aa r X i v : . [ m a t h . L O ] S e p INFINITESIMAL ANALYSIS WITHOUT THE AXIOMOF CHOICE
KAREL HRBACEK AND MIKHAIL G. KATZ
Abstract.
It is often claimed that analysis with infinitesimals re-quires more substantial use of the Axiom of Choice than traditionalelementary analysis. The claim is based on the observation that thehyperreals entail the existence of nonprincipal ultrafilters over N , astrong version of the Axiom of Choice, while the real numbers canbe constructed in ZF . The axiomatic approach to nonstandardmethods refutes this objection. We formulate a theory SPOT which suffices to carry out infinitesimal arguments, and prove that
SPOT is a conservative extension of ZF . Thus the methods ofCalculus with infinitesimals are just as effective as those of tradi-tional Calculus. This result and conclusion extend to large partsof ordinary mathematics and beyond. We also develop a strongeraxiomatic system SCOT , conservative over ZF + ADC , which issuitable for handling such features as an infinitesimal approach tothe Lebesgue measure. Proofs of the conservativity results com-bine and extend the methods of forcing developed by Enayat andSpector.
Contents
1. Introduction 21.1. Axiom of Choice in Mathematics 21.2. Countering the objection 41.3.
SPOT and
SCOT
52. Theory
SPOT and Calculus with infinitesimals 72.1. Some consequences of SPOT 72.2. Mathematics in SPOT 103. Theory
SCOT and Lebesgue measure 124. Conservativity of
SPOT over ZF SCOT over ZF c Date : September 9, 2020.
Key words and phrases. nonstandard analysis, axiom of choice, ultrafilter, forcing,extended ultrapower.
6. Standardization for parameter-free formulas 277. Idealization 307.1. Idealization over uncountable sets 317.2. Further theories 338. Final Remarks 358.1. Open problems 358.2. Forcing with filters 368.3. Zermelo set theory 368.4. Weaker theories 368.5. Finitistic proofs 378.6.
SPOT and CH 379. Conclusion 38References 391.
Introduction
Many branches of mathematics exploit the Axiom of Choice ( AC )to one extent or another. It is of considerable interest to gauge howmuch the Axiom of Choice can be weakened in the foundations ofnonstandard analysis. Critics of analysis with infinitesimals often claimthat nonstandard methods require more substantial use of AC thantheir standard counterparts. The goal of this paper is to refute such aclaim.1.1. Axiom of Choice in Mathematics.
We begin by consider-ing the extent to which AC is needed in traditional non-infinitesimalmathematics. Simpson [29] introduces a useful distinction between set-theoretic mathematics and ordinary or non-set-theoretic mathematics.The former includes such disciplines as general topology, abstract al-gebra and functional analysis. It is well known that fundamental the-orems in these areas require strong versions of AC . Thus • Tychonoff’s Theorem in general topology is equivalent to full AC . • Prime Ideal Theorem asserts that every ring with unit has a(two-sided) prime ideal.
PIT is an essential result in abstractalgebra and is “almost” as strong as AC (it is equivalent toTychonoff’s Theorem for Hausdorff spaces). It is also equivalentto the Ultrafilter Theorem: Every proper filter over a set S (i.e.,in P ( S )) can be extended to an ultrafilter. NFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 3 • Hahn-Banach Theorem for general vector spaces is equivalent tothe statement that every Boolean algebra admits a real-valuedmeasure, a form of AC that is somewhat weaker than PIT .Jech [15] and Howard and Rubin [11] are comprehensive references forthe relationships between these and many other forms of AC .Researchers in set-theoretic mathematics have to accept strong formsof AC as legitimate whether or not they use nonstandard methods.Our concern here is with ordinary mathematics, which, according toSimpson, includes fields such as Calculus, countable algebra, differ-ential equations, and real and complex analysis. It is often felt thatresults in these fields should be effective in the sense of not being de-pendent on AC . However, even these branches of mathematics cannotdo entirely without AC . There is a number of fundamental classicalresults that rely on it; they include • the equivalence of continuity and sequential continuity for real-valued functions on R ; • the equivalence of the ε - δ definition and the sequential definitionof closure points for subsets of R ; • closure of the collection of Borel sets under countable unionsand intersections; • countable additivity of Lebesgue measure.Without an appeal to AC one cannot even prove that R is not a unionof countably many countable sets, or that a strictly positive functioncannot have vanishing Lebesgue integral (Kanovei and Katz [18]). How-ever, these results follow already from ACC , the Axiom of Choice forCountable collections, a weak version of AC that many mathemati-cians use without even noticing. Nevertheless, it is true enough thatno choice is needed to define the real number system itself, or to developCalculus and much of ordinary mathematics.It has to be emphasized that objections to AC are not a matterof ontology, but of epistemology. In other words, the issue is not theexistence of objects, but proof techniques and procedures. For betteror worse, many mathematicians nowadays believe that the objects ofinterest to them can be represented by set-theoretic structures in auniverse that satisfies ZFC . Nevertheless, they may prefer results thatare effective, that is, do not use AC . For the purposes of this discussion,mathematical results are effective if they can be proved in ZF . Muchof ordinary mathematics is effective in this sense.We now consider whether nonstandard methods require anythingmore. A common objection to infinitesimal methods in the Calculus For example Halmos [9], p. 42; see [3], Sec. 5.7 for further discussion.
KAREL HRBACEK AND MIKHAIL G. KATZ is the claim that the mere existence of the hyperreals implies theexistence of a nonprincipal ultrafilter U over N . The proof is simple:Fix an infinitely large integer ν in N ∗ \ N and define U ⊆ P ( N ) by X ∈ U ←→ ν ∈ X ∗ , for X ⊆ N . It is easy to see that U is anonprincipal ultrafilter over N . For example, if X ∪ Y ∈ U , then ν ∈ ( X ∪ Y ) ∗ = X ∗ ∪ Y ∗ , where the last step is by the TransferPrinciple. Hence either ν ∈ X ∗ or ν ∈ Y ∗ , and so X ∈ U or Y ∈ U . If X is finite, then X = X ∗ , hence ν / ∈ X ∗ and so U is nonprincipal.By the well-known result of Sierpi´nski [28] (see also Jech [15], Prob-lem 1.10), U is a non-Lebesgue-measurable set (when subsets of N areidentified with real numbers in some natural way). In the celebratedmodel of Solovay [30], ZF + ACC holds (even the stronger
ADC , theAxiom of Dependent Choice, holds there), but all sets of real numbersare Lebesgue measurable, hence there are no nonprincipal ultrafiltersover N in this model. The existence of nonprincipal ultrafilters over N requires a strong version of AC such as PIT ; it cannot be proved in ZF (or even ZF + ADC ).1.2.
Countering the objection.
How can such an objection be an-swered? As in the case of the traditional mathematics, the key is tolook not at the objects but at the methods used. Analysis with in-finitesimals does not have to be based on hyperreal structures in theuniverse of
ZFC . It can be developed axiomatically; the monographby Kanovei and Reeken [19] is a comprehensive reference for such ap-proaches. Internal axiomatic presentations of nonstandard analysis,such as
IST or BST , extend the usual ∈ -language of set theory bya unary predicate st ( st ( x ) reads x is standard ). For reference, theaxioms of BST are stated in Section 7.It is of course possible to weaken
ZFC to ZF within BST or IST ,but this move by itself does not answer the above objection. It is easilyseen, by a modification of the argument given above for hyperreals, thatthe theory obtained from
BST or IST by replacing
ZFC by ZF proves PIT (Hrbacek [12]). This argument uses the full strength of the prin-ciples of Idealization and Standardization (see Section 7). However,Calculus with infinitesimals can be fully carried out assuming muchless. Examination of texts such as Keisler [21] and Stroyan [34] revealsthat only very weak versions of these principles are ever used there. Ofcourse one has to postulate that infinitesimals exist (Nontriviality), but By the hyperreals we mean a proper elementary extension of the reals, i.e., a properextension that satisfies Transfer. The definite article is used merely for grammaticalcorrectness. Subsets of N can be identified with real numbers; see SP ⇒ SP ′ inthe proof of Lemma 2.4 for one way to do that. NFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 5 stronger consequences of Idealization are not needed. As for Standard-ization, these textbooks only explicitly postulate a special consequenceof it, namely, the following principle: SP ( Standard Part ) Every limited real is infinitely close to a stan-dard real;see Keisler [21, 22], Axioms A - E. However, this is somewhat mis-leading. Keisler does not develop Calculus from his axioms alone; theydescribe some properties of the hyperreals, but the hyperreals are con-sidered to be an extension of the field R of real numbers in the universeof ZFC , and the principles of
ZFC can be freely used. In particular,the principle of Standardization is not an issue; it is automatically satis-fied for any formula. While Standardization for formulas about integersappears innocuous, Standardization for formulas about reals can leadto the existence of nonprincipal ultrafilters. On the other hand, someinstances of Standardization over the reals are unavoidable, for exam-ple to prove the existence of the function f ′ (the derivative of f ) definedin terms of infinitesimals for a given real-valued function f on R .1.3. SPOT and SCOT.
In the present text, we introduce a theory
SPOT in the st - ∈ -language, a subtheory of IST and
BST , and weshow that
SPOT proves Countable Idealization and enough Standard-ization for the purposes of the Calculus. We use ∀ st and ∃ st as quan-tifiers over standard sets. The axioms of SPOT are: ZF (Zermelo - Fraenkel Set Theory) T (Transfer) Let φ be an ∈ -formula with standard parameters. Then ∀ st x φ ( x ) → ∀ x φ ( x ) . O (Nontriviality) ∃ ν ∈ N ∀ st n ∈ N ( n = ν ). SP ′ (Standard Part) ∀ A ⊆ N ∃ st B ⊆ N ∀ st n ∈ N ( n ∈ B ←→ n ∈ A ) . Our main result is the following.
Theorem A
The theory
SPOT is a conservative extension of ZF . Thus the methods used in the Calculus with infinitesimals do notrequire any appeal to the Axiom of Choice.The result allows significant strengthenings. We let SN be the Stan-dardization principle for st - ∈ -formulas with no parameters. The princi-ple allows Standardization of much more complex formulas than SPOT alone.
KAREL HRBACEK AND MIKHAIL G. KATZ
Theorem B
The theory
SPOT + SN is a conservative extensionof ZF . It is also possible to add some Idealization. We let BI ′ be BoundedIdealization (see Section 7) for ∈ -formulas with standard parameters. Theorem C
The theory
SPOT + B + BI ′ is a conservative extensionof ZF . This is the theory
BST with
ZFC replaced by ZF , Standardizationweakened to SP and Bounded Idealization weakened to BI ′ ; we denoteit BSPT ′ . This theory enables the applicability of some infinitesimaltechniques to arbitrary topological spaces. It also proves that there isa finite set S containing all standard reals, a frequently used idea.As noted above, some important results in elementary analysis andelsewhere in ordinary mathematics require the Axiom of CountableChoice. On the other hand, ACC entails no “paradoxical” conse-quences, such as the existence of Lebesgue-non-measurable sets, or theexistence of an additive function on R different from f a : x ax for all a ∈ R . Many mathematicians find ACC acceptable. These considera-tions apply as well to the following stronger axiom.
ADC (Axiom of Dependent Choice) If R is a binary relation on aset A such that ∀ a ∈ A ∃ a ′ ∈ A ( aRa ′ ), then for every a ∈ A thereexists a sequence h a n | n ∈ N i such that a = a and a n Ra n +1 for all n ∈ N .This axiom is needed for example to prove the equivalence of the twodefinitions of a well-ordering (Jech [16], Lemma 5.2):(1) Every nonempty subset of a linearly ordered set ( A, < ) has aleast element.(2) A has no infinite decreasing sequence a > a > . . . > a n > . . . .We denote by ZF c (“ ZFC
Lite”) the theory ZF + ADC . This theoryis sufficient for axiomatizing ordinary mathematics (and many results ofset-theoretic mathematics as well). Let
SCOT be the theory obtainedfrom
SPOT by adding SN and strengthening SP to Countable st - ∈ -Choice ( CC ); see Section 3. Theorem D
The theory
SCOT is a conservative extension of ZF c .In SCOT one can carry out most techniques used in infinitesimaltreatments of ordinary mathematics. As examples, we give a proofof Peano’s Existence Theorem and an infinitesimal construction of
NFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 7
Lebesgue measure in Section 3. Thus the nonstandard methods used inordinary mathematics do not require any more choice than is generallyaccepted in traditional ordinary mathematics.Further related conservative extension theorems can be found in Sec-tions 5, 6 and 7.2.
Theory
SPOT and Calculus with infinitesimals
Some consequences of SPOT.
The axioms of
SPOT weregiven in Section 1.3.
Lemma 2.1.
The theory
SPOT proves the following: ∀ st n ∈ N ∀ m ∈ N ( m < n → st ( m )) . Proof.
Given a standard n ∈ N and m < n , let A = { k ∈ N | k < m } .By SP there is a standard B ⊆ N such that for all standard k , k ∈ B iff k ∈ A iff k < m . The set B ⊆ N is bounded above by n (Transfer), so ithas a greatest element k ( < is a well-ordering of N ) , which is standardby Transfer. Now we have k < m and k + 1 ≮ m , so k o + 1 = m and m is standard. (cid:3) Lemma 2.2. (Countable Idealization) Let φ be an ∈ -formula with ar-bitrary parameters. The theory SPOT proves the following: ∀ st n ∈ N ∃ x ∀ m ∈ N ( m ≤ n → φ ( m, x )) ←→ ∃ x ∀ st n ∈ N φ ( n, x ) . Proof. If ∀ st n ∈ N φ ( n, x ), then, for every standard n ∈ N , ∀ m ∈ N ( m ≤ n → φ ( m, x )), by Lemma 2.1.Conversely, assume ∀ st n ∈ N ∃ x ∀ m ∈ N ( m ≤ n → φ ( m, x )). Bythe Axiom of Separation of ZF , there is a set S = { n ∈ N | ∃ x ∀ m ∈ N ( m ≤ n → φ ( m, x )) } , and the assumption implies that ∀ st n ∈ N ( n ∈ S ).Assume that S contains standard integers only. Then N \ S = ∅ bythe axiom O . Let ν be the least element of N \ S . Then ν is nonstandardbut ν − µ be some nonstandard element of S . We have ∃ x ∀ m ∈ N ( m ≤ µ → φ ( m, x )); as n ≤ µ holds for all standard n ∈ N , we obtain ∃ x ∀ st n ∈ N φ ( n, x ). (cid:3) Countable Idealization easily implies the following more familiarform. We use ∀ st fin and ∃ st fin as quantifiers over standard finite sets. Corollary 2.3.
Let φ be an ∈ -formula with arbitrary parameters. Thetheory SPOT proves the following: For every standard countable set A ∀ st fin a ⊆ A ∃ x ∀ y ∈ a φ ( x, y ) ←→ ∃ x ∀ st y ∈ A φ ( x, y ) . KAREL HRBACEK AND MIKHAIL G. KATZ
The axiom SP ′ is often stated and used in the form( SP ) ∀ x ∈ R ( x limited → ∃ st r ∈ R ( x ≈ r ))where x is limited iff | x | ≤ n for some standard n ∈ N , and x ≈ r iff | x − r | ≤ /n for all standard n ∈ N , n = 0. The unique standard realnumber r is called the standard part of x or the shadow of x ; notation sh ( x ).We note that in the statement of SP ′ , N can be replaced by anycountable standard set A . Lemma 2.4.
The statements SP ′ and SP are equivalent (over ZF + O + T ).Proof. SP ′ ⇒ SP : Assume x ∈ R is limited by a standard n ∈ N . Let A = { q ∈ Q | q ≤ x } . Applying SP ′ with N replaced by Q , we obtaina standard set B ⊆ Q such that ∀ st q ∈ Q ( q ∈ B ←→ q ∈ A ). As ∀ st q ∈ B ( q ≤ n ) holds, the set B is bounded above (apply Transferto the formula q ∈ B → q ≤ n ) and so it has a supremum r ∈ R ,which is standard (Transfer again). We claim that x ≈ r . If not, then | x − r | > n for some standard n , hence either x < r − n or x > r + n .In the first case sup B ≤ r − n and in the second, sup B ≥ r + n ; eitherway contradicts sup B = r . SP ⇒ SP ′ : Given A ⊆ N , let χ A be the characteristic function of A . Define a real number x A = Σ ∞ n =0 χ ( n )10 n +1 ; as 0 ≤ x A ≤ , there isa standard real number r ≈ x A . Let r = Σ ∞ n =0 a n n +1 be the decimalexpansion of r where for every n there is k > n such that a k = 9. Notethat if n is standard, then there is a standard k with this property, byTransfer. If χ ( n ) = a n for all n , then A is standard and we let B = A .Otherwise let n be the least n where χ ( n ) = a n . From r ≈ x A itfollows easily that n is nonstandard. In particular, a n ∈ { , } holdsfor all standard n , hence, by Transfer, for all n ∈ N . Let B = { n ∈ N | a n = 1 } . Then B is standard and for all standard n ∈ N , n ∈ B iff a n = 1 iff χ ( n ) = 1 iff n ∈ A . (cid:3) As explained in the Introduction, Standardization over uncountablesets such as R , even for very simple formulas, implies the existence ofnonprincipal ultrafilters over N , and so it cannot be proved in SPOT (consider a standard set U such that ∀ st X ( X ∈ U ←→ X ⊆ N ∧ ν ∈ X ), where ν is a nonstandard integer). But we need to be able to provethe existence of various subsets of R and functions from R to R thatarise in Calculus and may be defined in terms of infinitesimals. Unlikethe undesirable example above, such uses generally involve Standard-ization for formulas with standard parameters. NFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 9 An st - ∈ -formula Φ( v , . . . , v n ) is ∆ st if it is of the form Q st x . . . Q st m x m ψ ( x , . . . , x m , v , . . . , v n )where ψ is an ∈ -formula and Q stands for ∃ or ∀ . Lemma 2.5.
Let Φ( v , . . . , v n ) be a ∆ st formula with standard param-eters. Then SPOT proves: ∀ st S ∃ st P ∀ st v , . . . , v n (cid:0) h v , . . . , v n i ∈ P ←→ h v , . . . , v n i ∈ S ∧ Φ( v , . . . , v n ) (cid:1) . Proof.
Let Φ( v . . . , v n ) be Q st x . . . Q st m x m ψ ( x , . . . , x m , v , . . . , v n ) and φ ( v . . . , v n ) be Q x . . . Q m x m ψ ( x , . . . , x m , v , . . . , v n ). By Transfer,Φ( v . . . , v n ) ←→ φ ( v . . . , v n ) for all standard v . . . , v n . The set P = {h v , . . . , v n i ∈ S | φ ( v , . . . , v n ) } exists by the Separation Principle of ZF , and has the required property. (cid:3) This result has twofold importance: • The meaning of every predicate that for standard inputs is de-fined by a Q st x . . . Q st m x m ψ formula with standard parametersis automatically extended to all inputs, where it it given by the ∈ -formula Q x . . . Q m x m ψ . • Standardization holds for all ∈ -formulas with additional predi-cate symbols, as long as all these additional predicates are de-fined by ∆ st formulas with standard parameters.In BST all st - ∈ -formulas are equivalent to ∆ st formulas (see Kanoveiand Reeken [19], Theorem 3.2.3). In SPOT the equivalence is true onlyfor certain classes of formulas, but they include the definitions of allthe basic concepts of Calculus and much beyond.We recall that h ∈ R is infinitesimal iff 0 < | h | < n holds for allstandard n ∈ N , n >
0. We use ∀ in and ∃ in for quantifiers ranging overinfinitesimals and 0. The basic concepts of Calculus have infinitesimaldefinitions that involve a single alternation of such quantifiers.The following proposition strengthens a result in Vopˇenka [35], p. 148.The variables x, y range over R and m, n, ℓ range over N \ { } . Proposition 2.6. In SPOT the following is true: Let φ ( x, y ) be an ∈ -formula with arbitrary parameters. Then ∀ in h ∃ in k φ ( h, k ) ←→∀ st m ∃ st n ∀ x [ | x | < /n → ∃ y ( | y | < /m ∧ φ ( x, y )) ] . By duality, we also have: ∃ in h ∀ in k φ ( h, k ) ←→∃ st m ∀ st n ∃ x [ | x | < /n ∧ ∀ y ( | y | < /m → φ ( x, y )) ] . Proof.
The formula ∀ in h ∃ in k φ ( h, k ) means: ∀ x [ ∀ st n ( | x | < /n ) → ∃ y ∀ st m ( | y | < /m ∧ φ ( x, y )) ] , where we assume that the variables m, n do not occur freely in φ ( x, y ).Using Countable Idealization (Lemma 2.2), we rewrite this as ∀ x [ ∀ st n ( | x | < /n ) → ∀ st m ∃ y ∀ ℓ ≤ m ( | y | < /ℓ ∧ φ ( x, y )) ] . We now use the observation that ∀ ℓ ≤ m ( | y | < /ℓ ) is equivalentto | y | < /m , and the rules ( α → ∀ st v β ) ←→ ∀ st v ( α → β ) and( ∀ st v β → α ) ←→ ∃ st v ( β → α ), valid assuming that v is not free in α (note that Transfer implies ∃ n st ( n )). This enables us to rewrite thepreceding formula as follows: ∀ x ∀ st m ∃ st n [ | x | < /n → ∃ y ( | y | < /m ∧ φ ( x, y )) ] . After exchanging the order of the first two universal quantifiers, weobtain the formula ∀ st m ∀ x ∃ st n [ | x | < /n → ∃ y | y | < /m ∧ φ ( x, y )) ] , to which we apply (the dual form of) Countable Idealization to get ∀ st m ∃ st n ∀ x ∃ ℓ ≤ n [ | x | < /ℓ → ∃ y ( | y | < /m ∧ φ ( x, y )) ] . After rewriting ∃ ℓ ≤ n [ | x | < /ℓ → . . . ] as [ ∀ ℓ ≤ n ( | x | < /ℓ ) → . . . ]and replacing ∀ ℓ ≤ n ( | x | < /ℓ ) by | x | < /n , we obtain ∀ st m ∃ st n ∀ x [ | x | < /n → ∃ y ( | y | < /m ∧ φ ( x, y )) ] , proving the proposition. (cid:3) Mathematics in SPOT.
We give some examples to illustratethe how infinitesimal analysis works in
SPOT . Example 2.7. If F is a standard real-valued function on an openinterval ( a, b ) in R and a, b, c, d are standard real numbers with c ∈ ( a, b ), we can define(1) F ′ ( c ) = d ←→ ∀ in h ∃ in k (cid:18) h = 0 → F ( c + h ) − F ( c ) h = d + k (cid:19) . Let Φ(
F, c, d ) be the formula on the right side of the equivalence in (1).Lemma 2.5 establishes that the formula Φ is equivalent to a ∆ st for-mula, and φ ( F, c, d ) provided by the proof of Lemma 2.5 is easily seento be equivalent to the standard ε - δ definition of derivative. For anystandard F , the set F ′ = {h c, d i | φ ( F, c, d ) } is standard; it is thederivative (function) F ′ of F .Proposition 2.6 generalizes straightforwardly to all formulas thathave the form A h . . . A h n E k . . . E k m φ ( h , . . . , k , . . . , v , . . . ) or E h . . . E h n A k . . . A k m φ ( h , . . . , k , . . . , v , . . . ) where each A is either ∀ or ∀ in , and each E is either ∃ or ∃ in . All such formulas are equivalentto δ st formulas. NFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 11
Formulas of the form Q h . . . Q n h n φ ( h , . . . , h n , v , . . . , v k ) whereeach Q is either ∀ or ∀ in or ∃ or ∃ in , but all quantifiers over infinitesimalsare of the same kind (all existential or all universal), are also ∆ st . Asan example, ∃ in h ∀ y ∃ in k φ ( h, k, x, y ) is equivalent to ∃ h ∀ y ∃ k ∀ st m ∀ st n ( | h | < /m ∧ | k | < /n ∧ φ ( h, k, x, y )) . The two quantifiers over standard elements of N can be replaced by asingle one: ∃ h ∀ y ∃ k ∀ st m ( | h | < /m ∧ | k | < /m ∧ φ ( h, k, x, y )) , and then moved to the front using Countable Idealization.Klein and Fraenkel proposed two benchmarks for a useful theory ofinfinitesimals (see Kanovei et al. [17]): • a proof of the Mean Value Theorem by infinitesimal techniques; • a definition of the definite integral in tems of infinitesimals.The theory SPOT easily meets these criteria. The usual nonstandardproof of the Mean Value Theorem (Robinson [26], Keisler [21, 22]) usesStandard Part and Transfer, and is easily carried out in
SPOT . Thefamiliar infinitesimal definition of the Riemann integral for standardbounded functions on a standard interval [ a, b ] also makes sense in
SPOT and can be expressed by a ∆ st formula. In the next examplewe outline a treatment inspired by Keisler’s use of hyperfinite Riemannsums in [22]. Example 2.8. Riemann Integral.
We fix a positive infinitesimal h and the corresponding “hyperfinitetime line” T = { t i | i ∈ Z } where t i = i · h . Let f be a standard real-valued function continuous on the standard interval [ a, b ]. Let i a , i b besuch that i a · h − h < a ≤ i a · h and i b · h < b ≤ i b · h + h . Then(2) Z ba f ( t ) dt = sh (cid:0) Σ i b i = i a f ( t i ) · h (cid:1) . It is easy to show that the value of the integral does not depend on thechoice of h . We thus have, for standard f, a, b, r : R ba f ( t ) dt = r iff ∀ in h ∃ in k (cid:0) Σ i b i = i a f ( t i ) · h = r + k (cid:1) iff ∃ in h ∃ in k (cid:0) Σ i b i = i a f ( t i ) · h = r + k (cid:1) . The formulas are of the form
A E and
E E respectively, and thereforeequivalent to ∆ st formulas.The approach generalizes easily to the Riemann integral of boundedfunctions on [ a, b ]. We say that T h = h t ′ i i i b i = i a is an h - tagging on [ a, b ] if i · h ≤ t ′ i ≤ ( i + 1) · h for all i = i a , . . . , i b − i b · h ≤ t ′ i b ≤ b . Thenfor standard f, a, b, r • f is Riemann integrable on [ a, b ] and R ba f ( x ) dx = r iff • ∀ in h ∀T h ∃ in k (cid:0) Σ i b i = i a f ( t ′ i ) · h = r + k (cid:1) iff • ∃ in h ∀T h ∃ in k (cid:0) Σ i b i = i a f ( t ′ i ) · h = r + k (cid:1) . These formulas are again equivalent to ∆ st formulas (the first one is ofthe form A A E and in the second one both quantifiers over standardsets are existential).The tools available in
SPOT enable nonstandard definitions andproofs in parts of mathematics that go well beyond the Calculus.
Example 2.9. Fr´echet Derivative.
Given standard normed vectorspaces V and W , a standard open subset U of V , a standard function f : U → W , a standard bounded linear operator A : V → W and astandard x ∈ U ; A is the Fr´echet derivative of f at x ∈ U iff ∀ z ∀ in h ∃ in k (cid:18) k z k V = h > → k f ( x + z ) − f ( x ) − A · z k W k z k V = k (cid:19) . This definition is equivalent to a ∆ st formula.In Section 6 we show that Standardization for arbitrary formulaswith standard parameters can be added to SPOT and the resultingtheory is still conservative over ZF . This result enables one to disposeof any concerns about the form of the defining formula.3. Theory
SCOT and Lebesgue measure
We recall (see Section 1.3) that
SCOT is SPOT + SN + CC , wherethe principle of Countable st - ∈ -Choice postulates the following: CC Let φ ( u, v ) be an st - ∈ -formula with arbitrary parameters. Then ∀ st n ∈ N ∃ x φ ( n, x ) → ∃ f ( f is a function ∧ ∀ st n ∈ N φ ( n, f ( n )) . The set N can be replaced by any standard countable set A . Weconsider also the principle of Countable Standardization. SC (Countable Standardization) Let ψ ( v ) be an st - ∈ -formula witharbitrary parameters. Then ∃ st S ∀ st n ( n ∈ S ←→ n ∈ N ∧ ψ ( n )) . Lemma 3.1.
The theory
SPOT proves that CC implies SC .Proof. Let φ ( n, x ) be the formula “( ψ ( n ) ∧ x = 0) ∨ ( ¬ ψ ( n ) ∧ x = 1)”.If f is a function provided by CC , let A = { n ∈ N | f ( n ) = 0 } . By SP there is a standard set S such that, for all standard n ∈ N , n ∈ S iff n ∈ A iff ψ ( n ) holds. (cid:3) NFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 13 CC st is the following principle: ∀ st n ∈ N ∃ st x φ ( n, x ) → ∃ st F ( F is a function ∧ ∀ st n ∈ N φ ( n, F ( n )) . CC st R is obtained from CC st by restricting the range of the variable x to R . Lemma 3.2.
The theory
SPOT + CC proves CC st R .Proof. First use the principle CC to obtain a function f : N → R suchthat ∀ st n ∈ N ( f ( n ) ∈ R ∧ st ( f ( n )) ∧ φ ( n, f ( n )) . Next define a relation r ⊆ N × N by h n, m i ∈ r iff m ∈ f ( n ). By SC there is a standard R ⊆ N × N such that h n, m i ∈ R iff h n, m i ∈ r holds for all standard h n, m i . Now define F : N → R by F ( n ) = { m | h n, m i ∈ R } . Thefunction F is standard and, for every standard n , the sets F ( n ) and f ( n ) have the same standard elements. As they are both standard, itfollows by Transfer that F ( n ) = f ( n ). (cid:3) The full principle CC st can conservatively be added to SCOT ; seeProposition 5.6.A useful consequence of SC is the ability to carry out external in-duction. Lemma 3.3. (External Induction) Let φ ( v ) be an st - ∈ -formula witharbitrary parameters. Then SPOT + SC proves the following: [ φ (0) ∧ ∀ st n ∈ N ( φ ( n ) → φ ( n + 1)) → ∀ st n φ ( n ) ] . Proof. SC yields a standard set S ⊆ N such that ∀ st n ∈ N ( n ∈ S ←→ φ ( n )) . We have 0 ∈ S and ∀ st n ∈ N ( n ∈ S → n + 1 ∈ S ). Then ∀ n ∈ N ( n ∈ S → n + 1 ∈ S ) by Transfer, and S = N by induction.Hence ∀ st n ∈ N φ ( n ) holds. (cid:3) In Example 3.6 it is convenient to use the language of external collec-tions. Let φ ( v ) be an st - ∈ -formula with arbitrary parameters. We usedashed curly braces to denote the external collection x ∈ A | φ ( x ) .We emphasize that this is merely a matter of convenience; writing z ∈ x ∈ A | φ ( x ) is just another notation for φ ( z ).Standardization in BST implies the existence of a standard set S such that ∀ st z ( z ∈ S ←→ z ∈ x ∈ A | φ ( x ) ). We do not haveStandardization over uncountable sets is SCOT , but one importantcase can be proved.
Lemma 3.4.
Let φ ( v ) be an st - ∈ -formula with arbitrary parameters.Then SCOT proves that inf st r ∈ R | φ ( r ) exists. The notation indicates the greatest standard s ∈ R such that s ≤ r for all standard r with the property φ ( r ) (+ ∞ if there is no such r ). Proof.
Consider S = q ∈ Q | ∃ st r ∈ R ( q ≥ r ∧ φ ( r )) . As Q iscountable, the principle SC implies that there is a standard set S suchthat ∀ q ∈ Q ( q ∈ S ←→ q ∈ S ). Therefore inf S exists (+ ∞ if S = ∅ )and it is what is meant above by inf st r ∈ R | φ ( r ) . (cid:3) We give two examples of mathematics in
SCOT . Example 3.5. Peano Existence Theorem.
Peano’s Theorem as-serts that every first-order differential equation of the form y ′ = f ( x, y )has a solution (not necessarily a unique one) satisfying the initial con-dition y (0) = 0, under the assumption that f is continuous in a neigh-borhood of h , i . The infinitesimal proof begins by constructing thesequences x = 0 , x k +1 = x k + h where h > y = 0 , y k +1 = y k + h · f ( x k , y k ) . One then shows that there is N ∈ N such that x k , y k are definedfor all k ≤ N , a = st ( x N ) >
0, and for some standard
M > | y k | ≤ M · a holds for all k ≤ N . The desired solution is a stan-dard function Y : [0 , a ] → R such that for all standard x ∈ [0 , a ],if x ≈ x k , then Y ( x ) ≈ y k . On the face of it one needs Standard-ization over R to obtain this function, but SC suffices. Consider thecountable set A = ( Q × Q ) ∩ ([0 , a ] × [ − M · a, M · a ]). By SC , thereis a standard Z ⊆ A such that for all standard h x, y i , h x, y i ∈ Z iff ∃ k ≤ N ( x ≈ x k ∧ ( y ≈ y k ∨ y ≥ y k ). Define a standard function Y on Q ∩ [0 , a ] by Y ( x ) = inf { y | h x, y i ∈ Z } . It is easy to verify that Y is continuous on Q ∩ [0 , a ] and that its extension Y to a continuousfunction on [0 , a ] is the desired solution. Example 3.6. Lebesgue measure.
In a seminal paper [24] Loebintroduced measures on the external power set of R ∗ which becameknown as Loeb measures, and used them to construct the Lebesguemeasure on R . Substantial use of external collections is outside thescope of this paper, but it is possible to eliminate the intermediate stepand give an infinitesimal definition `a la Loeb of the Lebesgue measurein internal set theory. We outline here how to construct the Lebesgueouter measure on R in SCOT .Let T be a hyperfinite time line (see Example 2.8) and let E ⊆ R bestandard. A finite set A ⊆ T covers E if ∀ t ∈ T ( ∃ st x ∈ E ( t ≈ x ) → t ∈ A ) . NFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 15
We define µµµ by setting(3) µµµ ( E ) = inf st r ∈ R | r ≈ | A | · h for some A that covers E .
The collection whose infimum needs to be taken is external, but theexistence of the infimum is justified by Lemma 3.4. It is easy to seethat the value of µµµ ( E ) is independent of the choice of the infinitesimal h in the definition of T . Thus the external function µµµ can be defined forstandard E ⊆ P ( R ) by an st - ∈ -formula with no parameters (prefacethe formula on the right side of (3) by ∀ in h or ∃ in h ). The principle SN yields a standard function m on P ( R ) such that m ( E ) = µµµ ( E ) for allstandard E ⊆ R . We prove that m is σ -subadditive.Let E = S ∞ n =0 E n where E and the sequence h E n | n ∈ N i arestandard. If Σ ∞ n =0 m ( E n ) = + ∞ the claim is trivial, so we assumethat m ( E n ) = r n ∈ R for all n . Fix a standard ε >
0. For everystandard n ∈ N there exists A such that φ ( n, A ): “ A covers E n ∧| A | · h < r n + ε/ n +1 ” holds. By Countable st - ∈ -Choice there is asequence h A n | n ∈ N i such that for all standard n φ ( n, A n ) holds. ByCountable Idealization (“Overspill”) there is a nonstandard ν ∈ N suchthat | A n | · h < r n + ε/ n +1 holds for all n ≤ ν . We let A = S νn =0 A n .Clearly A is finite and covers E . Thus for r = sh ( | A | · h ) we obtain m ( E ) ≤ r and | A | · h ≤ Σ νn =0 | A n | · h < Σ νn =0 r n + ε. Since the sequence Σ ∞ n =0 r n converges, we have sh (Σ νn =0 r n ) = Σ ∞ n =0 r n and m ( A ) ≤ Σ ∞ n =0 r n + ε . As this is true for all standard ε >
0, weconclude that m ( E ) ≤ Σ ∞ n =0 r n = Σ ∞ n =0 m ( E n ). (cid:3) For closed intervals [ a, b ], m ([ a, b ]) = b − a : Compactness of [ a, b ]implies that ∀ t ∈ T ∩ [ a, b ] ∃ st x ∈ E ( t ≈ x ). Thus if A covers [ a, b ]then A ⊇ T ∩ [ a, b ]; and for A = T ∩ [ a, b ] one sees easily that | A | · h ≈ ( b − a ). With more work, one can show that m ( E ) coincides with theconventionally defined Lebesgue outer measure of E for all standard E ⊆ R . See Hrbacek [13] Section 3 for more details and other equivalentnonstandard definitions of the Lebesgue outer measure. One can defineLebesgue measurable sets from m in the usual way. One can also defineLebesgue inner measure for standard E by µ − ( E ) = sup st r ∈ R | r ≈ | A | · h for some A such that ∀ t ∈ T ( t ∈ A → ∃ st x ∈ E ( t ≈ x )) In [13] Remark (3) on page 22 it is erroneously claimed that the statement m ( A ) = r is equivalent to an internal formula. The existence of the function m there followsfrom Standardization, just as in the case of m above. and prove that a standard bounded E ⊆ R is Lebesgue measurable iff m ( E ) = m − ( E ), and the common value is the Lebesgue measure of E ;see Hrbacek [14].4. Conservativity of
SPOT over
ZFTheorem 4.1.
The theory
SPOT is a conservative extension of ZF :If θ is an ∈ -sentence, then ( SPOT ⊢ θ ) implies that ( ZF ⊢ θ ) . Theorem 4.1 = Theorem A is an immediate consequence of the fol-lowing proposition. Proposition 4.2.
Every countable model M = ( M, ∈ M ) of ZF has acountable extension M ∗ = ( M ∗ , ∈ ∗ , st ) to a model of SPOT in which M is the class of all standard sets.Proof of Theorem 4.1 . Suppose SPOT ⊢ θ but ZF θ , where θ is an ∈ -sentence. Then the theory ZF + ¬ θ is consistent, therefore ithas a countable model M , by G¨odel’s Completeness Theorem. UsingProposition 4.2 one obtains its extension M ∗ (cid:15) SPOT , so in particular M ∗ (cid:15) θ and, by Transfer in M ∗ , M (cid:15) θ . This is a contradiction. (cid:3) The rest of this section is devoted to the proof of Proposition 4.2.4.1.
Forcing according to Enayat and Spector.
We combine theforcing notion used by Enayat [7] to construct end extensions of modelsof arithmetic, with the one used by Spector in [32] to produce extendedultrapowers of models M of ZF by an ultrafilter U ∈ M .In this subsection we work in ZF , define our forcing notion and proveits basic properties. The next subsection deals with generic extensionsof countable models of ZF and the resulting extended ultrapowers.The general reference to forcing and generic models in set theory isJech [16].The set of all natural numbers is denoted N and letters m, n, k, ℓ arereserved for variables ranging over N . The index set over which theultrapowers will eventually be constructed is denoted I . In this sectionwe assume I = N . A subset p of N is called unbounded if ∀ m ∃ n ∈ p ( n ≥ m ) and bounded if it is not unbounded. Of course unboundedis the same as infinite, and bounded is the same as finite. We usethis terminology with a view to Section 7, where the construction isgeneralized to I = P fin ( S ) for any infinite set S . The notation ∀ aa i ∈ p ( for almost all i ∈ p ) means ∀ i ∈ p \ c for some bounded c .As usual, the symbol V denotes the universe of all sets, and V α ( α ranges over ordinals) are the ranks of the von Neumann cumulativehierarchy. We let F be the class of all functions with domain I . Thenotation ∅ k stands for the k -tuple h∅ , . . . , ∅i . NFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 17
Definition 4.3.
Let P = { p ⊆ I | p is unbounded } . For p, p ′ ∈ P wesay that p ′ extends p (notation: p ′ ≤ p ) iff p ′ ⊆ p .Let Q = { q ∈ F | ∃ k ∈ N ∀ i ∈ I ( q ( i ) ⊆ V k ∧ q ( i ) = ∅ ) } . Thenumber k is the rank of q . We note that Q is a proper class. We let¯1 = q where q ( i ) = {∅} for all i ∈ I ; ¯1 is the only q ∈ Q of rank 0.The forcing notion H is defined as follows: H = P × Q and h p ′ , q ′ i ∈ H extends h p, q i ∈ H (notation: h p ′ , q ′ i ≤ h p, q i ) iff p ′ extends p , rank q ′ = k ′ ≥ k = rank q , and for almost all i ∈ p ′ and all h x , . . . , x k ′ − i ∈ q ′ ( i ), h x , . . . , x k − i ∈ q ( i ). Every h p, q i ∈ H extends h p, ¯1 i .The poset P is used to force a generic filter over I as in Enayat [7],and H forces an extended ultrapower of V by the generic filter U forcedby P . It is a modification of the forcing notion from Spector [32], withthe difference that in [32] U is not forced but assumed to be a givenultrafilter in V .A set D ⊆ P is dense in P if for every p ∈ P there is p ′ ∈ D such that p ′ extends p . We note that for any set S ⊆ I , the set D S = { p ∈ P | p ⊆ S ∨ p ⊆ I \ S } is dense in P .Similarly, a class E ⊆ H is dense in H if for every h p, q i ∈ H thereis h p ′ , q ′ i ∈ E such that h p ′ , q ′ i ≤ h p, q i .The forcing language L has a constant symbol ˇ z for every z ∈ V (which we identify with z when no confusion threatens), and a constantsymbol ˙ G n for each n ∈ N . Given an ∈ formula φ ( w , . . . , w r , v , . . . , v s ),we define the forcing relation h p, q i (cid:13) φ (ˇ z , . . . , ˇ z r , ˙ G n , . . . , ˙ G n s ) for h p, q i ∈ H by meta-induction on the logical complexity of φ . We use ¬ , ∧ and ∃ as primitives and consider the other logical connectivesand quantifiers as defined in terms of these. Usually, we suppress theexplicit listing in φ of the constant symbols ˇ z for the elements of V . Definition 4.4. (Forcing relation.)(1) h p, q i (cid:13) ˇ z = ˇ z iff z = z .(2) h p, q i (cid:13) ˇ z ∈ ˇ z iff z ∈ z .(3) h p, q i (cid:13) ˙ G n = ˙ G n iff rank h p, q i = k > n , n and ∀ aa i ∈ p ∀h x , . . . , x k − i ∈ q ( i ) ( x n = x n ) . (4) h p, q i (cid:13) ˙ G n ∈ ˙ G n iff rank h p, q i = k > n , n and ∀ aa i ∈ p ∀h x , . . . , x k − i ∈ q ( i ) ( x n ∈ x n ) . (5) h p, q i (cid:13) ˙ G n = ˇ z iff h p, q i (cid:13) ˇ z = ˙ G n iff rank h p, q i = k > n and ∀ aa i ∈ p ∀h x , . . . , x k − i ∈ q ( i ) ( x n = z ) . (6) h p, q i (cid:13) ˇ z ∈ ˙ G n iff rank h p, q i = k > n and ∀ aa i ∈ p ∀h x , . . . , x k − i ∈ q ( i ) ( z ∈ x n ) . (7) h p, q i (cid:13) ˙ G n ∈ ˇ z iff rank h p, q i = k > n and ∀ aa i ∈ p ∀h x , . . . , x k − i ∈ q ( i ) ( x n ∈ z ) . (8) h p, q i (cid:13) ¬ φ ( ˙ G n , . . . , ˙ G n s ) iff rank q = k > n , . . . , n s and thereis no h p ′ , q ′ i extending h p, q i such that h p ′ , q ′ i (cid:13) φ ( ˙ G n , . . . , ˙ G n s ) . (9) h p, q i (cid:13) ( φ ∧ ψ )( ˙ G n , . . . , ˙ G n s ) iff h p, q i (cid:13) φ ( ˙ G n , . . . , ˙ G n s ) and h p, q i (cid:13) ψ ( ˙ G n , . . . , ˙ G n s ).(10) h p, q i (cid:13) ∃ v ψ ( ˙ G n , . . . , ˙ G n s , v ) iff rank q = k > n , . . . , n s andfor every h p ′ , q ′ i extending h p, q i there exist h p ′′ , q ′′ i extending h p ′ , q ′ i and m ∈ N such that h p ′′ , q ′′ i (cid:13) ψ ( ˙ G n , . . . , ˙ G n s , ˙ G m ) . Lemma 4.5. (Basic properties of forcing)(1) If h p, q i (cid:13) φ and h p ′ , q ′ i extends h p, q i , then h p ′ , q ′ i (cid:13) φ .(2) h p, q i (cid:13) φ and h p, q i (cid:13) ¬ φ is impossible.(3) Every h p, q i extends to h p ′ , q ′ i such that h p ′ , q ′ i (cid:13) φ or h p ′ , q ′ i (cid:13) ¬ φ .(4) If h p, q i (cid:13) φ and p ′ \ p is bounded, then h p ′ , q i (cid:13) φ .Proof. (1) - (3) are immediate from the definition of forcing and (4)can be proved by induction on the complexity of φ . (cid:3) The following proposition establishes a relationship between thisforcing and ultrapowers.
Proposition 4.6. (“ Lo´s’s Theorem” ) Let φ ( v , . . . , v s ) be an ∈ -formulawith parameters from V .Then h p, q i (cid:13) φ ( ˙ G n , . . . , ˙ G n s ) iff rank q = k > n , . . . , n s and ∀ aa i ∈ p ∀h x , . . . , x k − i ∈ q ( i ) φ ( x n , . . . , x n s ) . Proof.
For atomic formulas (cases (1) - (7)) the claim is immediatefrom the definition. Case (9) is also trivial (union of two bounded setsis bounded).Case (8): Let c = { i ∈ p | ∀h x , . . . , x k − i ∈ q ( i ) ¬ φ ( x n , . . . , x n s ) } .We need to prove that h p, q i (cid:13) ¬ φ , iff p \ c is bounded.Assume that h p, q i (cid:13) ¬ φ and p \ c is unbounded. We let p ′ = p \ c and q ′ ( i ) = {h x , . . . , x k − i ∈ q ( i ) | φ ( x n , . . . , x n s ) } for i ∈ p ′ , q ′ ( i ) = {∅ k } for i ∈ I \ p ′ . Then h p ′ , q ′ i ∈ H extends h p, q i and, by the inductiveassumption, h p ′ , q ′ i (cid:13) φ , a contradiction.Conversely, assume h p, q i ¬ φ and p \ c is bounded. Then thereis h p ′ , q ′ i of rank k ′ extending h p, q i such that h p ′ , q ′ i (cid:13) φ . By theinductive assumption, there is a bounded set d such that ∀ i ∈ ( p ′ \ d ) ∀h x , . . . , x k ′ − i ∈ q ′ ( i ) φ ( x n , . . . , x n s ) . But ( p \ c ) ∪ d is a bounded set, so there exist i ∈ ( p ′ ∩ c ) \ d . Forsuch i and h x , . . . , x k ′ − i ∈ q ′ ( i ) one has both ¬ φ ( x n , . . . , x n s ) and φ ( x n , . . . , x n s ), a contradiction. NFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 19
Case (10):Let c = { i ∈ p | ∀h x , . . . , x k − i ∈ q ( i ) ∃ v ψ ( x n , . . . , x n s , v ) } . We needto prove that h p, q i (cid:13) ∃ v ψ iff p \ c is bounded.Assume that h p, q i (cid:13) ∃ v ψ and p \ c is unbounded. We let p ′ = p \ c and q ′ ( i ) = {h x , . . . , x k − i ∈ q ( i ) | ¬ ∃ v ψ ( x n , . . . , x n s , v ) } for i ∈ p ′ ; q ′ ( i ) = {∅ k } for i ∈ I \ p ′ . Then h p ′ , q ′ i extends h p, q i and, by thedefinition of (cid:13) , there exist h p ′′ , q ′′ i extending h p ′ , q ′ i with rank q ′′ = k ′′ ,and m < k ′′ such that h p ′′ , q ′′ i (cid:13) ψ ( ˙ G n , . . . , ˙ G n s , ˙ G m ). By the inductiveassumption, there is a bounded set d such that ∀ i ∈ ( p ′′ \ d ) ∀h x , . . . , x k ′′ − i ∈ q ′′ ( i ) ψ ( x n , . . . x n s , x m ) . Hence ∀ i ∈ ( p ′′ \ d ) ∀h x , . . . , x k ′′ − i ∈ q ′′ ( i ) ∃ v ψ ( x n , . . . x n s , v ) . But i ∈ p ′′ \ d implies i ∈ p ′ ; this contradicts the definition of q ′ .Assume that p \ c is bounded. By the Reflection Principle in ZF there is a least von Neumann rank V α such that for all i ∈ c and all h x , . . . , x k − i ∈ q ( i ) there exists v ∈ V α such that ψ ( x n , . . . x n s , v ).Let h p ′ , q ′ i be any condition extending h p, q i and let k ′ = rank q ′ . Welet p ′′ = p ′ ∩ c and q ′′ ( i ) = {h x , . . . , x k ′ − , x k ′ i |h x , . . . , x k ′ − i ∈ q ′ ( i ) ∧ ψ ( x n , . . . x n s , x k ′ ) ∧ x k ′ ∈ V α } for i ∈ p ′′ , q ′′ ( i ) = {∅ k ′ } otherwise. Then h p ′′ , q ′′ i extends h p ′ , q ′ i and, bythe inductive assumption, h p ′′ , q ′′ i (cid:13) ψ ( ˙ G n , . . . , ˙ G n s , ˙ G k ′ ). This provesthat h p, q i (cid:13) ∃ v ψ . (cid:3) We observe that if q is in Q and ℓ < k = rank q , then q ↾ ℓ definedby ( q ↾ ℓ )( i ) = {h x , . . . , x ℓ − i | ∃ x ℓ , . . . , x k − h x , . . . , x k − i ∈ q ( i ) } isin Q . Corollary 4.7. If rank q = k > n , . . . , n s , h p ′ , q ′ i extends h p, q i and h p ′ , q ′ i (cid:13) φ ( ˙ G n , . . . , ˙ G n s ) , then h p ′ , q ′ ↾ k i extends h p, q i and h p ′ , q ′ ↾ k i (cid:13) φ ( ˙ G n , . . . , ˙ G n s ) . Lemma 4.8.
Let z ∈ V . For every h p, q i there exist h p, q ′ i extending h p, q i and m < k ′ = rank q ′ such that h p, q ′ i (cid:13) ˇ z = ˙ G m .Proof. Let q ′ ( i ) = {h x , . . . , x k − , x k i | h x , . . . , x k − i ∈ q ( i ) ∧ x k = z } ,and m = k , k ′ = k + 1. (cid:3) We abbreviate the relation h p, q i (cid:13) ˇ z = ˙ G m as h p, q i (cid:13) z = ˙ G m , and h p, q i (cid:13) ˇ z ∈ ˙ G m as h p, q i (cid:13) z ∈ ˙ G m . We say that h p, q i decides φ if h p, q i (cid:13) φ or h p, q i (cid:13) ¬ φ . The following lemma is needed for the proofthat the extended ultrapower satisfies SP . Lemma 4.9.
For every h p, q i and m < k = rank q there is h p ′ , q ′ i thatextends h p, q i and is such that for every n ∈ N , h p ′ , q ′ i decides n ∈ ˙ G m .Proof. We first construct a sequence hh p n , q n i | n ∈ N i such that h p , q o i = h p, q i and for each n , rank q n = k , p n +1 ⊂ p n , h p n +1 , q n +1 i extends h p n , q n i , and h p n +1 , q n +1 i decides n ∈ ˙ G m .Given p n , let c = { i ∈ p n | ∀h x , . . . , x k − i ∈ q n ( i ) ( n ∈ x m ) } . If c is unbounded, we let p ′ n +1 = c and q n +1 = q n . Otherwise p n \ c isunbounded and we let p ′ n +1 = p n \ c and q n +1 ( i ) = {h x , . . . , x k − i ∈ q n ( i ) | n / ∈ x m } for i ∈ p ′ n +1 , q n +1 ( i ) = {∅ k } otherwise. We obtain p n +1 from p ′ n +1 by omitting the least element of p ′ n +1 . Proposition 4.6implies that h p n +1 , q n +1 i decides n ∈ ˙ G m .Let i n be the least element of p n . We define h p ′ , q ′ i as follows: i ∈ p ′ iff i ∈ p n and q ′ ( i ) = q n ( i ), where i n ≤ i < i n +1 ; q ′ ( i ) = {∅ k } otherwise.It is clear from the construction and Proposition 4.6 that h p ′ , q ′ i ∈ H ,it extends h p, q i , and it decides n ∈ ˙ G m for every n ∈ N . (cid:3) Extended ultrapowers.
In this subsection we define the ex-tended ultrapower of a countable model of ZF by a generic filter U ,prove some fundamental properties of this structure, and conclude thatit is a model of SPOT .In this section we take Zermelo-Fraenkel set theory as our metathe-ory, but the proof employs very little of its powerful machinery. Sub-section 8.5 explains how the proof given below can be converted into afinitistic proof.We use ω for the set of natural numbers in the metatheory, and r, s as variables ranging over ω .Let M = ( M, ∈ M ) be a countable model of ZF . Concepts definedin Subsection 4.1 make sense in M and all results of 4.1 hold in M .When M is understood, we use the notation and terminology from 4.1for the concepts in the sense of M ; thus N for N M , “unbounded” for“unbounded in the sense of M ”, P for P M , (cid:13) for “ (cid:13) in the sense of M ”, etc. The model M need not be well-founded, and ω is isomorphicto an initial segment of N which may be proper. Definition 4.10.
U ⊆ M is a filter on P if(1) M (cid:15) “ p ∈ P ” for every p ∈ U ;(2) If p ∈ U and M (cid:15) “ p ′ ∈ P ∧ p extends p ′ ”, then p ′ ∈ U ;(3) For eny p , p ∈ U there is p ∈ U such that M (cid:15) “ p extends p ∧ p extends p ”.A filter U on P is M - generic if for every D ∈ M such that M (cid:15) “ D is dense in P ” there is p ∈ U for which p ∈ M D . NFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 21
Since P has only countably many dense subsets in M , M -genericfilters are easily constructed by recursion. Let h p r | r ∈ ω i be anenumeration of P and h D r | r ∈ ω i be an enumeration of all densesubsets of P in M . Let q = p and for each s ∈ ω let q s +1 = p r for the least r such that M (cid:15) “ p r extends q s ∧ p r ∈ D s ”.Then let U = { p ∈ M | M (cid:15) “ p ∈ P ∧ q s extends p ” for some s ∈ ω } . M -generic filters G ⊆ M × M on H are defined and constructedanalogously. Lemma 4.11. If G is an M -generic filter on H , then U = { p ∈ P |∃ q h p, q i ∈ G} is an M -generic filter on P .Proof. In M : if D is dense in P , then {h p, q i | p ∈ D ∧ q ∈ Q } is densein H . (cid:3) We now define the extended ultrapower of M by U ; we follow closelythe presentation in Spector [32].Let Ω = { m ∈ M | M (cid:15) “ m ∈ N ” } . We define binary relations = ∗ and ∈ ∗ on Ω as follows: m = ∗ n iff there exists h p, q i ∈ G such that rank q = k > m, n and h p, q i (cid:13) ˙ G m = ˙ G n ; m ∈ ∗ n iff there exists h p, q i ∈ G such that rank q = k > m, n and h p, q i (cid:13) ˙ G m ∈ ˙ G n .It is easily seen from the definition of forcing and Proposition 4.6that = ∗ is an equivalence relation on Ω, and a congruence relation withrespect to ∈ ∗ . We denote the equivalence class of m ∈ Ω in the relation= ∗ by G m , define G m ∈ ∗ G n iff m ∈ ∗ n , and let N = { G m | m ∈ Ω } .The extended ultrapower of M by U is the structure N = ( N, ∈ ∗ ).There is a natural embedding j of M into N defined as follows: ByLemma 4.8 for every z ∈ M there exist h p, q i ∈ G and m < rank q such that h p, q i (cid:13) z = ˙ G m . We let j ( z ) = G m and often identify j ( z )with z . It is easy to see that the definition is independent of the choiceof representative from G m , and that j is an embedding of M into N . Proposition 4.12. (The Fundamental Theorem of Extended Ultrapow-ers) Let φ ( v , . . . , v s ) be an ∈ -formula with parameters from M .If G n , . . . , G n s ∈ N , then the following statements are equivalent:(1) N (cid:15) φ ( G n , . . . , G n s ) .(2) There is some h p, q i ∈ G such that h p, q i (cid:13) φ ( ˙ G n , . . . , ˙ G n s ) holds in M .(3) There exists some h p, q i ∈ G with rank q = k > n , . . . , n s suchthat M (cid:15) ∀ i ∈ p ∀h x , . . . , x k − i ∈ q ( i ) φ ( x n , . . . , x n s ) . Proof.
Statement (3) is just a reformulation of (2) using Proposition 4.6plus the fact that if d is bounded, then h p, q i ∈ G implies h ( p \ d ) , q i ∈ G .(Observe that D = {h p ′ , q ′ i | h p ′ , q ′ i ≤ h p, q i ∧ p ′ ≤ ( p \ d ) } is densein h p, q i ; for the definition of “dense in” see the sentence precedingLemma 5.3.)The equivalence of (1) and (2) is the Forcing Theorem. It is provedas usual, by induction on the logical complexity of φ . The cases v = v and v ∈ v follow immediately from the definitions of = ∗ and ∈ ∗ , andthe conjunction is immediate from (3) in the definition of a filter.We consider next the case where φ is of the form ¬ ψ . First assumethat N (cid:15) ¬ ψ ( G n , . . . , G n s ). Lemma 4.5 (3), implies that there ex-ists h p, q i ∈ G such that either h p, q i (cid:13) ψ ( ˙ G n , . . . , ˙ G n s ) or h p, q i (cid:13) ¬ ψ ( ˙ G n , . . . , ˙ G n s ). In the first case N (cid:15) ψ ( G n , . . . , G n s ) by the induc-tive assumption; a contradiction. Hence h p, q i (cid:13) ¬ ψ ( ˙ G n , . . . , ˙ G n s ).Assume that N ¬ ψ ( G n , . . . , G n s ); then N (cid:15) ψ ( G n , . . . , G n s )and the inductive assumption yields h p ′ , q ′ i ∈ G such that h p ′ , q ′ i (cid:13) ψ ( G n , . . . , ˙ G n s ). There can thus be no h p, q i ∈ G such that h p, q i (cid:13) ¬ ψ ( ˙ G n , . . . , ˙ G n s ).Finally, we assume that φ ( u , . . . , v s ) is of the form ∃ w ψ ( u , . . . , v s , w ).If N (cid:15) φ ( G n , . . . , G n s ) then N (cid:15) ψ ( G n , . . . , G n s , G m ) for some m ∈ Ω. By the inductive assumption there is some h p, q i ∈ G such that h p, q i (cid:13) ψ ( ˙ G n , . . . , ˙ G n s , ˙ G m ) and hence, by the definition of forcing, h p, q i (cid:13) ∃ w ψ ( ˙ G n , . . . , ˙ G n s , w ).Conversely, if h p, q i ∈ G and h p, q i (cid:13) ∃ w ψ ( ˙ G n , . . . , ˙ G n s , w ), thenby the definition of forcing there are h p ′ , q ′ i ∈ G and m ∈ Ω suchthat h p ′ , q ′ i (cid:13) ψ ( ˙ G n , . . . , ˙ G n s , ˙ G m ). By the inductive assumption, N (cid:15) ψ ( G n , . . . , G n s , G m ) holds, and hence N (cid:15) φ ( G n , . . . , G n s ). (cid:3) Corollary 4.13.
The embedding j is an elementary embedding of M into N . Corollary 4.14.
The structure N satisfies ZF . Proposition 4.15.
The structure b N = ( N, ∈ ∗ , M ) satisfies the princi-ples of Transfer, Nontriviality and Standard Part.Proof. Transfer is Corollary 4.13.Working in M , for every h p, q i with rank q = k define q ′ of rank k + 1by q ′ ( i ) = {h x , . . . , x k − , x k i | h x , . . . , x k − i ∈ q ( i ) ∧ x k = i } andnote that h p, q ′ i extends h p, q i . By M -genericity of G some such h p, q ′ i belongs to G . It is easily seen from the Fundamental Theorem that G k is an integer in N and that N (cid:15) “ G k = n ” for all n ∈ Ω. Hence O holds in b N . NFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 23
It remains to prove the Standard Part principle. Let G m ∈ N . ByLemma 4.9 there is h p, q i ∈ G which decides n ∈ ˙ G m for all n ∈ Ω.The set E = { n ∈ Ω | h p, q i (cid:13) n ∈ ˙ G m } is definable in M , hencethereis e ∈ M such that M (cid:15) “ n ∈ e ” iff n ∈ E iff N (cid:15) “ n ∈ G m ”. Thus b N (cid:15) “ st ( e ) ∧ ∀ st ν ∈ N ( ν ∈ e ←→ ν ∈ G m )”. (cid:3) This proves Proposition 4.2, and hence Theorem A . (cid:3) Conservativity of
SCOT over ZF c In this section we show that if ZF is replaced by ZF c , the StandardPart principle can be strengthened to Countable st - ∈ -Choice.We recall that ZF c implies ACC ; this provides enough choice toprove that the ordinary ultrapower of a countable model M of ZF c by an M -generic filter U on P satisfies Lo´s’s Theorem and thus yieldsan elementary extension of M . A proof that every countable model M = ( M, ∈ M ) of ZF c has an extension to a model of SPOT + SC in which M is the class of all standard sets can be obtained by astraightforward adaptation of the arguments in Enayat’s paper [7], inparticular, of the proofs of Theorems B and C there. Essentially, allone has to do is replace countable models of second-order arithmeticby countable models of ZF c . We followed this approach in an earlyversion of the present paper.Another proof of conservativity of SCOT over ZF c was suggestedto us by Kanovei in a private communication. Its basic idea is to useforcing to add to M a mapping of ω onto R without adding any reals.In the resulting generic extension there are nonprincipal ultrafilters over N , and one can take an ultrapower of M by one of them to obtain anextension that satisfies SCOT . However, this method does not seemadequate for handling Idealization over uncountable sets in Section 7.We first outline the simplification of the forcing that is possible inthe presence of
ACC , and then use it to prove that, assuming M is acountable model of ZF c , the structure b N from Proposition 4.15 satisfiesalso CC and SN . The proof can be viewed as a warm-up for similarbut more complex arguments of the following sections.We work in Zermelo-Fraenkel set theory with ADC added. Given I = N and q ∈ Q of rank k , ACC guarantees the existence of afunction f such that ∀ i ∈ I ( f ( i ) ∈ q ( i )); let b q ( i ) = { f ( i ) } where f ( i ) = h f ( i ) , . . . , f k − ( i ) i . For any h p, q i ∈ H the condition h p, b q i extends h p, q i . We could replace Q by e Q = { e q ∈ F | ∃ k ∀ i ∈ I ( e q ( i ) ∈ V k ) } . But there is no need at all for the symbols ˙ G k , k ∈ N , and Spector’s component Q of the forcing notion H , if our forcing languageallows names for all f ∈ F (see below for details).On the other hand, Standardization and Countable st - ∈ -Choice, un-like Transfer and Idealization, deal with st - ∈ -formulas, so we need togeneralize our definition of the forcing relation to such formulas.The forcing notion we use in this section is P . The forcing language e L has a constant symbol ˇ f for every function f ∈ F (the check is usuallysuppressed). Forcing is defined for arbitrary st - ∈ -formulas. Only thefollowing clauses in the definiton of the forcing relation are necessary: Definition 5.1. (Simplified forcing.)(1’) p (cid:13) f = f iff ∀ aa i ∈ p ( f ( i ) = f ( i )) . (2’) p (cid:13) f ∈ f iff ∀ aa i ∈ p ( f ( i ) ∈ f ( i )) . (8) p (cid:13) ¬ φ iff there is no p ′ extending p such that p ′ (cid:13) φ. (9) p (cid:13) ( φ ∧ ψ ) iff p (cid:13) φ and p (cid:13) ψ .(10’) p (cid:13) ∃ v ψ iff for every p ′ extending p there exist p ′′ extending p ′ and a function f ∈ F such that p ′′ (cid:13) ψ ( f ) . (11) h p, q i (cid:13) st ( f ) iff ∃ x ∀ aa i ∈ p ( f ( i ) = x ) iff ∀ aa i, i ′ ∈ p ( f ( i ) = f ( i ′ )).The basic properties of forcing from Lemma 4.5 remain valid, butProposition 4.6 (“ Lo´s’s Theorem”) of course holds only for ∈ -formulas,in the form p (cid:13) φ ( f , . . . , f r ) iff ∀ aa i ∈ p φ ( f ( i ) , . . . , f r ( i )) . Let now M be a countable model of ZF c , U an M -generic filter on P , and N = ( N, ∈ ∗ ) the ultrapower of M by U . If M (cid:15) “ f ∈ F ”, then[ f ] U is the equivalence class of f modulo U . We identify x ∈ M with[ c x ] U where c x is the constant function on I with value x in the senseof M , and let b N = ( N, ∈ ∗ , M ).Proposition 4.12 takes the following form. Proposition 5.2.
Let φ ( u , . . . , u r ) be an st - ∈ -formula with parame-ters from M . If M (cid:15) “ f , . . . , f r ∈ F ”, then the following statementsare equivalent:(1) b N (cid:15) φ ([ f ] U , . . . , [ f r ] U ) .(2) There is some p ∈ U such that p (cid:13) φ ( f , . . . , f r ) holds in M . Corollaries 4.13 and 4.14 and Proposition 4.15 remain valid in thismodified setting. For ∈ -formulas Proposition 5.2 is just a fancy way tostate the ordinary Lo´s’s Theorem, but for st - ∈ -formulas it provides auseful handle on the behavior of b N .We need the following corollary (which can also be proved more te-diously directly from the definition of the forcing relation). The state-ment “ D ⊆ P is dense in p ” means that ∀ p ′′ ≤ p ∃ p ′ ≤ p ′′ ( p ′ ∈ D ). NFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 25
Lemma 5.3. If p (cid:13) ∀ st m ∈ N φ ( m ) , then the set D m = { p ′ ∈ P | p ′ (cid:13) φ ( m ) } is dense in p for every m ∈ N .Proof. We show that the claim holds in every model M of ZF c . Forevery p ′′ ≤ p in M there is an M -generic filter U such that p ′′ ∈ U .By Proposition 5.2 b N (cid:15) ∀ st m φ ( m ), so, for every m ∈ N , b N (cid:15) φ ( m ).By 5.2 again, there exists p ′ ∈ U such that p ′ (cid:13) φ ( m ). We can take p ′ ≤ p ′′ . (cid:3) Lemma 5.4.
Let φ ( u, v ) be an st - ∈ -formula with parameters from e L . .Then ZF c proves the following: If p (cid:13) ∀ st m ∃ v φ ( m, v ) , then thereis p ′ ∈ P and a sequence h f m | m ∈ N i such that p ′ extends p and p ′ (cid:13) φ ( m, f m ) for every m ∈ N .Proof. By Lemma 5.3, the set D m = { p ′ ∈ P | p ′ (cid:13) ∃ v φ ( m, v ) } isdense in p for each m ∈ N . Clause (10’) in the definition of simplifiedforcing implies that also the set E m = { p ′ ∈ P | ∃ f ∈ F p ′ (cid:13) φ ( m, f ) } is dense in p . We let h m ′ , p ′ i R h m ′′ , p ′′ i iff p ′′ ⊂ p ′ ⊆ p ∧ m ′′ = m ′ + 1 ∧ p ′ ∈ E m ′ ∧ p ′′ ∈ E m ′′ . Applying
ADC to the relation R we obtain a sequence h p m | m ∈ N i such that p ⊆ p and, for each m , p m +1 ⊂ p m and ∃ f ∈ F ( p m (cid:13) φ ( m, f )). We next use ACC to obtain a sequence h f m | m ∈ N i suchthat p m (cid:13) φ ( m, f m ). Note that the Reflection Principle of ZF providesa set A = V α such that for all m ,( ∃ f ∈ F p m (cid:13) φ ( m, f )) → ( ∃ f ∈ F ∩ A p m (cid:13) φ ( m, f )) . As in the proof of Lemma 4.9, let i m be the least element of p m andlet p ′ = S ∞ m =0 p m ∩ ( i m +1 \ i m ). Then for every m the set p ′ \ p m isbounded, hence p ′ (cid:13) φ ( m, f m ). (cid:3) Proposition 5.5. If M satisfies ZF c , then b N satisfies CC .Proof. Assume that b N (cid:15) ∀ st m ∃ v φ ( m, v ). Then there is p ∈ U suchthat p (cid:13) ∀ st m ∃ v φ ( m, v ). By Lemma 5.4 there is p ′ ∈ U and a sequence h f m | m ∈ N i such that p ′ (cid:13) φ ( m, f m ) for every m ∈ N .We define a function g on I by g ( i ) = {h m, f m ( i ) i | m ∈ N } .Recall that ˇ g is the name for g in the forcing language. By Lo´s’sTheorem, p ′ (cid:13) “ˇ g is a function with domain N , ” and, for every m ∈ N , p ′ (cid:13) ˇ g ( m ) = f m . We conclude that b N (cid:15) “ˇ g is a function with domain N ”, and b N (cid:15) ∀ st m ∈ N φ ( m, ˇ g ( m )). (cid:3) Proposition 5.6. If M satisfies ZF c , then b N satisfies CC st . Proof.
The assumption b N (cid:15) ∀ st m ∃ st v φ ( m, v ) implies that we can take f m = c x m in Lemma 5.4 and the proof of Proposition 5.5. For thefunction g on I defined by g ( i ) = {h m, x m i | m ∈ N } we then have b N (cid:15) “ˇ g is standard”. (cid:3) Let p, p ′ ∈ P and let γ be an increasing mapping of p ′ onto p ; weextend γ to I = N by defining γ ( a ) = 0 for a ∈ I \ p . Lemma 5.7. p (cid:13) φ ( f , . . . , f r ) iff p ′ (cid:13) φ ( f ◦ γ, . . . , f r ◦ γ ) .Proof. This follows by induction on the cases in the definition of sim-plified forcing using the observation that the mapping p ′′ → p ′ ∩ γ − [ p ′′ ]is an isomorphism of the posets ( { p ′′ ∈ P | p ′′ ≤ p } , ≤ ) and ( { p ′′ ∈ P | p ′′ ≤ p ′ } , ≤ ). (cid:3) Corollary 5.8.
Let φ be an st - ∈ -formula with parameters from V .Then p (cid:13) φ iff p ′ (cid:13) φ . Proposition 5.9.
The structure b N satisfies the principle of Standard-ization for st - ∈ -formulas with no parameters.Proof. For standard x , b N (cid:15) φ ( x ) iff p (cid:13) φ ( x ) for every p ∈ P . Theright side is expressible by an ∈ -formula. (cid:3) This completes the proof of Theorem D . Another principle that can be added to
SCOT is Dependent Choicefor st - ∈ -formulas. DC Let φ ( u, v ) be an st - ∈ -formula with arbitrary parameters.If b ∈ B and ∀ x ∈ B ∃ y ∈ B φ ( x, y ), then there is a sequence h b n | n ∈ N i such that b = b and ∀ st n ∈ N ( b n ∈ B ∧ φ ( b n , b n +1 )). Theorem 5.10. SCOT + DC is a conservative extension of ZF c .Proof. We show that DC holds in the structure b N .Assume that p (cid:13) b ∈ B and p (cid:13) ∀ x ∈ B ∃ y ∈ B φ ( x, y ). We now let A = {h p ′ , f ′ i | p ′ ≤ p ∧ p ′ (cid:13) f ′ ∈ B } , note that h p ′ , b i ∈ A , and define R on A by h p ′ , f ′ i R h p ′′ , f ′′ i iff p ′′ ≤ p ′ ∧ p ′′ (cid:13) φ ( f ′ , f ′′ ) . It is clear from the properties of forcing that for every h p ′ , f ′ i ∈ A there is h p ′′ , f ′′ i ∈ A such that h p ′ , f ′ i R h p ′′ , f ′′ i . Using ADC we obtaina sequence hh p n , f n i | n ∈ N i such that p ≤ p , f = b , and for all n ∈ N h p n , f n i ∈ A , p n +1 ≤ p n , and p n +1 (cid:13) φ ( f n , f n +1 ).The rest of the proof imitates the arguments in the last paragraphsof the proofs of Lemma 5.4 and Proposition 5.5. (cid:3) NFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 27 Standardization for parameter-free formulas
In this section we prove that the structure b N = ( N, ∈ ∗ , M ) satisfiesthe principle of Standardization for parameter-free formulas assumingonly that the model M satisfies ZF . Explicitly, the principle postulates: SN Let φ ( v ) be an st - ∈ -formula with no parameters. Then ∀ st A ∃ st S ∀ st x ( x ∈ S ←→ x ∈ A ∧ φ ( x )) . Lemma 6.1.
The principle SN is equivalent to Standardization for st - ∈ -formulas with standard parameters.Proof. Given φ ( v, p , . . . , p ℓ ) where p , . . . , p ℓ are standard, we fix astandard P such that h p , . . . , p ℓ i ∈ P and apply SN to the formula ψ ( w ) with no parameters expressing “ ∃ w , . . . , w ℓ ( w = h v, w , . . . , w ℓ i ∧ φ ( v, w , . . . , w ℓ ))” and to the set A × P . We get a standard S such thatfor all standard inputs h x, y , . . . , y ℓ i ∈ S ←→ h x, y , . . . , y ℓ i ∈ A × P ∧ φ ( x, y , . . . , y ℓ ) holds. The set T = { x ∈ A | h x, p , . . . , p ℓ i ∈ S } standardizes φ ( v, p , . . . , p ℓ ). (cid:3) With the exception of the last proposition, in this section we workin ZF . As in Section 5, we begin with generalizing forcing to st - ∈ -formulas.We add the following clauses to Definition 4.4:(11) h p, q i (cid:13) st ( z ) for every z ∈ V .(12) h p, q i (cid:13) st ( ˙ G n ) iff rank q = k > n and ∃ x ∀ aa i ∈ p ∀h x , . . . , x k − i ∈ q ( i ) ( x n = x ) or, equivalently, ∀ aa i, i ′ ∈ p ∀h x , . . . , x k − i ∈ q ( i ) ∀h x ′ , . . . , x ′ k − i ∈ q ( i ′ ) ( x n = x ′ n ) . The basic properties of forcing from Lemma 4.5 in Subsection 4.1remain valid, but “ Lo´s’s Theorem” does not hold for st - ∈ -formulas.However, Proposition 4.6 is instrumental in the proof of Lemma 4.9.The technical lemma that follows is a simple consequence of Proposi-tion 4.6 for forcing of ∈ -formulas, but it remains valid even for forcingof st - ∈ -formulas. Definition 6.2.
For σ = h n , . . . , n s i where n , . . . , n s < k are mutu-ally distinct let π kσ : V k → V s be the “projection” of V k onto V s : π kσ ( h x , . . . , x k − i ) = h x n , . . . , x n s i . For q ∈ Q of rank k , q ↾ σ of rank s is defined by ( q ↾ σ )( i ) = π kσ [ q ( i )]. Lemma 6.3.
Let φ be an st - ∈ -formula with parameters from V .Assume that rank q = k , σ = h n , . . . , n s i where n , . . . , n s < k ,and rank q = k , σ = h m , . . . , m s i with m , . . . , m s < k . If ( q ↾ σ )( i ) = ( q ↾ σ )( i ) for all i ∈ p (we write q ↾ σ = p q ↾ σ ), then h p, q i (cid:13) φ ( ˙ G n , . . . , ˙ G n s ) if and only if h p, q i (cid:13) φ ( ˙ G m , . . . , ˙ G m s ) .Proof. The proof is by induction on the logical complexity of φ .For ∈ -formulas the assertion follows immediately from Proposition 4.6.In particular, it holds for all atomic formulas involving ∈ and = (cases(1) - (7) in the definition of forcing). It is also clear for st (cases (11)and (12)) and for conjunction (case (9)).Case (8):Assume that the statement is true for φ , h p, q i (cid:13) ¬ φ ( ˙ G n , . . . , ˙ G n s ),and h p, q i ¬ φ ( ˙ G m , . . . , ˙ G m s ). Then there exists a condition h p ′ , q ′ i ≤h p, q i such that h p ′ , q ′ i (cid:13) φ ( ˙ G m , . . . , ˙ G m s ). From the inductive as-sumption it follows that h p ′ , q ′ ↾ σ i (cid:13) φ ( ˙ G , . . . , ˙ G s − ). Let ¯ q = q ′ ↾ σ ≤ q ↾ σ = p q ↾ σ . We define q ′ = ( π k σ ) − [¯ q ] ∩ q ≤ q .Now h p ′ , q ′ i ≤ h p, q i and q ′ ↾ σ = p ′ q ′ ↾ σ , so by the inductiveassumption h p ′ , q ′ i (cid:13) φ ( ˙ G n , . . . , ˙ G n s ). This is a contradiction with h p, q i (cid:13) ¬ φ ( ˙ G n , . . . , ˙ G n s ). The reverse implication follows by ex-changing the roles of h p, q i and h p, q i .Case (10):Assume that h p, q i (cid:13) ∃ v ψ ( ˙ G n , . . . , ˙ G n s , v ), Let h p ′ , q ′ i ≤ h p, q i ,where rank q ′ = k ′ = k + r . We need to find h p ′′ , q ′′ i extending h p ′ , q ′ i and m such that h p ′′ , q ′′ i (cid:13) ψ ( ˙ G m , . . . , ˙ G m s , ˙ G m ). This will prove that h p, q i (cid:13) ∃ v ψ ( ˙ G m , . . . , ˙ G m s , v ).We let ¯ q = π k ′ σ [ q ′ ] ⊆ π k σ [ q ] = p π k σ [ q ] and ¯¯ q = ( π k σ ) − [¯ q ] ∩ q ≤ q .We define q ′ of rank k ′ = k + r by q ′ ( i ) = {h x , . . . , x k − , y , . . . , y r − i | h x , . . . , x k − i ∈ ¯¯ q ∧h x n , . . . , x n s i = π k ′ σ ( h x ′ , . . . , x ′ k − , y , . . . , y r − i ) for some h x ′ , . . . , x ′ k − , y , . . . , y r − i ∈ q ′ } for i ∈ p ′ ; q ′ ( i ) = {∅ k ′ } otherwise. We observe that q ′ ( i ) = ∅ and π k ′ σ [ q ′ ] = p ′ π k ′ σ [ q ′ ].We have h p ′ , q ′ i ≤ h p, q i ; hence there are h p ′′ , q ′′ i ≤ h p ′ , q ′ i withrank q ′′ = k ′′ and n such that h p ′′ , q ′′ i (cid:13) ψ ( ˙ G n , . . . , ˙ G n s , ˙ G n ). Finallywe construct q ′′ of rank k ′′ = k ′ + 1 such that h p ′′ , q ′′ i ≤ h p ′′ , q ′ i and,for some m , π k ′′ σ n ( q ′′ ) = p ′′ π k ′′ σ m ( q ′′ ), where σ n = h n , . . . , n s , n i and σ m = h m , . . . , m s , m i . By the inductive assumption, this es-tablishes h p ′′ , q ′′ i (cid:13) ψ ( ˙ G m , . . . , ˙ G m s , ˙ G m ).We start with b q = π k ′′ σ [ q ′′ ] ⊆ π k ′ σ [ q ′ ] = p ′ π k ′ σ [ q ′ ] and define q ′′ ( i ) = {h x , . . . , x k ′ − , z i | h x , . . . , x k ′ − i ∈ ( π k ′ σ ) − [ b q ] ∩ q ′ ∧h x n , . . . , x n s , z i ∈ π k ′′ σ n ( q ′′ ) } NFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 29 for i ∈ p ′′ ; q ′′ ( i ) = {∅ k ′ +1 } otherwise.We have h p ′′ , q ′′ i ≤ h p ′′ , q ′ i and rank q ′′ = k ′′ = k ′ + 1. Let m = k ′ . Itfollows from the construction that π k ′′ σ n ( q ′′ ) = p ′′ π k ′′ σ m ( q ′′ ). (cid:3) Corollary 6.4.
Let φ be an st - ∈ -sentence with parameters from V .Then h p, q i (cid:13) φ iff h p, ¯1 i (cid:13) φ . As in Section 5, let p, p ′ ∈ P and let γ be an increasing mapping of p ′ onto p ; we extend γ to I by defining γ ( a ) = 0 for a ∈ I \ p . Lemma 6.5.
Let φ ( v , . . . , v s ) be an st - ∈ -formula with parametersfrom V . Then h p, q i (cid:13) φ ( ˙ G n , . . . , ˙ G n s ) iff h p ′ , q ◦ γ i (cid:13) φ ( ˙ G n , . . . , ˙ G n s ) .Proof. As for Lemma 5.7. (cid:3)
Corollary 6.6.
Let φ ( v , . . . , v r ) be an st - ∈ -formula. For z , . . . , z r ∈ V h p, q i (cid:13) φ ( z , . . . , z r ) iff h p ′ , q ′ i (cid:13) φ ( z , . . . , z r ) . Proposition 6.7.
The structure b N = ( N, ∈ ∗ , M ) satisfies the principleof Standardization for st - ∈ -formulas with no parameters.Proof. For standard z , b N (cid:15) φ ( z ) iff h p, q i (cid:13) φ ( z ) for every h p, q i ∈ H .The right side is expressible by an ∈ -formula. (cid:3) This completes the proof of Theorem B . There is yet another principle that can be added to
SPOT and keepit conservative over ZF . One of its important consequences is the im-possibility to uniquely specify an infinitesimal. UP (Uniqueness Principle) Let φ ( v ) be an st - ∈ -formula with stan-dard parameters. If there exists a unique x such that φ ( x ), then this x is standard. Theorem 6.8. SPOT + UP is a conservative extension of ZF .Proof. If b N (cid:15) ∃ x [ φ ( x ) ∧ ∀ y ( φ ( y ) → y = x ) ∧ ¬ st ( x )], then there is h p, q i ∈ G and m < k = rank q such that h p, q i (cid:13) φ ( ˙ G m ) ∧ ¬ st ( ˙ G m ) ∧∀ y ( φ ( y ) → y = ˙ G m ).Let r = q ↾ { m } , i.e., for all i ∈ I , x ∈ r ( i ) ←→ ∃h x , . . . , x k − i ∈ q ( i ) ( x m = x ). Claim 1. For every x ∈ S i ∈ p r ( i ) the set p x = { i ∈ p | x ∈ r ( i ) } isbounded; we let i x denote its greatest element.Proof of Claim 1. Let x be such that p x is unbounded. Define q x ( i ) = {h x , . . . , x k − i ∈ q ( i ) | ( x k = x ) } for i ∈ p x ; q x ( i ) = {∅ k } otherwise . Then h p x , q x i ≤ h p, q i and h p x , q x i (cid:13) ˙ G m = x , i.e., h p x , q x i (cid:13) st ( ˙ G m ), acontradiction. (cid:3) Claim 2. There exist unbounded mutually disjoint sets p , p ⊂ p andnonempty sets s ( i ) ⊆ r ( i ) for all i ∈ p ∪ p such that (cid:16)S i ∈ p s ( i ) (cid:17) ∩ (cid:16)S i ∈ p s ( i ) (cid:17) = ∅ . We postpone the proof of Claim 2 and complete the proof of thetheorem.We let e q ( i ) = {h x , . . . , x k − i ∈ q ( i ) | x k ∈ s ( i ) } for i ∈ p ∪ p , e q ( i ) = {∅ k } otherwise. We have h p , e q i ≤ h p, q i , h p , e q i ≤ h p, q i , andconsequently h p , e q i (cid:13) φ ( ˙ G m ), h p , e q i (cid:13) φ ( ˙ G m ).Let γ be an increasing mapping of p onto p extended by γ ( i ) = 0for i ∈ I \ p . By Lemma 6.5 h p , e q ◦ γ i (cid:13) φ ( ˙ G m ). We “amalga-mate” h p , e q i and h p , e q ◦ γ i to form a condition of rank 2 k as follows: h x , . . . , x k − , y , . . . , y k − i ∈ b q ( i ) ←→h x , . . . , x k − i ∈ e q ( i ) ∧ h y , . . . , y k − i ∈ e q ( γ ( i ))and observe that h p , b q i ≤ h p, e q i . Let σ = k and σ = { k + ℓ | ℓ < k } .We have b q ↾ σ = e q and b q ↾ σ = e q ◦ γ . By Lemma 6.3 h p , b q i (cid:13) φ ( ˙ G m ) ∧ φ ( ˙ G k + m ). But x m = y m holds for all h x , . . . , x k − , y , . . . , y k − i ∈ b q ( i )and all i ∈ p , so h p , b q i (cid:13) ˙ G m = ˙ G k + m . This contradicts h p , b q i (cid:13) ∀ y ( φ ( y ) → y = ˙ G m ) . Proof of Claim 2.
W.l.o.g. we can assume p = I = N (map p onto N in an increasing way). Define sequences h n ℓ | ℓ ∈ N i , h α ℓ | ℓ ∈ N i and h s ( n ℓ ) | ℓ ∈ N i by recursion as follows:Let n = 0, α = min { i x | x ∈ r (0) } and s ( n ) = s (0) = { x ∈ r (0) | i x = α } .At stage ℓ + 1 let n ℓ +1 = α ℓ + 1, α ℓ +1 = min { i x | x ∈ r ( n ℓ +1 ) } and s ( n ℓ +1 ) = { x ∈ r ( n ℓ +1 ) | i x = α ℓ +1 } .We observe that s ( n ℓ ) ∩ s ( n ℓ ′ ) = ∅ for all ℓ = ℓ ′ . It remains to let p = { n ℓ | ℓ ∈ N } and p = { n ℓ +1 | ℓ ∈ N } . (cid:3) (cid:3) Idealization
We recall the axioms of the theory
BST ; see the references Kanoveiand Reeken [19] and Fletcher et al. [8] for motivation and more detail.In addition to the axioms of
ZFC , they are: B (Boundedness) ∀ x ∃ st y ( x ∈ y ). T (Transfer) Let φ ( v ) be an ∈ -formula with standard parameters.Then ∀ st x φ ( x ) → ∀ x φ ( x ) . NFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 31 S (Standardization) Let φ ( v ) be an st - ∈ -formula with arbitrary pa-rameters. Then ∀ st A ∃ st S ∀ st x ( x ∈ S ←→ x ∈ A ∧ φ ( x )) . BI (Bounded Idealization) Let φ be an ∈ -formula with arbitraryparameters. For every set A ∀ st fin a ⊆ A ∃ y ∀ x ∈ a φ ( x, y ) ←→ ∃ y ∀ st x ∈ A φ ( x, y ) . Idealization over uncountable sets.
In order to obtain modelswith Bounded Idealization, the construction of Subsections 4.1 and 4.2needs to be generalized from I = N to I = P fin ( A ), where A is anyinfinite set. The key is the right definition of “unbounded” subsets of I . We work in ZF .We use the notation P ≤ m ( A ) for { a ∈ P fin ( A ) | | a | ≤ m } . Definition 7.1.
A set p ⊆ P fin ( A ) is thick if ∀ m ∈ N ∃ n ∈ N ∀ a ∈ P ≤ m ( A ) ∃ b ∈ p ∩ P ≤ n ( A ) ( a ⊆ b ) . We let ν p ( m ) denote the least n with this property. The set p is thin if it is not thick.Clearly { a ∈ P fin ( A ) | x ∈ a } is thick for every x ∈ A (with ν p ( m ) = m + 1). We now carry out the developments of Subsections 4.1 and 4.2with unbounded and bounded replaced by thick and thin , respectively.The definition of forcing and proofs of Lemmas 4.5 and 4.8 are asbefore. The following observation allows the proof of Proposition 4.6to go through as well. Lemma 7.2. If p is thick and S ⊆ P fin ( A ) , then either p ∩ S or p \ S is thick.Proof. Otherwise there is m such that(*) ∀ n ∈ N ∃ a ∈ P ≤ m ( A ) ∀ b ∈ ( p ∩ S ) ∩ P ≤ n ( A ) ( a * b )and there is m such that(**) ∀ n ∈ N ∃ a ∈ P ≤ m ( A ) ∀ b ∈ ( p \ S ) ∩ P ≤ n ( A ) ( a * b ) . Let m , m be as above, and let m = m + m . Since p is thick,(***) ∃ n ∈ N ∀ a ∈ P ≤ m ( A ) ∃ b ∈ p ∩ P ≤ n ( A ) ( a ⊆ b ) . Fix such n ; for a ∈ P ≤ m ( A ) such that ∀ b ∈ ( p ∩ S ) ∩ P ≤ n ( A ) ( a * b )and a ∈ P ≤ m ( A ) such that ∀ b ∈ ( p \ S ) ∩ P ≤ n ( A ) ( a * b ) we have a ∪ a ∈ P ≤ m ( A ). By ( ∗ ∗ ∗ ) there is b ∈ p ∩ P ≤ n ( A ) such that a ∪ a ⊆ b . Depending on whether b ∈ p ∩ S or b ∈ p \ S , thiscontradicts (*) or (**). (cid:3) The next lemma enables a generalization of Lemma 4.9.
Lemma 7.3.
Let h p n | n ∈ N i be such that, for all n , p n ∈ P and p n ⊇ p n +1 . Then there is p ∈ P with the property that for every n thereis k ∈ N such that ∀ a ∈ ( p \ p n ) ( | a | ≤ k ) . In particular, p \ p n is thin.Proof. Define p = S ∞ m =0 { a ∈ p m | | a | ≤ ν p m ( m ) } . Since p \ p n ⊆ S n − m =0 { a ∈ p m | | a | ≤ ν p m ( m ) } , a ∈ p \ p n implies | a | ≤ k for k =max { ν p (0) , . . . , ν p n − ( n − } .We show that p is thick. Given m ∈ N , we let n = ν p m ( m ). If a ∈ P fin ( A ) and | a | ≤ m , then there is b ∈ p m such that a ⊆ b and | b | ≤ n . By the definition of p , b ∈ p . So n has the requiredproperty. (cid:3) With these changes, the rest of the development of Sections 4.1and 4.2 goes through and establishes the following strengthening ofProposition 4.15.
Proposition 7.4.
The structure b N A = ( N A , ∈ ∗ , M ) for I = P fin ( A ) satisfies the principles of Transfer, Nontriviality, Boundedness, Stan-dard Part, and Bounded Idealization over A for ∈ -formulas with stan-dard parameters.Proof. To prove that b N (cid:15) O one can take d ∈ M such that M (cid:15) “ d isa function on I = P fin ( A ) ∧ ∀ a ∈ P fin ( A ) ( d ( a ) = | a | )”. Nontrivialityalso follows from Bounded Idealization.Let G m ∈ N and let h p, q i ∈ G have rank q = k > m . There is some X ∈ M such that M (cid:15) “ ∀ i ∈ I ∀h x , . . . , x k − i ∈ q ( i ) ( x m ∈ X ).” ByProposition 4.6 h p, q i (cid:13) ∀ v ( v ∈ ˙ G m → v ∈ ˇ X ), so ∀ v ∈ G m ( v ∈ X )holds in N . This proves Boundedness in b N .It remains to prove that Bounded Idealization over A holds in b N .Let φ ( u, v ) be an ∈ -formula with parameters from M . Assume that M (cid:15) ∀ a ∈ P fin ( A ) ∃ y ∀ x ∈ a φ ( x, y ). By the Reflection Principle in ZF , M (cid:15) “ ∃ an ordinal α ∀ a ∈ P fin ( A ) ∃ y ∈ V α ∀ x ∈ a φ ( x, y )”. Wework in the model M .Let h p, q i ∈ H be a forcing condition and k = rank q . For i = a ∈ p we let q ′ ( a ) = {h x , . . . , x k − , x k i | h x , . . . , x k − i ∈ q ( a ) ∧ x k ∈ V α ∧ ∀ x ∈ a φ ( x, x k ) } ; q ′ ( a ) = {∅ k +1 } otherwise. Then h p, q ′ i extends h p, q i . For every x ∈ A the set c = p ∩ { a ∈ P fin ( A ) | x ∈ a } is in P because p \ c is thin(Lemma 7.2), and so, by Proposition 4.6, h p, q ′ i (cid:13) φ ( x, ˙ G k ).By the genericity of G there exist h p, q i ∈ G and k ∈ Ω such that h p, q i (cid:13) φ ( x, ˙ G k ) for all x ∈ A . By the Fundamental Theorem 4.12 NFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 33 N (cid:15) φ ( x, G k ) for all x ∈ A . This means that the → implication ofIdealization over A for ∈ -formulas with standard parameters is satisfiedin b N . The opposite implication follows from SP ; see Lemma 2.1 inSection 2. (cid:3) Further theories.
Nelson’s
IST postulates a form of Idealizationthat is even stronger than Bounded Idealization (but it contradictsBoundedness). I Let φ be an ∈ -formula with arbitrary parameters. ∀ st fin a ∃ y ∀ x ∈ a φ ( x, y ) ←→ ∃ y ∀ st x φ ( x, y ) . We let
ISPT ′ be the theory obtained from BSPT ′ by deletingBoundedness and replacing Bounded Idealization for formulas withstandard parameters by Nelson’s Idealization for formulas with stan-dard parameters.Principle I implies the existence of a finite set that contains all stan-dard sets as elements, and has certain undesirable consequences fromthe metamathematical point of view. Kanovei and Reeken [19], The-orem 4.6.23, prove that there are countable models M = ( M, ∈ M ) of ZFC that cannot be extended to a model of
IST in which M would bethe class of all standard sets (assuming ZFC is consistent). We do notknow whether the same is the case for
ISPT ′ . Nevertheless we havethe following result. Theorem 7.5. ISPT ′ is a conservative extension of ZF .Proof. Let us assume that
ISPT ′ ⊢ θ but ZF θ , for some ∈ -sentence θ . Let ZF r be ZF with the Axiom Schema of Replacement restrictedto Σ r -formulas, and let ISPT ′ r be ISPT ′ with ZF replaced by ZF r .There is r ∈ ω for which ISPT ′ r ⊢ θ .Let M = ( M, ∈ M ) be a model of ZF + ¬ θ . By the Reflection Principleof ZF , valid in M , there is α ∈ M such that M (cid:15) “ α is a limit ordinal”, M (cid:15) φ V α for every axiom φ of ZF r , and M (cid:15) ( ¬ θ ) V α .We let A = V α and use Proposition 7.4 to extend M to a model b N A .We define N α = { x ∈ N A | b N A (cid:15) x ∈ V α } . It is easy to verify that b N α = ( N α , ∈ ∗ ↾ N α , M ∩ N α ) is a model of ISPT ′ r ; hence θ holds in b N α . On the other hand, ( ¬ θ ) V α holds in M and hence, by Transfer, ¬ θ holds in b N α . A contradiction. (cid:3) Kanovei and Reeken [19], Theorem 3.4.5, showed that the class ofbounded sets in
IST satisfies the axioms of
BST . This result holdsalso for
ISPT ′ and BSPT ′ , respectively, and establishes the followingtheorem. Theorem 7.6.
The theory
BSPT ′ is a conservative extension of ZF .This concludes the proof of Theorem C . Finally, we prove that if M satisfies ADC , then the model b N A con-structed in Proposition 7.4 satisfies CC . We note that the definitionof forcing for st - ∈ -formulas in Section 6 and Lemma 6.3 extend to I = P fin ( A ). Proposition 7.7. If M is a countable model of ZF c , then the extendedultrapower b N A satisfies Countable st - ∈ -Choice (both CC and CC st ).Proof. We work in ZF c . CC : Let us assume that h p, q i has rank k and h p, q i (cid:13) ∀ st m ∈ N ∃ v φ ( m, v ). Let E m = {h p ′ , q ′ i ∈ H | h p ′ , q ′ i ≤ h p, q i ∧ h p ′ , q ′ i (cid:13) φ ( m, ˙ G n ) for some n > k } . By an argument like the one in Lemma 5.3 it follows that for every m and every h p ′′ , q ′′ i ≤ ∗ h p, q i there is h p ′ , q ′ i ≤ h p ′′ , q ′′ i such that h p ′ , q ′ i ∈ E m . [We note that the E m may be proper classes, but bythe Reflection Principle there is a set S such that h p, q i ∈ S and forevery m and every h p ′′ , q ′′ i ∈ S there is h p ′ , q ′ i ≤ h p ′′ , q ′′ i such that h p ′ , q ′ i ∈ S ∩ E m . The classes E m can be replaced by the sets S ∩ E m in the argument below.]We let h m ′ , h p ′ , q ′ ii R h m ′′ , h p ′′ , q ′′ ii iff h p ′′ , q ′′ i ≤ h p ′ , q ′ i ≤ h p, q i ∧ m ′′ = m ′ +1 ∧ h p ′ , q ′ i ∈ E m ′ ∧ h p ′′ , q ′′ i ∈ E m ′′ . Applying
ADC to the relation R we obtain a sequence hh p m , q m i | m ∈ N i such that h p , q i ≤ h p, q i and, for each m , h p m +1 , q m +1 i ≤h p m , q m i and h p m , q m i (cid:13) φ ( m, ˙ G n ) for some n ∈ N , n > k . Letrank q m = ℓ m and let n m < ℓ m , n m > k , be the least such n .As in the proof of Lemma 7.3, let p ∞ = S ∞ m =0 C m where C m = { a ∈ p m | | a | ≤ ν p m ( m ) } . We recall that p ∞ \ p m is thin for every m ; hence h p ∞ , q m i (cid:13) φ ( m, ˙ G n m ). We define a function q ∞ ∈ Q of rank k + 1 asfollows: If a ∈ C m \ S m − j =0 C j then q ∞ ( a ) = {h x , . . . , , x k i | x k is a function ∧ dom x k = N ∧ ∀ j ≤ m ∃ y k , . . . , y ℓ j − ( h x , . . . , x k − , y k , . . . , y ℓ j − i ∈ q j ( a ) ∧ x k ( j ) = y n j ) ∧∀ j > m ( x k ( j ) = 0) } . By “ Lo´s’s Theorem”, h p ∞ , q ∞ i (cid:13) “ ˙ G k is a function with dom ˙ G k = N .”Now assume that b N A (cid:15) ∀ st m ∃ v φ ( m, v ). Then there is h p, q i ∈ G such that h p, q i (cid:13) ∀ st m ∃ v φ ( m, v ). By the above discussion, thereis a condition of the form h p ∞ , q ∞ i such that h p ∞ , q ∞ i ∈ G ; hence NFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 35 b N (cid:15) “ G k is a function with dom G k = N .” Fix m ∈ M ; let b N (cid:15) G k ( m ) = G ℓ ; we can assume ℓ > k . Then there is some h p ′ , q ′ i ∈ G , h p ′ , q ′ i ≤ h p ∞ , q ∞ i , such that h p ′ , q ′ i (cid:13) ˙ G k ( m ) = ˙ G ℓ .Let σ m = k ∪ { n m } and σ ′ = k ∪ { ℓ } .From h p ∞ , q m i (cid:13) φ ( m, ˙ G n m ) and Lemma 6.3 it follows that h p ∞ , q m ↾ σ m i (cid:13) φ ( m, ˙ G k ). We see from the construction that h p ′ ∩ p m , q ′ ↾ σ ′ i ≤h p ′ ∩ p m , q m ↾ σ m i . As p ′ \ p m is thin, we have h p ′ , q ′ ↾ σ ′ i (cid:13) φ ( m, ˙ G k ),and by Lemma 6.3 again, h p ′ , q ′ i (cid:13) φ ( m, ˙ G ℓ ). From h p ′ , q ′ i ∈ G , weconclude that b N (cid:15) φ ( m, G ℓ ) and hence b N (cid:15) φ ( m, G k ( m )). CC st : Assuming h p, q i (cid:13) ∀ st m ∈ N ∃ st v φ ( m, v ), we can require inthe definition of E m that h p ′ , q ′ i (cid:13) st ( ˙ G n ), i.e., h p ′ , q ′ i (cid:13) ˙ G n = c z n for a uniquely determined z n . In the definition of q ∞ ( a ) we can let x k = h z n j | j ∈ N i . Then h p ∞ , q ∞ i (cid:13) st ( ˙ G k ) and b N (cid:15) st ( G k ). (cid:3) Let
BSCT ′ be the theory obtained from BSPT ′ by adding ADC and strengthening SP to CC ; analogously for ISCT ′ . The last twotheorems of this section follow by the same arguments as those used toprove Theorems 7.5 and 7.6. Theorem 7.8.
The theory
ISCT ′ is a conservative extension of ZF c . Theorem 7.9.
The theory
BSCT ′ is a conservative extension of ZF c . Final Remarks
Open problems. (1) Are the theories
BSCT ′ + SN and ISCT ′ + SN (defined above)conservative extensions of ZF c ?We do not know whether b N A for I = P fin ( A ) with uncountable A satisfies SN . In the absence of AC , a way to formulate and prove asuitable analog of Lemma 5.7 is not obvious.(2) Are the theories BSPT and
ISPT conservative extensions of ZF ? Are the theories BSCT and
ISCT conservative extensions of ZF c ?Here BSPT is obtained from
BSPT ′ by strengthening (Bounded)Idealization to allow arbitrary parameters; similarly for the other the-ories. The likely answer is yes; the obvious approach is to iterate theforcing used to prove the primed versions. Spector develops iterated ex-tended ultrapowers in [33]. His method would require nontrivial adap-tations in our framework, but it is likely to work provided the answerto problem (1) is yes. The ultimate result would be that BSCT + SN and ISCT + SN are conservative extensions of ZF c . (3) Does every countable model of ZF have an extension to a modelof BSPT ′ ?The likely answer is again yes, using a suitable iteration of extendedultrapowers.(4) Is SPOT + SC a conservative extension of ZF ?8.2. Forcing with filters.
A more elegant and potentially more pow-erful notion of forcing is obtained by replacing P with e P = {P | P is a filter of unbounded subsets of I } where P ′ extends P iff P ⊆ P ′ . “ Lo´s’s Theorem” 4.6 then takes theform: hP , q i (cid:13) φ ( ˙ G n , . . . , ˙ G n s ) iff rank q = k > n , . . . , n s and ∃ p ∈ P ∀ i ∈ p ∀h x , . . . , x k − i ∈ q ( i ) φ ( x n , . . . , x n s ) . The forcing notion P we actually use amounts to restricting oneself toprincipal filters.8.3. Zermelo set theory.
Similar results can be obtained for theo-ries weaker than ZF . Let Z ∞ = ZF − Replacement be the Zermeloset theory, and let BT denote Transfer for bounded formulas. In theproof of Proposition 4.2 the extended ultrapower can be replaced bythe extended bounded ultrapower (see Chang and Keisler [5], Sec. 4.4,for a discussion of ordinary bounded ultrapowers). This proves that SPOT − = Z ∞ + O + BT + SP is a conservative extension of Z ∞ .With some modifications, this theory can be taken as an axiomatiza-tion of the internal part of nonstandard universes of Keisler [5, 22](the superstructure framework for nonstandard analysis). Analogousresults can be obtained for SCOT − , BSPT − and BSCT − .8.4. Weaker theories.
Reverse Mathematics has as its goal the cal-ibration of the exact set-theoretic strength of the principal results inordinary mathematics. One of its chief accomplishments is the discov-ery that, with a few exceptions, every theorem in ordinary mathematicsis logically equivalent (over S ) to one of the five subtheories S − S ofsecond-order arithmetic Z , known collectively as “The Big Five.” Here S is the weakest of the five theories, the second-order arithmetic withrecursive comprehension axiom, also denoted RCA , and S , known as WKL , is obtained by adding the Weak K¨onig’s Lemma to the axiomsof RCA . We refer to Simpson [29] for a comprehensive introductionto Reverse Mathematics.Keisler and others extended the ideas of Reverse Mathematics to thenonstandard realm. In Keisler’s paper [23] it is established that if S isany of the “Big Five” theories above, then S has a conservative exten-sion ∗ S to a theory in the language with an additional unary predicate NFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 37 st ; the axioms of ∗ S include O , SP and, for the theories stronger than WKL , also FOT (First-Order Transfer). There is an extensive bodyof work by van den Berg, Sanders and others (see [4], [27] and thereferences therein) devoted to determining the exact proof-theoreticstrength of particular results in infinitesimal ordinary mathematics.Substantial parts of it can be carried out in these and other elemen-tary systems for nonstandard mathematics, for example Nelson [25]and Sommer and Suppes [31]. However, these systems do not enablethe natural reasoning as practiced in analysis. They are usually formal-ized in the language of second-order arithmetic or type theory. Basicobjects of ordinary analysis, such as real numbers, continuous functionsand separable metric spaces, have to be represented in these theoriesvia suitable codes, and the results may have to be presented “up to in-finitesimals,” because the full strength of the Transfer principle or theStandard Part principle is not available. The focus of this paper is ontheories like
SPOT or SPOT − , which axiomatize both the traditionaland the nonstandard methods of ordinary mathematics in the way theyare customarily practiced. Rather than looking for the weakest princi-ples that enable a proof of a given mathematical theorem, we formulatetheories that are as strong as possible while still effective (conservativeover ZF ) or semi-effective (conservative over ZF c ).8.5. Finitistic proofs.
The model-theoretic proof of Proposition 4.2as given here is carried out in ZF . Using techniques from Simpson [29],Chapter II, esp. II.3 and II.8, it can be verified that the proof goesthrough in RCA (w.l.o.g. one can assume that M ⊆ ω ).The proof of Theorem A from Proposition 4.2 requires the G¨odelCompleteness Theorem and therefore WKL ; see [29], Theorem IV.3.3.We conclude that Theorem A can be proved in WKL .Theorem A , when viewed as an arithmetical statement resulting fromidentifying formulas with their G¨odel numbers, is Π . It is well-knownthat WKL is conservative over PRA (Primitive Recursive Arith-metic) for Π sentences ([29], Theorem IX.3.16); therefore Theorem A is provable in PRA . Since
PRA is generally considered to correctlycapture finitistic reasoning as envisioned by Hilbert [10] (see e.g. Simp-son [29], Remark IX.3.18), we conclude that Theorem A has a finitisticproof.These remarks apply equally to Theorems B - D .8.6. SPOT and CH.
Connes (see for example [6], pp. 20–21) objectsto the use of ultrafilters but approves of the Continuum Hypothesis( CH ). In the absence of full AC , it is important to distinguish (atleast) two versions of CH . CH : Every infinite subset of R is either countable or equipotentto R . CH + : R is equipotent to ℵ (often written 2 ℵ = ℵ ).The axioms CH and CH + are equivalent in ZFC , but not in ZF c . Itis known that ZF c + CH does not imply the existence of any nonprin-cipal ultrafilters over N ( CH holds in the Solovay model). We have: SPOT + CH is a conservative extension of ZF + CH .Proof. Let φ be an ∈ -sentence. Then SPOT + CH ⊢ φ iff SPOT ⊢ ( CH → φ ) iff ZF ⊢ ( CH → φ ) iff ZF + CH ⊢ φ . (cid:3) However, it seems clear that Connes has CH + in mind. But CH + implies that R has a well-ordering (of order type ℵ ). From this iteasily follows that there exist nonprincipal ultrafilters over N (for ex-ample, Jech [16], p. 478 proves a much stronger result). Thus Connes’sposition on this matter is incoherent.Apart from the issue of CH , Connes’s repeated criticisms of Robin-son’s framework starting in 1994 are predicated on the premise thatinfinitesimal analysis requires ultrafilters on N (which are incidentallyfreely used in some of the same works where Connes criticizes Robin-son). Our present article shows that Connes’s premise is erroneousfrom the start. 9. Conclusion
In this paper we observe that infinitesimal methods in ordinary math-ematics require no Axiom of Choice at all, or only those weak formsof AC that are routinely used in the traditional treatments. This con-clusion follows from the fact that the theory SPOT and its variousstrengthenings, which do not imply the existence of nonprincipal ultra-filters over N , or other strong forms of AC , are sufficient to carry outinfinitesimal arguments in ordinary mathematics (and beyond).But most users of nonstandard analysis work with hyperreals, andthe existence of hyperreals does imply the existence of nonprincipalultrafilters over N . So it would seem that ultrafilters are needed, afterall. However, this view implicitly assumes that set theory like ZFC ,based exclusively on the membership predicate ∈ , is the only correctframework for Calculus.Historically, the Calculus of Newton and Leibniz was first made rig-orous by Dedekind, Weierstrass and Cantor in the 19th century usingthe ε - δ approach. It was eventually axiomatized in the ∈ -language as ZFC . After Robinson’s development of nonstandard analysis it was See for example Katz and Sherry [20].
NFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 39 realized that Calculus with infinitesimals also admits a rigorous for-mulation, closer to the ideas of Leibniz, Bernoulli, Euler (see [1]) andCauchy (see [2]). It can be axiomatized in a set theory using the st - ∈ -language. The primitive predicate st can be thought of as a formaliza-tion of the Leibnizian distinction between assignable and inassignablequantities. Such theories are obtained from ZFC by adding suitableversions of Transfer, Idealization and Standardization.Now that it has been established that the infinitesimal methods donot carry a heavier foundational burden than their traditional counter-parts, one can ask the following question. Which foundational frame-work constitutes a more faithful formalization of the techniques of the17–19 century masters? For all the achievements of Cantor, Dedekindand Weierstrass in streamlining analysis, built into the transformationthey effected was a failure to provide a theory of infinitesimals whichwere the bread and butter of 17–19 century analysis, until Weierstrass.By the yardstick of success in formalization of classical analysis, ar-guably
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Department of Mathematics, City College of CUNY, New York, NY10031,
E-mail address : [email protected] Department of Mathematics, Bar Ilan University, Ramat Gan 52900Israel,
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