analysis by Bashir Abdel-Fattah [email protected] Abstract The basic idea of nonstandard analysis is to extend the reals to a field that includes "infinite” and "infinitesimal” elements in order to simplify proofs and con- cepts by replacing limits and epsilon-delta proofs by expressions involving infinites- imals. This paper discusses the construction and properties of the hyperreals and some basic results results in differential calculus before moving on to logic and cul- minating in theL´o^sTheorem/transfer principle. The contents are largely based on the papers (Davis, 2009) and (Rayo, 2015), with lesser contributions from (Fletcher et al., 2017), (Claassens, 2016), (Marker, 2010), (Keef and Guichard, 2015), and (Murnaghan, 2015). Also note that this paper assumes little background knowledge outside of analysis (the first section does begin with a brief high-level overview in terms of abstract algebra, but if the reader is unfamiliar with that material they can safely proceed knowing that everything else they need will be introduced throughout the paper). Contents 1 Basics of Nonstandard Analysis3 1.1 Constructing Infinitesimals........................3 1.1.1 Definitions and Properties of Filters...............3 1.1.2 Ultrafilter Construction of the Hyperreals...........6 1.1.3 Properties of the Hyperreals...................9 1.1.4 Enriching Sets and Functions to the Hyperreal Setting.... 12 1.2 Basic Analysis using the Hyperreals................... 13 2 The Transfer Principle 16 2.1 Formal Language and First-Order Logic................ 16 2.2 Basics of Transfer............................. 19 2.2.1 Ultraproducts and Ultrapowers................. 19 2.2.2L´o^s'Theorem ........................... 21 References 25 2 1 Basics of Nonstandard Analysis 1.1 Constructing Infinitesimals Although there are several different schemes of varying degrees of sophistication and utility in rigorously introducing infinite and infinitesimal elements to analysis, in this paper we will focus on the ultrafilter construction developed by Abraham Robinson in the 1960s. Robinson's method, being both one of the oldest nonstandard schemes and relatively straightforward, is well developed in the mathematical literature, and the basic idea of the method is as follows: for any set X and ring R, the set RX of all function f : X ! R is a ring under pointwise addition and multiplication (f + g)(x) = f(x) + g(x)(f · g)(x) = f(x) · g(x) However, even if R is an integral domain or a field, RX won't be an integral domain in general due to the presence of zero divisors. For example, let X be any set and let R be any ring. Then, for any subset S ⊆ X, let χS(x) denote the characteristic function ( 1R if x 2 S χS(x) = 0R if x 62 S Then, letting A and B are two disjoint nonempty subsets of X, neither χA(x) nor χB(x) are uniformly zero, yet (χA · χB)(x) = χA(x) · χB(x) = χA\B(x) = χ;(x) is uniformly zero. In the case of the hyperreals ∗R, our aim is the quotient the ring RN of real-valued sequences by a maximal ideal MAX so that RN=MAX is a field. In a heuristic sense, this amounts to finding an equivalence relation =MAX that declares two sequences hani and hbni equal if the set fn 2 N : an = bng is "too large", so that if hani · hbni = hanbni =MAX h0i, then at least one of hani and hbni contained "enough" zeros to be equivalent to zero to begin with. This scheme of eliminating zero divisors is accomplished with the notion of filters. 1.1.1 Definitions and Properties of Filters We begin with a definition of filters in terms of their properties. For many of the properties below, there are several equivalent definitions, many of which are more general or more sophisticated than those stated here, but we shall proceed with following in the interests of simplicity. Definition 1.1.1. A filter on a set X is a subset F of the power set P(X) satisfying the following properties: 3 1. Proper Filter: ; 62 F 2. Finite Intersection Property: If A; B 2 F, then A \ B 2 F 3. Superset Property: If A 2 F and B ⊃ A, then B 2 F Additionally, F is said to be a ultrafilter if it also satisfies: 4. Maximality: For any A ⊆ X, either A 2 F or X n A 2 F F is further said to be a free ultrafilter if it satisfies: 5. Freeness: F contains no finite subsets of X Note that ultrafilters actually obey a stronger version of the maximality property (of which the above definition is a special case), which we prove below in order to familiarize the reader with the style of reasoning associated with filters: Lemma 1.1.1. Let U be an ultrafilter on a set X, and let fA1;A2;:::;Ang be a finite Sn collection of disjoint subsets of S such that i=1 Ai = X. Then there is exactly one set Aj 2 fA1;A2;:::;Ang such that Aj 2 U. Proof. First, suppose that U contains none of the sets A1;A2;:::An. Then, by the maximality property of ultrafilters, (X n A1); (X n A2);:::; (X n An) 2 U, so by the finite intersection property n n \ [ (X n Am) = X n ( Am) = X n X = ; m=1 m=1 is an element of U. This is a contradiction since U must be a proper filter, so we must have that at least one of the sets A1;A2;:::An is in U. Now suppose that U contains more than one of the sets A1;A2;:::;An. Let Ai and Aj be two distinct sets such that Ai;Aj 2 U. Then, by the intersection property, we have Ai \ Aj = ; 2 U This is also a contradiction, so we must have that U contains at most one of the sets A1;A2;:::;An. Therefore U contains exactly one of the sets A1;A2;:::;An. Examples of filters on a set X include: 1. The trivial filter F = X. 2. The principle filter FA, which is defined in terms of any set A ⊂ X as FA = fY ⊂ X : Y ⊃ Ag In the case that A = fag contains only a single element, then the principle filter Fa is in fact an ultrafilter. 3. The cofinite/Fr´echet filter F co = fY ⊂ X : X n Y is finiteg. Fr´echet filters are non-principal. 4 All of the above examples are relatively simple and uninteresting for our pur- poses, and it turns out that every free ultrafilter is non-principle, and in fact cannot be explicitly constructed. One might then doubt whether any collection of sets that obeys all five of the given properties even exists, but we can prove the existence of free ultrafilters on any infinite set using Zorn's Lemma. We will proceed to that proof in short order, but first we need the following helpful lemma. Lemma 1.1.2. Let F be a filter on a set X, and let A ⊂ X be a set such that A 62 F and (X n A) 62 F. Then F can be extended to a filter F 0 on X containing both F and A. 0 Proof. To begin with, assume F 6= ; (if F = ;, then we can just take F = FA, the principle filter generated by the set A, and the result follows trivially), and consider the collection F 0 = fY 0 ⊂ X : there exists some Y 2 F such that Y 0 ⊃ (Y \ A)g We will now verify that F 0 satisfies each of the properties of a filter in order. 1. Proper Filter: First, note that X 2 F by the superset property (given any Y 2 F, then X ⊃ Y , so X 2 F). This means that A 6= ;, because in the case that A = ; we have (X n A) = X 2 F, which contradicts the hypotheses of the lemma. Furthermore, for any Y 2 F we have Y \ A 6= ;, because if Y \ A = ; then (X n A) ⊃ Y , which implies that (X n A) 2 F by the superset property of F, a contradiction. Therefore, given any Y 0 2 F 0, then there is some Y 2 F such that Y 0 ⊃ (Y \ A) 6= ;, thus Y 0 6= ;. 0 0 0 2. Finite Intersection Property: Suppose Y1 ;Y2 2 F . Then there exist some 0 0 sets Y1;Y2 2 F such that Y1 ⊃ (Y1 \ A) and Y2 ⊃ (Y2 \ A). Since Y1 \ Y2 2 F 0 0 by the finite intersection property of F, then Y1 \ Y2 ⊃ (Y1 \ Y2) \ A is an element of F 0 by definition. 0 0 3. Superset Property: This follows fairly trivially. Suppose Y1 2 F and that 0 0 0 Y2 ⊃ Y1 . Since there exists some Y1 2 F such that Y1 ⊃ (Y1 \ A), in which 0 0 0 case Y2 ⊃ (Y1 \ A) as well, so Y2 2 F by definition. Therefore F 0 is a filter, and we can readily verify that A 2 F 0 and F ⊂ F 0: for any 0 Y 2 F we have that Y; A ⊃ (Y \ A), hence Y; A 2 F by definition. Note that the above lemma also justifies the choice of terminology for that max- imality property: any set that does not obey the maximality property of ultrafilters is not maximal in the sense that there is a strictly larger filter containing it. We will prove in the following theorem that a maximal filter (in the sense that there is no strictly larger filter containing it) obeys the maximality property of ultrafilters, and thus the two notions are equivalent.
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