CH04

What properties are preserved in passing from \(\mathbb{R}\) to \({}^*\mathbb{R}\)? We have seen a number of examples, and will now consider some more in order to illustrate the powerful logical transfer principle that underlies them. To formulate this principle we will need to develop a precise language in which to describe transferable properties. Ultimately this will allow us to abandon the ultrapower description of \({}^*\mathbb{R}\) and ultrafilter calculations, in the same way that the Dedekind completeness principle allows us to abandon the view of real numbers as cuts or equivalence classes of Cauchy sequence of rationals.

Later it will be seen that the strength of nonstandard analysis lies in the ability to transfer properties back from \({}^*\mathbb{R}\) to \(\mathbb{R}\), providing a new technique for exploring real analysis.

4.1 Transforming Statements

1. The Eudoxus-Archimedes Principle. The statement \[ \forall x\; \exists m(x < m\; and\; m \in \mathbb{N}) \] is true when the variable x ranges over \(\mathbb{R}\), but is no longer true when \(x\) ranges over \({}^*\mathbb{R}\) (e.g., let \(x = [(1, 2, 3, ... )]\)). But if \(\mathbb{N}\) is replaced by its "*-transform" \({}^*\mathbb{N}\), the result is the statement \[ \forall x\; \exists m(x < m\; and\; m \in {}^*\mathbb{N}) \] which is true when \(x\) ranges over all of \({}^*\mathbb{R}\). This example shows that in order to determine the truth value of a sentence, we need to explain what values a quantified variable is allowed to take. We can achieve this by using bounded quantifiers, a notational device that displays the range of quantification explicitly. Thus the first sentence can be conveniently written as \[ \forall x \in \mathbb{R}\; \exists m \in \mathbb{N} (x \lt m) \] which is simply true. Its *-transform \[ \forall x \in {}^*\mathbb{R}\; \exists m \in {}^*\mathbb{N} (x \lt m) \] is also true. On the other hand, \[ \forall x \in {}^*\mathbb{R}\;\exists m \in \mathbb{N}(x < m) \] is false.

2. Density of the Rationals. This is expressed by the true statement \[ \forall x,y \in \mathbb{R}(x < y\; implies\; \exists q \in {}^*\mathbb{Q}(x < q < y)) \] The *-transform \[ \forall x, y \in {}^*\mathbb{R}(x < y\; implies\; \exists q \in {}^*\mathbb{Q}(x < q < y)) \] is also true.

3. Finiteness. Let \(A = \{r_1, ... , r_k\}\) be a finite subset of \mathbb{R}\) Then the statement \[ \forall x \in A (x = r_1\; or\; x = r_2\; or \cdots or x = r_k) \] is true, and so is its *-transform \[ \forall x \in {}^*\mathbb{A} (x = {}^*r_1\; or\; x = {}^*r_2\; or \cdots or\; x = {}^*r_k) \] Since we identify \(r_i\) with \({}^*r_i\) in regarding \(\mathbb{R}\) as a subset of \({}^*\mathbb{R}\), this implies that \({}^*A = A\). Hence finite sets of standard numbers have no nonstandard elements (Ex. 3.10(1)).
Question: why does this argument not work for infinite sets (Theorem 3.9.1)?

4. Finitary Set Operations. If \(A, B \subseteq \mathbb{R}\), then the statement \[ \forall x\in\mathbb{R}(x \in A\cup B　iff　x \in A　or　x \in B) \] transforms to the true statement \[ \forall x \in {}^*\mathbb{R}(x \in {}^*(A \cup B)　iff　x \in {}^*A　or　x \in {}^*B) \] which shows that *(AnU B) =*AU *B. Similarly for the other results of Exercise 3.10(3). Question: why does the argument not work for unions of infinitely many sets (3.10(4))?

5. Discreteness of N. If n EN, then the statement Vx EN (nn::; x::; n + 1 implies x =nor x = n + 1) transforms to Vx E *N (*n::; x ::; *(n + 1) implies x =n*nor x = *(n + 1)), which again is true. Since n = *n and likewise *(n + 1) = n + 1, this shows that there are no nonstandard members of *N occurring between any standard natural numbers. Also, there are no members of *N smaller than 1, i.e., Vx E *N (x 2:: 1); hence any member of *N -N must be greater than all members of N, and so is unlimited, i.e., infinitely large (Section 3.8).

6. Unbounded Sets of Reals. If we assume that there is an unlimited N E *N, then we can deduce the Eudoxus-Archimedes principle in the following way. If r is any real number, then r < N, since N is unlimited, and so the statement 3n E *N (r < n) is true. This is the *-transform of the statement 3n E N(r < n), and as we shall see, a statement must be true if its *-transform is. This shows that there is a positive integer greater than r. More generally, this argument can be used to show that if the enlargement *A of a set of reals has an unlimited member, then A itself must be unbounded in JR. in the sense that for any real r there is a member of A that is greater than r. In brief: if A has an unlimited nonstandard member, then it has arbitrarily large standard members.

It appears from these examples that the *-transform of a statement arises by attaching the "*" prefix to symbols that name particular entities, but not attaching it to variable symbols. The precise definition of *-transform will be laid out in Section 4.4.

4.2 Relational Structures

The examples just given used a semiformal logical symbolism to express statements that were asserted to be true or false of the structures JR and *R This symbolism will now be explicitly described.

A relational structure is a system of the form \[ S = (S, Rels, Funs), \] where S is a nonempty set, RelS is a collection of finitary relations on S, and Funs is a collection of finitary functions on S (possibly including partial functions). For instance, associated with any set Sis the full structure \[ (S, Rels, Funs) \] based on S, where Rels consists of all the finitary relations on S, and Funs consists of all the finitary functions on S. Since sets are unary relations, a full structure includes all subsets of S in Rels.

The full structure based on JR will be denoted by 9t. Associated with it is the structure \[ *9t = (\({}^*\mathbb{R}\), {*Po: P E RelJR}, {*f: f E FunJR}). \] Thus *9t consists of the extensions * P and * f of all relations and functions on JR, as defined in Sections 3.9, 3.11, and 3.14. *9t is not, however, a full structure, since there are relations on \({}^*\mathbb{R}\) that are not of the form * P for any P E Rel!Jl.

Exercise 4.2.1
Show that none of the sets N, Z, Q, JR, and indeed no infinite subset of JR, can belong to Rel*!Yl.·

4.3 The Language of a Relational Structure

Associated with each relational structure S is a language .Cs based on the following alphabet: • Logical Connectives: A. and V or ...., not --+ implies +--t if and only if • Quantifier Symbols: 't/ for all 3 there exists 4.3 The Language of a Relational Structure • Parentheses: (, ) , [, ] • Variables: A countable collection of symbols, for which we use letters '( like x, y, z, x1, x,etc. Terms of Cg These are strings of symbols defined inductively by the following rules: • Each variable is an .Cg-term. • Each elementns of Sis an .Cg-term, called a constant. • Ifnf E Fung is an m-ary function, and 71, ... , 7m are .Cg-terms, then /(71, ... , 7m) is an .Cg-term. We will adopt the customary conventions of notation that depart from this formal definition. For instance, we continue to use the usual "infix" notation for binary operations, writing 71+72or 71·72for f( 7t, 72) when f is addition or multiplication, etc. We will also retain such standard notations as 1/x, 􀄧' x2, JxJ, ex, etc. What Does a Term Name ? A closed term is one that has no variables and therefore is made up of constants and function symbols. Such a term is intended to name a particular element of the structure S. But there are many opportunities in mathematics to write down symbolic expressions that have no meaning because the element they purport to name does not exist, as in tan(7r'/2). (In ordinary language there is the similar phenomenon of syntactically wellformed expressions that do not denote anything, such as Chomsky's famous "green ideas"n.) A closed term is undefined if it does not name anything. Here are the rules that determine when, and what, a closed term names: • The constant s names itself. • If 71, ... , 7m name the elements s1, ... , sm, respectively, and thenmtuple (st, ... , sm) is in the domain of J, then f(7t, ... , 7m) names the element f(s1, ... , sm)· • f ( 71, ... , 7 m) is undefined if one of 71, ... , 7 mis undefined, or if they are all defined but name an m-tuple that is not in the domain of f.

Atomic Formulae of \(mathcal{£}_{s}\)

These are strings of the form P(7t, ... ,7k) where P E Rels is k-ary, and the 7i are .Cs-terms. Such strings assert basic relationships between elements of S and serve as the building blocks for more complex expressions. We also use conventional notation for atomic formulae where appropriate. For binary relations (k = 2) there is the usual infix notation: P(71, 72) is written 71 = 72 when P is the identity relation { (a, b) E S x S : a = b}, and as 71 < 72 when P = {(a,b)(: a< b}. Similarly for 7t > 72, 7t(:::; 72, 7t 2::(72. When k = 1 we have unary, or monadic, atomic formulae of the form P(7), with P being a subset of S. Such a formula expresses membership of P and so will usually be written in the form 7E P. Formulae • Each atomic .Cs-formula is an .Cs-formula. • If cp and '1/J are .Cg-formulae, then so are cp 1\ '1/J, cp V '1/J, •cp, cp ---+ '1/J, cp 􀊍 '1/J. • If cp is an .Cs-formula, x is any variable symbol, and P E Rels ts unary, i.e., P is a subset of S, then (Vx E P) cp, (::3x E P) cp are .Cs-formulae. Here Pis the bound of the quantifier in question. A formula is said to be defined if and only if all of its closed terms are defined. Parentheses will be inserted or deleted in formulae where convenient to aid legibility. Various abbreviations and informalities will be used, such as writing xsysz for the formula (x :::; y)1\ (y :::; z), or collapsing a string of similar quantifiers with the same bound like (Vx E P) (Vy E P) (Vz E P) to the form (Vx, y, z E P). 4.3 The Language of a Relational Structure Sentences An occurrence of the variable x within a formula '1/J is called bound if it is located within a formula of the form (Vx E P)

y), the first occurrence of x is free, while the others are bound, and the only occurrence of y is free. If a formula contains free variables, then it has no particular meaning until we assign some values to those free variables. Thus the above formula makes a true assertion if x = y = 0, but if x = 2, then it cannot be true whatever the value of y is. A sentence is a formula in which all variables are bound. The role of each symbol in a sentence is determined. There are no free variables that need to be assigned a value, and if the closed terms of the sentence are all defined then it has a fixed meaning and makes a definite assertion. A defined sentence is either true or false. An atomic sentence is just an atomic formula P(71 , ..􀀂. , 7k•) that is a sentence. This means that the terms 71 , ... , 7k are all closed, i.e., the formula has no variables at all. Truth and Quantification Suppose that there is only one variable, say x, that has any free occurrence in a certain formula

4.4 *-Transforms

A formula in the language LfR of the real-number structure 􀊛 has symbols P, f for relations and functions of􀊛. It can be turned into a formula of the language L•fR of the hyperreal structure *􀊛 by replacing P by * P, and f by *f. Any constant r naming a real number is left as is, since we identify r inn􀊛 with *r in fs. More precisely, we first define the *-transform *7 of an LfR-term T. This is obtained by replacing each function symbol f occurring in T by * f, leaving the variables and constants ofnT alone. Even more formally, we can give the definition by induction on the formation of T, using the following rules: • If T is a variable or an LfR-constant, then *7 is just T. 4.4 *-Transforms • If 7 is /(71, ... , 7m), then *7 is */(*71, ... , *7m)· The *-transform *cp of an .C!R-formula cp is obtained by • replacing each term 7 occurring in cp by *7; • replacing the relation symbol P of any atomic formula occurring in cp by *P; and • replacing the "bound" P of any quantifier (\:fx E P) or (3x E P) occurring in cp by * P. Again we can spell this out by induction on the formation of cp: *P(*71, ... , *7k) *cp 1\ *'¢ *cp v *'¢ *(P(7I, ... , 7k)) *(cp 1\ '¢) *(cpov'¢) *(•cp) ·-•(*cp) *(Vx E P) cp ·*(:J x E P) cp .- *cpo-+ *'¢ *cp 􀊍 *'¢ (Vx E *P) *cp (:Jx E *P) *cp. We tend to drop the * symbol when referring to the transforms of some of the more well-known relations like =, #, <, 2:, etc., and well-known mathematical functions like sin, cos, log, ex, etc. For instance, *(1r < f(x + 1)) *(sin ex E Q) (1r < *f(x + 1)), (sin ex E *Q), and so on. Even further, we noted in Section 3.13 that it would do no harm to drop the * symbol in referring to the extension * f of any function f. If this practice is adopted systematically, then the transform *7 of each term 7 will just be 7 itself. Then atomic formulae like etc. that express basic equalities and inequalities will be left alone under *-transformation, while a membership formula 7 E P becomes 7 E *P. With all these conventions in place, the general procedure for "adding the stars" reduces simply to replacing P(7I, ... 'o7k) by *P(7I, ... 'o7k), Vx E P by Vx E * P, :Jx E P by :Jx E *P. To summarise all of this in words; the essence of *-transformation is to 44 4. The Transfer Principle (i) replace the bound P of any quantifier by its enlargement * P; and (ii) replace relations appearing in atomic formulae by their enlargements, but only in the (unary) case of a membership formula (T E P), or for relations of arity greater than one other than the common relations =, =/-, <, 2::, etc. Exercise 4.4.1 Review the examples of Section 4.1, formalising them precisely in L>.R., and verify that they conform to our definition of *-transform.

4.5 The Transfer Principle

The notion of an L>.R sentence and its *-transform provides an explanation of the notion of an "appropriately formulated statement" as discussed in Section 1.2, and hence provides a first answer to the question as to which properties are subject to transfer between IR and \({}^*\mathbb{R}\): any property expressible by an L>.R.-sentence is transferable. Formally, the transfer principle is stated as follows: • A defined L>.R-sentence cp is true if and only if *cp is true. As a first illustration of this, beyond the examples of Section 4.1, consider the proof that \({}^*\mathbb{R}\) is an ordered field (Theorem 3.6.1). Now, the fact that IR is an ordered field can be expressed in a finite number of L>.R.-sentences, like (Vx, y E \(\mathbb{R}\) )( x +y = y + x), (Vx E IR)o(x · 1 = x), (Vx, y E \(\mathbb{R}\)) (X < y V x = y V y < X), and so on. By transfer we can immediately conclude that the *-transforms of these sentences are true, showing that \({}^*\mathbb{R}\) is an ordered field. In particular, to show that multiplicative inverses exist in \({}^*\mathbb{R}\), instead of making an ultrapower construction of the inverses as in the proof of Theorem 3.6.1 we simply observe that it is true that (Vx E IR) [x =/-0 -t (3y E IR) x y = 1] · and conclude by transfer that (Vx E \({}^*\mathbb{R}\)) [x =/-0 -t (3y E \({}^*\mathbb{R}\)) x y = 1]. · For another example, consider the closed interval [a, b] = {x E IR: a􀂪 x 􀂪 b} 4.5 The Transfer Principle in the real line defined by points a, b E JR. Then it is true that ('Vx E JR)(x E [a, b] +--+a::; x ::; b), so by transfer we see that the enlargement of [a, b] is the hyperreal interval defined by a and b (Exercise 3.10(6)): *[a,ob] = {x E \({}^*\mathbb{R}\)o: a :S x :S b}. Similarly, we can transfer to \({}^*\mathbb{R}\) many familiar facts about standard mathematical functions. Thus the following are true: ('Vx E \({}^*\mathbb{R}\)) sin(7r-x) = sinnx, ('Vx E \({}^*\mathbb{R}\)) cosh x +sinh x = ex, ('Vx,y E \({}^*\mathbb{R}\)+) lognxy = logx +logy. All of the above examples involve taking a universally quantified LfJl.sentence of the form ('Vx, y, ... E JR)

4.6 Justifying Transfer

In constructing the ordered field \({}^*\mathbb{R}\) we repeatedly used the criterion that a particular property was to hold of hyperreals [r), [s), ... iff the corresponding property held of the real numbers rn, Sn, ... for almost all n. In fact, this almost-all criterion works for any property expressible by an L!Jt-formula, and that ultimately is why the transfer principle holds. To spell this out some further technical notation is needed. For a formula cp we write cp(x1, ... , xp) to indicate that the list x1, ... , Xp includes all the variables that occur free in the formula cp. Then cp(sb ... , sp) is the sentence obtained by replacing each free occurrence of Xi in cp by the constant si· For example, if cp(x1, x2) is the formula (3y E Q) (xi+ X􀛬 < y), then cp( 1r, .J2) is the sentence (3y E Q)((1r2 + (vl2)2 < y). Now, if cp(x1, ... , xp) is a formula of L!R, and r1, ... , rP E IRN, put [cp(r1, ... , rP)] = {n EN: cp(r;.,, ... , r􀇞) is true}. This extends the definitions of [r = s], [r < s], etc. to L!R-formulae in general. Then such statements as [r] = [s] iff [r = s] E :F, [r] < [s] iff [r < s] E :F, [r] E *A iff [rEA] E :F *P((r1], ... , [rk]) iff [P(r1, •.., rk)] E :F 4. 7 Extending Transfer 4 7 (cf. Sections 3.6, 3.9, 3.14) are seen to be cases of the following fundamental result. For any £􀛭-formula

4.7 Extending Transfer

We defined general relational structures S and their languages £s0, but applied these ideas only to the language [,fJt in describing the transfer principle. In fact, it is possible to use the ultrapower construction to build an "enlargement" of any structurenS and obtain a transfer principle for it. For instance, by replacing \(\mathbb{R}\) by C this would give us a way of embarking on the nonstandard study of complex analysis. It is important also to realise that the language [,fJt is limited by the fact that its quantifiable variables can range only over elements of IR, and not over more complicated entities like subsets of IR, sequences, real-valued functions, etc. For example, the Dedekind completeness principle, every subset of \(\mathbb{R}\) that is nonempty and bounded above has a least upper bound, cannot be formulated in Lr.Jt because the language does not allow quantifiers of the type Vx E P(IR) 48 4. The Transfer Principle that apply to a variable (x) whose range of values is the set of all subsets of JR. Later on (Chapter 13), a language will be introduced that does have such "higher-order" quantifiers and for which an appropriate transfer principle exists. Before then we will see that .C'.R is still powerful enough to develop a great deal of the standard theory of \(\mathbb{R}\), including the convergence of sequences and series, differential and integral calculus, and the basic topology of the real line. Indeed, for the next half-dozen chapters we will forget about the ultrapower construction and explore all t :Iese topics using only the fact that \({}^*\mathbb{R}\) is an ordered field that