NSA, IST, ZFC MATTHEW TOWERS

Abstract. An introduction to Nelson-style IST. Expanded notes, originally for a Kinderseminar talk on non-standard analysis.

Contents 1. Introduction 2. Internal Set Theory 3. Some non-standard real analysis 4. Towards a model of IST (ultrapowers) References

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1. Introduction Non-standard analysis is about doing rigorous infinitesimal calculus. Although non-standard methods can be used in many areas (see [LG81]: topology, differential equations, functional analysis, perturbation theory, deformation theory of algebras. . . ) all I have time for in this talk is (a) to attempt to convince you that they involve some interesting mathematics, and (b) to do a little non-standard real analysis. Here are some reasons to do (b). Motivation 1: Infinitesimals were used by mathematicians (and physicists) for thousands of years — at least since Archimedes (see Supplement - The Method in [Hea02] ). (Even if you trust Euler and Cauchy’s infinitesimal manipulations without a rigorous theory of NSA, you might not trust mine. . . ) Example 1.1. (Euler, Introductio in Analysin Infinitorum (1748), see [Gol98]) For x ∈ R define ex by (1 + x/j)j for j infinitely large. Expanding binomially, x2 j(j − 1) + ··· j2 2! =1 + x + x2 /2! + · · ·

ex =1 + x +

since, as j is infinitely large, (j − 1)/j = 1 etc. Now define log as the inverse of exp, what is its power series at zero? Write ex = 1 + y = (1 + x/j)j . Then 1 + x/j =(1 + y)1/j (1/j)(1/j − 1) 2 (1/j)(1/j − 1)(1/j − 2) 3 y + y + ··· 2! 3! x = log(1 + y) =y − y 2 /2 + y 3 /3 − · · · =1 + y/j +

as, j being infinitely large, 1/j = 0. See the paper Is mathematical history written by the victors? by Bair et al. sections 3.5 onwards for a description of Euler’s derivation of the product formula 1

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for sin and sinh using infinitely large integers and references to work on formalizing Euler’s arguments in NSA. Following Euler’s manipulations of this kind feels something like being given a lift to the shops by Michael Schumacher. Example 1.2. (Cauchy, from his Cours d’analyse [Cau09, p.34]) A function f is continuous at x if f (x + h) − f (x) is infinitesimal whenever h is. Part of the aim of this talk is to convice you that this is a definition to be taken seriously. Motivation 2: One amongst many possible examples from a first year analysis course: say someone asks you to prove that if g : (a, b] → R is uniformly continuous and xn , yn → a in (a, b] then g(xn ) and g(yn ) tend to the same limit. You probably think “for large n, xn and yn are close to a, so xn is close to yn , so g(xn ) is close to g(yn ) by uniform continuity, therefore g(xn ) − g(yn ) → 0”. The translation of this into epsilon-delta language adds irrelevant complications, whereas the NSA proof follows exactly the thought process above. There are many ways to build mathematical structures in which you can do infinitesimal analysis: from simple ones like R[]/2 , to more sophisticated constructions like the hyperreals (using ultrapowers and ultralimits) smooth infinitesimal analysis (a topos-theoretical approach in which you give up the law of the excluded middle [Bel08], and infinitesimals are numbers which are not not equal to zero), . . . Our method, following Nelson [Nel77], is to introduce a new set theory. 2. Internal Set Theory IST is a set theory consisting of ZFC with three new axiom schemes. We do not change the existing axioms of ZFC at all. But we do add a new unary predicate to the formal language used in stating them: standard. This is undefined, just as the binary predicate ∈ is undefined (but of course we think we know what it means. . . ). A formula/statement in the formal language will be called classical or internal if it does not involve the new term ‘standard’. We introduce a bit of notation: ∀s x means for all standard x, and ∀sf x means for all standard finite x. Finite has the same meaning in IST as it does in ZFC: “in bijection with a (possibly non-standard) natural number”. Here is the first axiom scheme: Idealisation (I). Fix a classical binary relation. Then in order for there to exist x related to every standard y, it’s enough that for each standard finite set F we can find x = xF related to every element of F . Formally, let R be an internal formula with free variables x, y, t1 , . . . , tn (allowing n = 0). Then ∀t1 · · · ∀tn (∃x∀s yR(x, y) ⇐⇒ ∀sf F ∃x∀(f ∈ F )R(x, f )). Let’s apply (I). Choose an infinite set X and let R(x, y) be (x 6= y) ∧ (x ∈ X) ∧ (y ∈ X). Certainly for each standard finite subset F of X there exists v ∈ X with v 6= f for all f ∈ F , otherwise X would not be infinite. So by (I), there exists v ∈ X such that v 6= x for every standard x ∈ X. This v is necessarily non-standard. We have shown: Lemma 2.1. Any infinite set contains non-standard elements. Apply the same reasoning to the relation < on the natural numbers, to get: Lemma 2.2. There exists ν ∈ N larger than every standard natural number. Definition 2.3. x ∈ R is called infinitesimal if |x| is smaller than the reciprocal of every standard natural number.

NSA, IST, ZFC

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The reciprocal of the real provided by Lemma 2.2 would be an example of a non-zero infinitesimal. Sets like R, Z, N in IST are constructed in exactly the same way as usual — we have all the ordinary ZFC axioms to work with. On to the next axiom scheme: Standardisation (S). Let E be a standard set and P any property (in the formal language, possibly using the predicate ‘standard’). Then there exists a standard set A whose standard elements are exactly the standard elements of E satisfying P . Formally, for any formula P (x) of the formal language with free variables x, t1 , . . . , tn (allowing n = 0): ∀t1 · · · ∀tn ∀s E∃s A∀s x(x ∈ A ⇐⇒ (x ∈ E ∧ P (x))) We write the set A defined above as S

{x ∈ E : P (x)}

but remember that this is only notation, shorthand for the set whose existence is given by (S) and not an application of set comprehension. We cannot form {x ∈ E : P (x)} using the axiom scheme of comprehension because, the ZFC axioms being unchanged, we only have comprehension axioms with classical formulas in them. For example there’s no guarantee that {x ∈ R : x is infinitesimal} 1

is a set. Note that we still can’t prove anything is standard. The final axiom scheme remedies this: Transfer (T). Let P be an internal formula with free variables x, t1 , . . . , tn . Then ∀s t1 · · · ∀s tn (∀s xP (x, t1 , . . . , tn ) =⇒ ∀xP (x, t1 , . . . , tn )) A classical formula is true for all values as soon as it is true for all standard values. We can now prove things are standard, since (T) allows us to say that if there is a unique object satisfying P , then there’s a unique standard object with P , so that object is standard (take the contrapositive to get ∃xP (x) =⇒ ∃s xP (x), so by building uniqueness into the formula, ∃!xP (x) =⇒ ∃!s xP (x)). √ Example 2.4. ∅, 0, 1, 2, 2, π, R, N, . . . are standard. 3. Some non-standard real analysis Let us do some undergraduate real analysis. Remember nothing has changed about the set R, it is still a complete ordered archimedean field. Write a ' b if a − b is infinitesimal. Recall Cauchy’s definition of continuity mentioned earlier. Definition 3.1. f : R → R is said to be S-continuous at x if x ' y implies f (x) ' f (y). 1It isn’t: such a set would have a least upper bound s by the completeness property — which still holds as any theorem of ZFC is a theorem of IST. This least upper bound is obviously infinitesimal, but then 2s > s is still infinitesimal. The same problem with sets failing to exist occurs if you try to prove by induction that all natural numbers are standard.

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It seems as if this should be related to ordinary continuity, but it clearly isn’t the same in general. Take the standard function f (x) = 1/x, continuous at all points of (0, 1). If δ > 0 is infinitesimal then f is not S-continuous at δ, for δ ' 2δ but f (δ) − f (2δ) = −1/(2δ) is certainly not infinitesimal. Proposition 3.2. Let f, x be standard. Then f is S-continuous at x if and only if it is continuous at x. Proof. Suppose f is continuous at x, and let  > 0 be standard. Then ∃δ > 0∀y : |x − y| < δ =⇒ |f (x) − f (y)| < . This statement is exactly the sort of thing to which we can apply transfer. The δ that arises must therefore be standard. Now if y ' x then |y − x| < δ for any standard . Hence |f (x) − f (y)| <  for any standard , so we have f (x) ' f (y). Conversely suppose f is S-continuous at x. Then ∀s  > ∃δ > 0 : |y − x| < δ =⇒ |f (x) − f (y)| <  (take δ infinitesimal independent of ). Apply transfer to get continuity.  By transfer, if I is a standard interval then f is continuous on I as soon as it is continuous at all standard points of I. So: Proposition 3.3. Let f be a standard function on a standard interval I. Then f is continuous on I if and only if it is S-continuous at all standard points of I. What if we insist on S-continuity at all points, not just the standard ones? Proposition 3.4. Let f be a standard function on a standard interval I. Then f is uniformly continuous on I if and only if it is S-continuous at all points of I. Proof. Let f be uniformly continuous and x ' y both in I. We need f (x) ' f (y). ∀s  > 0∃s δ > 0 : ∀a, b ∈ I : |a − b| < δ =⇒ |f (a) − f (b)| <  The ∃s is an application of (T) to the definition of uniform continuity, like before. We conclude ∀s  > 0, |f (x) − f (y)| <  and hence f (x) ' f (y). Conversely suppose f is S-continuous at all points of I. We have ∀s  > 0∃δ > 0 : |x − y| < δ =⇒ |f (x) − f (y)| <  by choosing δ infinitesimal for any standard . An application of (T) gives uniform continuity.  Returning to our example of the function f (x) = 1/x, the reason it fails to be uniformly continuous on (0, 1) is that something goes wrong near the left hand endpoint. As we saw before, S-continuity fails at infinitesimal points of (0, 1). Here’s another example of how standard -δ definitions can be written in nonstandard language. Lemma 3.5. Let an be a standard sequence of reals. Then an → 0 as n → ∞ if and only if aν ' 0 for all non-standard ν ∈ N. This also appears (in an equivalent form for partial sums) in [Cau09, p.125] 4. Towards a model of IST (ultrapowers) Anyone can write down a bunch of axioms that extend ZFC and investigate their consequences. Why should IST be an interesting extension? Might it be inconsistent? The following result is proved by William Powell in the appendix of [Nel77]. Theorem 4.1. IST is a conservative extension of ZFC, that is, any internal statement that can be proved in IST also has a proof in ZFC.

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In particular if IST is inconsistent we can prove 0 = 1 there, so we can prove 0 = 1 in ZFC which is therefore also inconsistent. The result either shows IST is useful or useless, according to your taste. The proof is rather involved, and certainly too long for this talk. Instead we look at an important part of it: building a model for IST within ZFC. We won’t even do this, but we’ll get started. The construction uses ultrafilters. From here onwards we work in ZFC. Definition 4.2. Let X be a set. An ultrafilter on X is a set U of subsets of X such that (1) U is closed under finite intersections and supersets (2) X ∈ U, ∅ ∈ /U (3) ∀E ⊂ X((E ∈ U) ∧ (X \ E ∈ U)) Without the last condition U is just called a filter. An ultrafilter is said to be principal if it is of the form {V ⊂ X : v ∈ V } for some v ∈ X. Proposition 4.3. Infinite sets admit non-principal ultrafilters. We think of elements of a non-principal ultrafilter on X as the “big” subsets of X (relative to that particular ultrafilter). Proposition 4.3 is stronger than ZF, but weaker than ZFC. It is implied by the ultrafilter lemma (UF, that every filter can be extended to an ultrafilter). UF ⇐⇒ Boolean prime ideal theorem ⇐⇒ compactness for first order logic ⇐⇒ completeness for FOL etc. These results are strictly weaker than the axiom of choice. Definition 4.4. Let U be an ultrafilter on X, and let Y be a set. The ultrapower ∗ Y of Y is Y X modulo the equivalence relation f ∼ g ⇐⇒ {x ∈ X : f (x) = g(x)} ∈ U. Note that if U is principal, the ultrapower looks like Y again. Also, there is a natural injection ι : Y → ∗ Y sending y to the constant function. We want to relate properties of Y to those of ∗ Y . Definition 4.5. Let A ⊂ Y . The enlargement ∗ A of A consists of the ∼-classes of functions f : X → Y such that f −1 (A) ∈ U. We can identify ∗ (Y × Y ) with ∗ Y × ∗ Y , so things like relations on Y have enlargements to relations on ∗ Y . One reason the ultrapower construction not just some random algebraic gadget is Lo´s’s theorem [BS06]: Theorem 4.6. ι is an elementary embedding. That is, any first order statement P about Y is true if and only if the corresponding ‘enlarged’ statement is true in ∗ Y . (actually this is a weak version of what Lo´s proved). Definition 4.7. Let U be a non-principal ultrafilter on N. The ultrapower ∗ R, with +, ×, < defined by extension, is called ‘the’ hyperreals. This construction is independent, up to isomorphism, of the choice of U — so long as you believe GCH! [KSTT05]. We write functions N → R as sequences. The enlarged order relation is that [(an )] ≤ [(bn )] means {n : an ≤ bn } ∈ U. Here [ ] denotes a ∼-equivalence class. The archimedean principle is that every real is less than a natural. This is a first order statement about R and provides a a good illustration of Lo´s’s theorem. It is not true that every element of ∗ R is bounded above by an element of ιN, hence ∗ R is non-archimedean. But this doesn’t contradict Lo´s — what should be true is that the enlargement of the archimedean principle holds for ∗ R, that is every element of ∗ R should be less than an element of ∗ N. This is easy to see: [(an )] ≤ [(dan e)] ∈ ∗ N.

BPI is that every Boolean algebra contains a prime ideal

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Note that ∗ N contains things like (1, 2, 3, . . .), a “non-standard natural number”, clearly larger than ι(r) for any r ∈ R Have we built a model of the IST reals in ZFC in which this kind of thing plays the role of the non-standard naturals? Not quite: as it stands, ∗ R doesn’t satisfy unrestricted (I), (S) and (T) (though there is a “transfer principle” coming from Lo´s’s theorem). See [Gol98] for an account of NSA using ∗ R. Second order properties like completeness can’t be transferred to ∗ R — and in fact completeness fails there2, the first indication that things haven’t gone exactly as planned. More complicated constructions (adequate ultralimits) are needed, see [Nel77] . . .

Acknowledgement: I first learned about NSA from Robert’s [Rob03], and these notes are influenced by his presentation. References [Bel08]

J. L. Bell, A primer of infinitesimal analysis, second ed., Cambridge University Press, Cambridge, 2008. [BS06] J. L. Bell and A. B. Slomson, Models and ultraproducts: An introduction, Dover, New York, 2006. ´ [Cau09] Augustin-Louis Cauchy, Cours d’analyse de l’Ecole Royale Polytechnique, Cambridge Library Collection, Cambridge University Press, Cambridge, 2009. [Gol98] Robert Goldblatt, Lectures on the hyperreals, Graduate Texts in Mathematics, vol. 188, Springer-Verlag, New York, 1998. [Hea02] T. L. Heath (ed.), The works of Archimedes, Dover, New York, 2002. [KSTT05] Linus Kramer, Saharon Shelah, Katrin Tent, and Simon Thomas, Asymptotic cones of finitely presented groups, Adv. Math. 193 (2005), no. 1, 142–173. MR 2132762 (2006d:20075) [LG81] Robert Lutz and Michel Goze, Nonstandard analysis, Lecture Notes in Mathematics, vol. 881, Springer-Verlag, Berlin, 1981, A practical guide with applications, With a foreword by Georges H. Reeb. [Nel77] Edward Nelson, Internal set theory: a new approach to nonstandard analysis, Bull. Amer. Math. Soc. 83 (1977), no. 6, 1165–1198. [Rob03] Alain M. Robert, Nonstandard analysis, Dover Publications Inc., Mineola, NY, 2003. E-mail address: [email protected]

2This time we are entitled to form {x ∈ ∗ R : x is infinitesimal}