ZFCΩ: A NONSTANDARD SET THEORY WHERE ALL SETS HAVE NONSTANDARD EXTENSIONS MAMLS at CCNY December 2nd, 2000 Mauro Di Nasso Universit`a di Pisa, Italy and CityCollege/CUNY email: [email protected] 1 Foundations of Nonstandard Methods AXIOMATIC SET THEORIES: Include nonstandard methods within a uni- fied general axiomatic system for mathe- matics. ELEMENTARY PRESENTATIONS: Provide “elementary” introductions to non- standard analysis which avoid (as far as possible) technical notions from mathemat- ical logic. 2 Superstructure Approach • A standard universe, namelya super- structure V (X)=∪n∈N Vn(X) where V0(X)=X and Vn+1(X)=P(Vn(X)). X contains a copyof N. • A nonstandard universe V (Y ). • A nonstandard embedding ∗ : V (X) → V (Y ) (i) ∗X = Y (typically ∗x = x for all x ∈ X) (ii) Transfer principle: ∗ preserves all “elementary” properties. (iii) N is a proper subset of ∗N. 3 In the nonstandard universe: • x is standard if x = ∗y for some y. • x is internal if x ∈ ∗y for some y. • x is external if x is NOT internal. By transfer: “Every(nonempty)bounded subset of N has a greatest element” ⇐⇒ “Every(nonempty) A ∈ ∗P(N) has a greatest element”. Hence N is external. 4 • κ-saturation (κ an infinite cardinal): If |F| <κis a familyof internal sets with the finite intersection propertyFIP (i.e. ∩F = ∅ for all finite F ⊆F) then ∩F = ∅. Whyonlythefinite levels in the hierarchy ? If ω is in the standard model, then ∗ω is NOT wellfounded. Why ∗ is only a bounded elemen- tary embedding ? Otherwise, transfer the following: “∀x ∈ N ∃Ax ∃f :[0,x] →Ax, ∈ iso- morphism of ordered structures”. For any ∗ infinite ξ ∈ N, the corresponding Aξ is NOT wellfounded. Give up the regularity axiom. 5 Limitations of the Superstructure Approach • In principle, nonstandard methods are NOT related to the notion of cumula- tive hierarchy. • Different superstructures are needed for different problems. • Superstructures onlymodel a fragment of ZFC. • Transfer is formulated byusing the tech- nical notion of bounded elementaryem- bedding. • Onlya limited amount of saturation is available NONSTANDARD SET THEORIES: Axiomatize nonstandard methods in the full generalityof set theory. 6 Internal viewpoint Independentlyintroduced byKarel Hrb`a˘cek and Edward Nelson in the 70’s. • Use the standardness predicate st and the internalness predicate int. Infinitesimals and other idealized objects are alreadythere. S = {x : st(x)}; I = {x : int(x)} S is a proper subset of I. • TRANSFER For each ∈-formula ϕ and x1,...,xn ∈ S: S I ϕ (x1,...,xn) ⇐⇒ ϕ (x1,...,xn) • SATURATION (Idealization) 7 External viewpoint Original view of Abraham Robinson. ∗ : V → M • Everyobject A in the standard uni- verse V has a “nonstandard” exten- sion ∗A ∈ M in the nonstandard uni- verse M: S = {∗x : x ∈ V} I = {y : ∃x ∈ V with y ∈ ∗x} S is a proper subset of I. • TRANSFER For each (bounded) ∈-formula ϕ and x1,...,xn ∈ V: V M ∗ ∗ ϕ (x1,...,xn) ⇐⇒ ϕ ( x1,..., xn) • κ-SATURATION (for a suitable κ) 8 What properties a nonstandard set theory NST should satisfy? • All principles of mathematics ZFC− (without regularity). • Transfer principle for all sets of ordi- narymathematics ( TP). • Unlimited levels of saturation (US). HRBA` CEK’s˘ paradox. ZFC− + TP + US =⇒ ∗A is a proper class for each infinite A. Proof. Consider the familyof internal sets F = {∗A \{ξ} : ξ ∈ A} and apply saturation. 9 Nelson’s Internal Set Theory IST The paradox is avoided because all sets are internal. There is no external notion of cardinality. Hrb`a˘cek’s NS1 and NS2 • TP + US +ZFC− except powerset and choice, is a conservative extension of ZFC. • TP + US +ZFC− except replace- ment, is a conservative extension of ZFC. 10 Motivations in formulating NST “FORMALISTIC” VIEW: Nonstandard methods are “short-cuts” for standard methods. “REALISTIC” VIEW: Look for axioms which are true in the “real” world. • In the “real” world, ZFC− holds, with separation and collection for all formu- las in the extended language. E.g. standardization should be true: ∀X oX = {x ∈ X : ∃yx= ∗y} is a set • In the “real” world, unlimited levels of saturation are available. 11 Research in nonstandard set theories can be seen as an attempt to answer the following question: How to avoid Hrb`a˘cek’s paradox? A (partial) list of proposals: Nelson’s Internal Set Theory IST; Hrb`a˘cek’s Nonstandard Set Theories NS1, NS2 and NS3; Kawai’s Nonstandard Set Theory NST; Fletcher’s Stratified Nonstandard Set Theory SNST; Ballard’s Enlargement Set Theory EST; Zermelo-Fraenkel-Boffa ZFBC, proposed byBal- lard and Hrb`ac˘cek; Kanovei-Reeken’s Bounded Set Theory BST and Hrb`a˘cekSet Theory HST; P´eraire’s Relative Set Theory RST; Di Nasso’s ∗ZFC; Gordon-Andreyev Nonstandard Class Theory NCT; etc. .... 12 ∗ZFC • ZFC−, where separation and replace- ment are assumed for all ∈-∗-formulas (∗ is a new symbol). • The nonstandard embedding ∗ : S → I is defined for all wellfounded sets, that is the standard universe is S = WF. • κ-saturation for all ∈-definable cardi- nals. Theorem. Let ϕ be any ∈-formula. Assume that κ-saturation ⇒ ϕS(x) for all x ∈ S with |x| <κ.ThenϕS(x) for all x ∈ S. Proof. Bycontradiction, let κ =min{|x| : x ∈ S and ¬ϕS(x)} 13 κ is ∈-definable, thus κ+-saturation im- plies ϕS(x) for all standard |x|≤κ,a contradiction. Example. A typical proof in nonstan- dard analysis shows that: “If |X|+-saturation holds and X is the topological product of a familyof compact spaces, then X is compact”. As a consequence, ∗ZFC proves Tychonov’s theorem for all topological products X. Theorem. Every model A |= ZFC has an elemen- tary end extension A ≺e A such that A =(WF)B for some B |= ∗ZFC. Corollary. For any ∈-sentence σ: ZFC σ ⇔ ∗ZFC σWF 14 We want more: • Nonstandard methods should be applied to all objects of mathematics (not only to the standard ones). • κ-saturation should be available for each fixed κ. • A presentation to nonstandard analy- sis should be possible in truly“elemen- tary” terms. 15 ZFCΩ • ZFC− where separation and collection are assumed for all ∈-Ω-formulas (Ω is a new symbol). • Ω is a function defined on the class of sequences, i.e. on the class of those functions f whose domain is an infinite cardinal κ. For κ-sequences, denote byΩ( f)=f(ακ). • If f is a sequence of nonemptysets: f(ακ)={g(αk):g(i) ∈ f(i) for all i} 16 • Point-wise unions and intersections are preserved byΩ, i.e. 1. f(i)=g(i) ∪ h(i) for all i ⇒ f(ακ)=g(ακ) ∪ h(ακ); 2. f(i)=g(i) ∩ h(i) for all i ⇒ f(ακ)=g(ακ) ∩ h(ακ). • If f(ακ)=g(ακ)then (ϕ ◦ f)(ακ)=(ϕ ◦ g)(ακ) (provided such compositions are defined) Call κ-internal anyelement of the form f(ακ), and denote by Iκ the collection of κ-internal sets. • κ-saturation: If |F| ≤ κ is a familyof κ-internal sets with the FIP, then ∩F = ∅. 17 Define Jκ(A)={f(ακ):f : κ → A}, the κ-nonstandard embedding. Theorem. + For each κ, Jκ : V → Iκ is a κ -saturated bounded elementary embedding of the uni- versal class of all sets V. If we also assume the following minimal- ity axiom,thenJκ is fully elementary. • Minimality Axiom Everyset is obtained from the empty- set bytransfinite iterations of the pow- erset and nonstandard extension oper- ations. That is V = ∪αWα where W0 = ∅; Wα+1 = P( ∪κ≤α Jκ(Wα) ∪ Wα); Wλ = ∪α<λWα if λ limit. 18 Theorem. Every model A |= ZFC is the wellfounded part A = WFB of some B |= ZFCΩ. Corollary. For any ∈-sentence σ: ZFC σ ⇔ ZFCΩ σWF If we reduce to countable sequences, we get the Alpha-Theory, a truly“elemen- tary” approach to nonstandard analysis which, in principle, can even be suitable at the freshman level. References: M. Di Nasso, On the foundations of nonstan- dard mathematics, Mathematica Japonica, vol.50, 1999, pp. 131-160. V. Benci and M. Di Nasso, The Alpha- Theory: a truly elementary axiomatic approach to nonstandard analysis, submitted. 19.