# Ultraproducts and Applications I

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1 Ultraproducts and Applications I Brent Cody Virginia Commonwealth University September 2, 2013

2 Outline Background of the Hyperreals Filters and Ultrafilters Construction of the Hyperreals The Transfer Principal Sample Theorems

3 History Liebniz proved that d(xy) = xdx + ydy as follows. d(xy) = (x + dx)(x + dy) xy = xy + xdy + ydx + dxdy xy = xdy + ydx + dxdy = xdy + ydx Can this proof be made rigorous? Without talking about limits (or ɛ and δ)?

4 History Around 1670 Newton and Leibniz used infinitesimal quantities, but the foundations of calculus were controversial until the mid 1800 s. ɛ-δ arguments were in use in the early 1800 s. Dedekind, Cantor and others: gave an explicit construction of the reals from the rationals around The foundations of calculus were solid by the 1870 s. Robinson showed, around 1960, that...the concepts and methods of mathematical logic are capable of providing a suitable framework for the development of the Differential and Integral Calculus by means of infinitely small and infinitely large quantities.

5 The Idea of Hyperreals What is an infintesimal? Idea: A sequence of real numbers a = a 0, a 1,... satisfying lim n a n = 0 will be regarded as an infinitesimal, while if lim n a n = then the sequence a will be an infinitely large quantity. E.g. 1, 1 2, 1 3,... will be twice 1 2, 1 4, 1 6,....

6 The Idea of Hyperreals R R is a new number system containing infinitesimal and infinite elements. Q R = equivalence classes of Cauchy sequences of rationals R R = equivalence classes of sequences of reals. But these equivalence relation are very different.

7 The Idea of Hyperreals In building R, sequences of reals a = a 0, a 1,... and b = b0, b 1,... will be identified if they agree at a Large number of places. In other words, a b if {n N : a n = b n } is large. What properties should largeness satisfy?

8 Filters Largeness (These properties can be derived) N should be large and should not be large. If A and B are large subsets of N then A B should be large. If A N is large and A B N then B should be large. Definition If X is a set then F is a filter on X if F is a collection of subsets of X such that (i) / F and X F, (ii) if A, B F then A B F, and (iii) if A F and A B X then B F. Idea: A X is large if and only if A F

9 Filters Examples: The cofinite filter on N F co = {A N : N \ A is finite} {2n : n N} / F co {2, 4, 6, 7, 8, 9, 10, 11, 12, 13...} F co If x X then the following is called the principal filter generated by x. F x = F {x} = {Y X : x Y }

10 Ultrafilters A filter U on X is an ultrafilter if (iv) for every A X either A F or X \ A U. Examples: If n N then the principal filter generated by n, namely F n = {Y N : n Y } is an ultrafilter but not a very useful one. Question: Are there any nonprincipal ultrafilters on N?

11 Theorem If X is an infinite set then there is a nonprincipal ultrafilter U on X. Zorn s Lemma tells us that maximal filters exist on X, and maximal filters are ultrafilters. The ultrafilter U given by Zorn s Lemma can be assumed to be nonprinciple because we can assume F co (X ) = {A X : X \ A is finite} U.

12 Definition of Hyperreals We want to expand the domain of the structure The new domain will be R = R, 0, 1, +,,. R N = set of infinite sequences of reals For sequences a = a 0, a 1,... and b = a 0, a 1,... define 0 = 0, 0,... a b = a 0 a 1, b 0 b 1,... 1 = 1, 1,... a b iff n N (a n b n ) a b = a 0 + b 0, a 1 + b 1,...

13 This new structure is not a field R N, 0, 1,,, because 0, 1, 0, 1, 0 has no inverse and it is not linearly ordered But this can be fixed... because 0, 1, 0, 1, is not -comparable to 1, 0, 1, 0,.

14 From here on, fix U to be a nonprinciple ultrafilter on N with U F co. Define an equivalence relation on R N by a 0, a 1,... b 0, b 1,... a n equals b n for a large set of n The equivalence class {n N : a n = b n } U. [ a] U = [ a] = [ a n ] = { b R N : b a} is just the set of all sequences of reals that are almost equal to a. So for example, 3, 7, 2, 1, 1, 1, 1,... is equivalent to 1, 1, 1, 1, 1, 1, 1,...

15 We now define a structure with domain the set of equivalence classes: R N /U. Adding equivalence classes: [ a] + [ b] := [ a b] = [ a n + b n : n N ] (Well-defined) If c, d R N and we have {n : a n = c n } U and {n : d n = b n } U, then {n : a n + b n } = {n : a n = c n } {n : b n = d n } U. Multiplying equivalence classes: [ a] [ b] = [ a b] = [ a n b n : n N ] (well defined) Ordering equivalence classes: [ a] [ b] : {n N : a n b n } U [ a] < [ b] : {n N : a n < b n } U This defines a new structure: R N /U, [ 0], [ 1], +,,

16 Ordered Field Theorem R N /U, [ 0], [ 1], +,, is a linearly ordered field. (Additive Inverses) [ a n : n N ] = [ a n : n N ] (Multiplicative Inverses) If [ a] = [ a n : n N ] [ 0] then {n N : a n 0} U and we can define b by { (a n ) 1 if a n 0 b n := 0 otherwise Then [ b] [ a] = [ 1]. (Linearity) this is where we need the ultra part of ultrafilter N = {n : a n < b n } {n : a n = b n } {n : b n < a n }

17 R This structure is called the hyperreals and is denoted by R = R N /U, [ 0], [ 1], +,, In Summary, elements of R are equivalence classes of sequences of reals defined using U, +,, are all defined using U, R is a linearly ordered field, and there is a copy of R sitting nicely inside of R.

18 Fitting R inside R Theorem There is a natural order-preserving field isomorphism from R into R. f :R R r r := [ r, r, r,... ] R will be identified with the copy of R sitting inside R, in the same way that Q is often identified with the copy of Q sitting inside R. In other words, if r R we sometimes mean r, r, r,... when we write r.

19 Infinitesimals and Infinities R contains infinitesimal and infinite elements! Infintesimals Let a = 1, 1 2, 1 3,.... Then [ a] R. If r R >0 we have {n N : a n < r} is cofinite and is thus in U F co. So we have [ a] < r. In other words, [ a] is less than every positive real number. Infinities Let b = 0, 1, 2, 3,.... Then [ b] R. If r R we have {n N : b n > r} is cofinite and is thus in U F co. So we have r < [ b]. In other words, [ b] is greater than every positive real number.

20 Definitions A hyperreal a R is... limited if there are real numbers r, s R such that r < a < s. infinitesimal if for all positive reals r R >0 one has r < a < r. unlimited if either for all reals r R one has r < a or for all reals r R one has a < r.

21 Definition Two hyperreals a, b R are said to be infinitely close to each other if their difference a b is infinitesimal. The halo of a hyperreal a R is the set hal(a) = {b R : a and b are infintesimaly close}

22 The -operation Extending sets If A R then A denotes the collection of hyperreals that are represented by a sequence a whose components are almost always in A i.e., {n N : a n A} U. N N contains unlimited (infinite) elements. [Draw pictures of N and R.]

23 The -operation Extending functions If f : R R is a function then f : R R denotes the function defined by f ([ a]) := [ f (a n ) : n N ]. We have f R = f.

24 The -operation Extending sequences If f : N R is a sequence then f : N R denotes the hypersequence defined as follows. If [ a] N what should f ([ a]) be? Since [ a] N we have {n N : a n N} U. Define b R N by b n := { f (a n ) if a n N 0 otherwise Then {n N : b n = f (a n )} U. Define f ([ a]) = [ b] (well-defined)

25 The -operation Example: The statement x m(x < m and m N) is true about R but false about R (b/c there are hyperreals larger than every natural number). We can apply this -operation to statements. E.g. ( x R m N(x < m)) = x R m N(x < m) Fix a natural number n N. Consider the statement: ( x N(n x n + 1 implies x = n or x = n + 1)) = x N( n x (n + 1) implies x = n or x = (n + 1))

26 The Transfer Principle The -operation preserves the truth of first-order statements. Theorem (The Transfer Principle) Suppose ϕ is a first-order statement in the language of R. Then ϕ is true about R if and only if ϕ is true about R. The statements asserting that R is a linearly ordered field ( x, y R(x + y = y + x), etc.) are true about R and so by the theorem R is a linearly ordered field.

27 Los Theorem Theorem ( Los Theorem) For any first-order formula ϕ(x 1,..., x k ) in the language of R and any r 1,..., r k R N, the sentence ϕ([r 1 ],..., [r k ]) is true if and only if ϕ(r 1 n,..., r k n ) is true for almost all n N. more about this in the next lecture.

28 Sample Theorems Theorem A real-valued sequence s n : n N converges to L R if and only if s n is infinitely close to L for all unlimited n. Recall that a sequence of reals s n : n N can be extended to a hyper sequence s n : n N = s n : n N of hyperreals s n R. Convergence to L amounts to the requirement that the extended tail of the sequence is contained in the halo of L. The role of the standard tails is taken over by the extended tail and the standard open neighborhoods (L ɛ, L + ɛ) are replaced by the infinitesimal neighborhood hal(l).

29 Sample Theorems Recall that for a, b R, we say a is infinitely close to b if and only if a b is infinitesimal. Theorem A function f : R R is continuous at the real point c R if and only if f (x) is infinitely close to f (c) for all x R such that x is infinitely close to c, i.e. if and only if f (hal(c)) hal(f (c)).

30 Sample Theorems Every hyperreal a R has a shadow in R, denoted by sh(r), defined to be the unique real that is infinitesimally close to a. Theorem (Intermediate Value Theorem) If a function f : R R is continuous on [a, b] where a, b R, then for every real d with f (a) < d < f (b) there is a real c (a, b) such that f (c) = d. Idea of proof using infintesimals: Partition [a, b] into intervals of equal infinitesimal width. Find an interval whose end points have f -values on either side of d. Then c will be the common shadow of these end-points.

31 Next Time Ultraproducts! More about Los Theorem. Robinson s Theorem. Let ϕ be a first-order statement in the language of fields (i.e., referring only to 0, 1, + and ). Then ϕ is true about C if and only if there are arbitrarily large primes p such that the statement ϕ is true of some algebraically closed field of characteristic p. Ax-Grothendieck Theorem. Every injective polynomial map f : C n C n is bijective.

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