An Introduction to Nonstandard Analysis Jeffrey Zhang Directed Reading Program Fall 2018 December 5, 2018 Motivation • Developing/Understanding Differential and Integral Calculus using infinitely large and small numbers • Provide easier and more intuitive proofs of results in analysis Filters Definition Let I be a nonempty set. A filter on I is a nonempty collection F ⊆ P(I ) of subsets of I such that: • If A; B 2 F , then A \ B 2 F . • If A 2 F and A ⊆ B ⊆ I , then B 2 F . F is proper if ; 2= F . Definition An ultrafilter is a proper filter such that for any A ⊆ I , either A 2 F or Ac 2 F . F i = fA ⊆ I : i 2 Ag is called the principal ultrafilter generated by i. Filters Theorem Any infinite set has a nonprincipal ultrafilter on it. Pf: Zorn's Lemma/Axiom of Choice. The Hyperreals Let RN be the set of all real sequences on N, and let F be a fixed nonprincipal ultrafilter on N. Define an (equivalence) relation on RN as follows: hrni ≡ hsni iff fn 2 N : rn = sng 2 F . One can check that this is indeed an equivalence relation. We denote the equivalence class of a sequence r 2 RN under ≡ by [r]. Then ∗ R = f[r]: r 2 RNg: Also, we define [r] + [s] = [hrn + sni] [r] ∗ [s] = [hrn ∗ sni] The Hyperreals We say [r] = [s] iff fn 2 N : rn = sng 2 F . < is defined similarly. ∗ ∗ A subset A of R can be enlarged to a subset A of R, where ∗ [r] 2 A () fn 2 N : rn 2 Ag 2 F : ∗ ∗ ∗ Likewise, a function f : R ! R can be extended to f : R ! R, where ∗ f ([r]) := [hf (r1); f (r2); :::i] The Hyperreals A hyperreal b is called: • limited iff jbj < n for some n 2 N. • unlimited iff jbj > n for all n 2 N. 1 • infinitesimal iff jbj < n for all n 2 N. 1 • appreciable iff n < jbj < n for some n 2 N. Transfer Principle ∗ Statement: A defined LR sentence φ is true iff φ is true. Examples: 8x; y 2 R; x < y ) 9q 2 Q(x < q < y): gets transferred to ∗ ∗ 8x; y 2 R; x < y ) 9q 2 Q(x < q < y): 8x; y 2 R; sin(x + y) = sin(x) cos(y) + cos(x) sin(y) gets transferred to ∗ ∗ ∗ ∗ ∗ ∗ 8x; y 2 R; sin(x + y) = sin(x) cos(y) + cos(x) sin(y) Shadows and Halos We say a hyperreal b is infinitely close to hyperreal c if b − c is infinitesimal and denote this by b ' c. One can show that ' is an equivalence relation. We define ∗ hal(b) = fc 2 R : b ' cg: Theorem Every limited hyperreal b is infinitely close to exactly one real number, called the shadow of b, denoted by sh(b). Convergence Note that a real-valued sequence is a function from N ! R, so it ∗ ∗ extends to a hypersequence mapping N ! R. Theorem A real valued sequence hsni converges to L 2 R iff sn ' L for all unlimited n. Theorem A real valued sequence hsni is Cauchy in R iff for all m,n unlimited hypernaturals, sm ' sn. Using these concepts, we can prove that a real-valued sequence s convergent in R ) s is Cauchy. Convergence Pf: Suppose hsni converges in R. Then by the first theorem, sn ' L for all unlimited n. So for all l; m unlimited hypernaturals, sl ' L ' sm ) sl ' sm because ' an equivalence relation. Then by the second theorem, hsni is Cauchy. Continuity Theorem ∗ ∗ ∗ f is continuous at c 2 R iff f (x) ' f (c) for all x 2 R such that x ' c. Example: f (c) = c2. Let c be real and x ' c. Then x = c + for some infinitesimal , and f (x) − f (c) = x2 − c2 = (c + )2 − c2 = c2 + 2c + 2 − c2 = 2c + 2 which is infinitesimal because c is a real number and so it is limited. Thus, c2 is continuous. Continuity Another Application: Theorem Let f be a real function defined on some open neighborhood of ∗ c 2 R, and let f be constant on hal(c). Then f is constant on some open interval (c − , c + ) ⊆ R. Pf: Note that for some positive infinitesimal d, we have the ∗ ∗ ∗ statement 8x 2 R such that (jx − cj) < d, f (x) = f (c) = L for ∗ + ∗ some L. This implies that 9y 2 R , 8x 2 R such that ∗ ∗ ∗ (jx − cj) < y, f (x) = f (c) = L 2 R. By transfer, we have the + sentence 9y 2 R , 8x 2 R such that (jx − cj) < y, f (x) = f (c) = L 2 R. Thus, f is constant on the interval (c − y; c + y) ⊆ R. Differentiation Theorem If f is defined at x 2 R, then L 2 R is the derivative of f at x iff for every nonzero infinitesimal , ∗f (x + ) is defined and ∗f (x + ) − ∗f (x) ' L. Example: Consider the real-valued function sin(x), where x 2 R. sin(x + ) − sin(x) Now consider for some an infinitesimal. Then by sum of sines, we get sin(x + ) − sin(x) sin(x) cos() + cos(x) sin() − sin(x) = Differentiation cos(x) is continuous, so cos() ' cos(0) = 1 and so sin(x) cos() ' sin(x). Thus, sin(x) cos() + cos(x) sin() − sin(x) cos(x) sin() ' Also, sin(x) is continuous, so sin() ' sin(0) = 0, so sin() ' and cos(x) sin() ' cos(x): By the theorem, this implies that the derivative of sin(x) at x 2 R is cos(x). Overview • The Transfer Principle is key. • Nonstandard Analysis makes analysis easier!.