preserves information

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1 The usual model construction for NFU preserves information M. Randall Holmes January 29, 2009 Abstract The usual model construction for NFU (Quine s New Foundations with urelements, shown to be consistent by Jensen) starts with a model of the usual set theory with an automorphism that moves a rank (this rank is the domain of the model). Most elements of the resulting model of NFU are urelements (it appears that information about their extensions is discarded). The surprising result of this paper is that this information is not discarded at all: the membership relation of the original model (restricted to the domain of the model of NFU ) is definable in the language of NFU. A corollary of this is that the urelements of a model of NFU obtained by the usual construction are inhomogeneous: this was the question the author was investigating initially. Other aspects of the mutual interpretability of NFU and a fragment of ZFC are discussed in sufficient detail to place this result in context. Contents 1 Introduction 1 2 Introduction to the set theory NFU 3 3 Mac Lane set theory with the Axiom of Rank 6 4 The Boffa model construction for NFU 8 1

2 5 The Axiom of Endomorphism and the type level membership relation 13 6 Recovering the lost information 17 7 The theory of a rank with an endomorphism and the theory of a Boffa model 20 8 Boffa models of NFU from general models of NFU 25 9 Inhomogeneity of urelements 28 1 Introduction This paper discusses a new result about the model theory of NFU, a variation of New Foundations (NF ), the notorious set theory proposed by Quine in [19], whose consistency remains an open question. This paper is not a contribution to the literature on that vexed question, as NFU, proposed by R. B. Jensen in [15], was there shown to be consistent relative to quite weak fragments of the usual set theory. What we call the usual construction of models for NFU in the title is not the construction of Jensen (though it is closely related) but a refinement of Jensen s construction which is attributed to Maurice Boffa (Thomas Forster makes the attribution to Boffa s [2] in his [6]. The construction described in the Boffa paper a bit more general, whereas that described in [6] is exactly what we give here; however, Boffa did describe this exact construction to the author in person in 1991). Briefly (full details are given below), the Boffa model construction begins with a model of a fragment of Zermelo set theory with an external automorphism which moves a rank of the cumulative hierarchy. The nonstandard rank which is moved by the automorphism is the domain of the model of NFU, and the redefinition of the membership relation which yields the model of NFU converts most of the elements of the nonstandard rank to urelements (it is the presence of urelements that distinguishes NFU from the problematic NF ). So it appears that the construction discards a vast amount of information about extensions of objects which are in effect treated as junk, in order to get a sensible theory of sets in a small corner of the 2

3 model. The main result of this paper is that this is an illusion. The restrictions to the domain of the Boffa model of the membership relation of the original model and of the automorphism of the original model are definable in the language of NFU in the Boffa model (and the definition is the same in every Boffa model). This is an unexpected (to us at least) particular strengthening of a general philosophical thesis of ours that Zermelo-style set theory (Zermelo set theory, ZFC and extensions) and Quine-style set theory (NFU and extensions thereof, rather than the original theory NF ) are mutually interpretable in interesting ways (the bare mathematical fact that theories of these two kinds are mutually interpretable is not in dispute). We have discussed this in various papers (in [11], [12] and most recently in [13]). We give an overview of the remaining sections of this paper. In the second section, we will introduce the set theories NF and NFU. We will mostly avoid mathematical reasoning inside these theories in this paper. In the third section we describe the exact fragment of ZFC to which we relate NFU (Mac Lane set theory with the Axiom of Rank). In the fourth section, we describe the Boffa construction of models of NFU and prove that it works. In the fifth section, we introduce an assertion (the Axiom of Endomorphism) which holds in Boffa models of NFU but not necessarily in general models of the theory. The Axiom of Endomorphism asserts the existence of a function with special properties, and is witnessed in Boffa models by a function which in effect codes the automorphism of the underlying model of Zermelo-style set theory. In the sixth section we prove the main result by showing that if the Axiom of Endomorphism holds it is witnessed by a unique (and therefore definable) object which can then be used to define the restriction to the domain of the Boffa model of the membership relation and automorphism of the underlying model. In the seventh section we give a theory satisfied by any nonstandard rank moved downward by an automorphism (the domain of a Boffa model) and explain how to recover a model of Mac Lane set theory from a Boffa model of NFU. The eighth section provides a brief overview of a way in which a Boffa model of NFU can be discovered inside any model of NFU at all. In the last section we briefly describe a specialized question about homogeneity of urelements in models of NFU which we were investigating when we established this result, and also the prior history of the Axiom of Endomorphism, of which we have been aware for some time. 3

4 2 Introduction to the set theory NFU New Foundations and NFU are set theories motivated by a streamlining of simple type theory. Simple type theory (TST ) is a many-sorted first order theory with sorts (called types ) indexed by the natural numbers. The well-formedness conditions for atomic sentences are briefly indicated by the templates x n = y n, x n y n+1 (where the superscripts indicate type; note below that we do not always supply explicit type indices on variables). The axiom schemes of TST are as follows: Axiom 2.1. (Extensionality) ( AB.A = B ( x.x A x B)) [for any assignment of types to the variables which results in a well-formed formula] Axiom 2.2. (Comprehension) The universal closure of ( A.( x.x A φ)) is an axiom [for any formula φ of the language of TST in which the variable A does not occur, and with the requirement that the type of A is one higher than the type of x] Definition 2.1. If φ is a formula (in which A, one type higher than x, does not occur free), we define {x φ} as the unique object A such that ( x.x A φ). This object exists by Comprehension and is unique by Extensionality. These are the axioms of naive set theory modified only by imposing type discipline. Axioms of Infinity and Choice are usually adjoined to TST : a discussion of their precise form at this point would take us too far afield. This theory has a very high degree of what Russell called systematic ambiguity, more recent set theorists tend to call typical ambiguity, and computer scientists call polymorphism. For any sentence φ, let φ + be the formula obtained by replacing each variable in φ with a variable one type higher (without creating identifications between variables). Note that for any axiom φ, φ + is also an axiom, and that if ψ follows from φ logically, ψ + will follow from φ + in the same way. It follows that if φ is a theorem, so is φ +. Further, if we define a mathematical object using notation {x n φ}, then {x n+1 φ + } (where x n+1 replaces x n in φ + ) will be a precisely analogous object in the next higher type. A concrete example of this: the number 3 is naturally defined in TST using Frege s definition: 3 is the set of all sets with three elements (a first-order definable concept). But there is a 3 2 which 4

5 is the type 2 set of all type 1 sets with three type 0 elements, a 3 3 which is the type 3 set of all type 2 sets with three type 1 elements, and so forth ad infinitum. Quine s suggestion which motivates New Foundations is that perhaps the hall of mirrors effect here is redundant: perhaps all the types are actually the same. NF is the first-order single sorted theory whose axioms are Axiom 2.3. (Extensionality) ( AB.A = B ( x.x A x B)) Axiom 2.4. (Comprehension) The universal closure of ( A.( x.x A φ)) is an axiom [for any formula φ of the language of set theory in which the variable A does not occur and which can be converted to a formula of TST by a suitable assignment of types to the variables appearing in it] Definition 2.2. If φ is a formula (in which A does not occur free), we define {x φ} as the unique object A such that ( x.x A φ). This object exists by Comprehension and is unique by Extensionality. Axioms of Infinity and Choice are not adjoined, because Infinity and the negation (!) of Choice are theorems of NF, as was shown by Specker in [22]. The axioms of NF are precisely the axioms of TST with indications of type dropped from the variables (in such a way as not to introduce new identifications between variables). It is more usual to define NF in a way which does not directly admit the connection to type theory. Definition 2.3. A formula φ is said to be stratified iff there is a function σ from variables to natural numbers (or integers) such that in each atomic subformula x = y of φ we have σ( x ) = σ( y ) and in each atomic formula x y of φ we have σ( x )+1 = σ( y ). The function σ is called a stratification of φ. It is easy to see that a formula φ is stratified iff there is an assignment of types to the variables in φ which makes φ a well-formed formula of the language of simple type theory. For this reason, the comprehension scheme of NF is called stratified comprehension. Axiom 2.5. (Stratified Comprehension) The universal closure of ( A.( x.x A φ)) is an axiom [for any stratified formula φ of the language of set theory in which the variable A does not occur] 5

6 Note that the Extensionality axiom of NF is an axiom, not an axiom scheme. Although this is less obvious, the same is true of stratified comprehension, because the stratified comprehension scheme is equivalent to the conjunction of finitely many of its instances (the canonical reference for this, though the specific implementation given is terrible, is [7]). The theory NFU of Jensen differs from NF only in the form of its axiom of extensionality, and also usually in an inessential but convenient notational addition. Axiom 2.6. (Empty Set) ( x.x ) [ is a new primitive constant] Axiom 2.7. (Weak Extensionality) ( ABx.x A ( y.y A y B) A = B) Definition 2.4. set(x) def x = ( y.y x) [one can also take the sethood predicate as primitive with an appropriate axiom and define the empty set]. Axiom 2.8. (Stratified Comprehension) The universal closure of ( A.( x.x A φ)) is an axiom [for any stratified formula φ of the language of set theory in which the variable A does not occur] Definition 2.5. If φ is a formula (in which A does not occur free), we define {x φ} as the unique object A such that set(a) and ( x.x A φ). This object exists by Comprehension and is uniquely determined by Extensionality (if it has elements, in which case it is certainly a set) and the definition of sethood (if it has no elements: the only elementless set is ). 3 Mac Lane set theory with the Axiom of Rank In this section we introduce a weak fragment of the usual set theory which is closely related to NFU. We call it Mac Lane set theory because Saunders Mac Lane advocated its use as a foundational theory in [16] (it is also called bounded Zermelo set theory ). An elegant and comprehensive discussion of this theory is found in Mathias s [17]. We include the Infinity axiom in the list below because Mac Lane did, but for our purposes Mac Lane set theory does not include Infinity unless this is specifically mentioned. 6

7 Mac Lane set theory is a first-order theory with equality and membership as primitive relations. Its axioms as originally proposed by Mac Lane are as follows. Axiom 3.1. (Extensionality) ( AB.A = B ( x.x A x B)) Axiom 3.2. (Pairing) ( ab.( c.( x.x c x = a x = b))): i.e., for any a and b, {a, b} exists. Axiom 3.3. (Union) ( a.( b.( x.x b ( y.x y y a)))): i.e, for any a, a exists. Axiom 3.4. (Power Set) ( a.( b.( x.x b ( y.y x y a)))): i.e., for any a, P(a) exists. Axiom 3.5. (Separation) For any formula φ in which each quantifier is restricted to a set, the universal closure of ( a.( b.( x.x b x a φ))) is an axiom: i.e., if each quantifier in φ is restricted to a set, then {x a φ} exists for each set a. Axiom 3.6. (Infinity) The first infinite von Neumann ordinal ω exists. Axiom 3.7. (Foundation) Any set x has an element y such that x y =. Axiom 3.8. (Choice) Any pairwise disjoint collection of nonempty sets has a choice set. This is distinguished from a modern version of Zermelo set theory only by the restriction to bounded formulas in the Separation scheme. This theory is not quite satisfactory for our purposes because it does not have a sensible relationship to the usual picture of the universe of set theory as generated by the iteration of the power set operation along the ordinals (the cumulative hierarchy of sets). We rectify this situation by defining the ranks of the cumulative hierarchy and giving a new axiom which adds no essential consistency strength but ensures that every set belongs to a rank. An excellent discussion of these issues is found in the previously mentioned [17]. Proofs are omitted. They are quite standard. Definition 3.1. A hierarchy is a set H such that ( I H.I = I I) (H is well-ordered by inclusion), ( I H. I H) (H is closed under set union of its subsets) and for each h H such that h H, the immediate successor of h in the inclusion order on H is P(h). 7

8 Definition 3.2. A set r is a rank iff there is a hierarchy H such that r H. Theorem 3.1. For any hierarchies H 1 and H 2, either H 1 H 2 or H 2 H 1. Corollary 3.1. All ranks (not just those in a particular hierarchy) are wellordered by inclusion. Axiom 3.9. (Rank) For any set x, there is a rank r such that x r. (Note that x P(r) will hold and P(r) will also be a rank: it is more convenient to stipulate that everything is included in a rank rather than that everything belongs to a rank, but the two formulations are equivalent). Definition 3.3. For any set x, we define the rank of x as the minimal rank r in the inclusion order such that x r. An amusing fact is that Mac Lane set theory with the Axiom of Rank (and with Infinity) is adequately axiomatized as follows: Axiom (Extensionality) ( AB.A = B ( x.x A x B)) Axiom (Power Set) ( a.( b.( x.x b ( y.y x y a)))): i.e., for any a, P(a) exists. Axiom (Separation) For any formula φ in which each quantifier is restricted to a set, the universal closure of ( a.( b.( x.x b x a φ))) is an axiom: i.e,, if each quantifier in φ is restricted to a set, then {x a φ} exists for each set a. Axiom (Rank) For any set x, there is a rank r such that x r. Axiom (Infinity) The first infinite von Neumann ordinal ω exists. Axiom (Choice) Any pairwise disjoint collection of nonempty sets has a choice set. That Foundation is not needed should not be surprising. That Pairing and Union are redundant may be slightly less obvious. It is a by no means original observation that the von Neumann definition of ordinals as transitive sets strictly well-ordered by membership is not satisfactory for weak set theories such as Zermelo set theory or Mac Lane set theory. The difficulty is that the von Neumann ordinal ω 2 cannot be shown to exist, while much longer well-orderings are readily constructed. We use Scott s trick ([20]) to solve this problem. 8

9 Definition 3.4. Let X be a set. For any set A, we define the local cardinal A X as the collection of all subsets of X equinumerous with A, and for any well-ordering we describe the local ordinal ot( ) X as the collection of all well-orderings of subsets of X which are similar to. Now we eliminate the locality of these definitions (the parameter X) by an appeal to the hierarchy: define A (resp. ot( )) as A r (resp. ot( ) r ), where r is the inclusion minimal rank s such that A s (resp. ot( ) s ) is nonempty. Definition 3.5. For any ordinal α we define V α as the rank r (if there is one) such that the order type of the inclusion order on the set of ranks strictly included in r is α. An extensive development of set theory in Zermelo (not Mac Lane) set theory with the Axiom of Rank, formulated in a way which resembles our minimal axiomatization just given and making extensive use of Scott s trick, is found in [18]. 4 The Boffa model construction for NFU We consider a model (M, E) of Mac Lane set theory with the Rank Axiom (and without any need to assume Infinity as yet). The metatheory may be supposed to be Mac Lane set theory with Infinity and Rank. M is the domain of objects which are sets of the model and E is the membership relation of the model. Following a common convention, we will write M = φ for the sentence φ is satisfied in the model M where we should ideally write (M, E) = φ. Definition 4.1. We give the recursive definition of M = φ, purely for the sake of precision. We assume that our language contains a name for each element of M (this simplifies the definition). If c is a name (or any term), c is its referent in M. 1. (atomic formulas) M = c = d iff c = d ; M = c d iff c E d. 2. (logical connectives) M = φ iff it is not the case that M = φ; M = φ ψ iff M = φ or M = ψ. 3. (quantifiers) M = ( x.φ) iff M = φ[c/x] for every name c. 9

10 The satisfaction relation for other models introduced below will be defined in the same way (we may add clauses to the definition of satisfaction for other models, but we will not repeat these clauses). In any model (M, E) of Mac Lane set theory with Rank, the collection {r M = r is a rank } is an infinite set (in the metatheory) and M = the inclusion order on ranks is a linear order. It follows by well-known results of model theory (see [5]) that we can stipulate further that the model M has a automorphism which moves an M-rank downward in the inclusion order. We select such an automorphism J and note that we now have a structure (M, E, J) for an expanded language: for any term t we let j(t) be a term representing J(t ) (this can be iterated, of course). We continue to stipulate that every element of M has an atomic name. The following will hold: 1. for all x, y M, J(x) = J(y) x = y 2. for all x, y M, J(x) E J(y) x E y 3. for some r such that M = r is a rank, (and so M = j(r) is a rank, where j(r) denotes J(r ) in our expanded language, as stipulated above) we have M = j(r) r j(r) r. The structure (M, E, J) satisfies the following additional axioms involving the new symbol j: Axiom 4.1. (Automorphism) ( x.( y.j(y) = x)), ( xy.j(x) = j(y) x = y), and ( xy.j(x) j(y) x y) Axiom 4.2. (Nontriviality) There is a rank r such that j(r) r and j(r) r. It must be noted that the axiom scheme of Separation satisfied by (M, E, J) does not apply to formulas that include the symbol j, except where j is applied to a term containing no bound variable: in this latter case each term involving j can be replaced with a free variable and the universal closure taken to obtain a j-free instance of Separation of which it is a case. We can now present the usual model construction for NFU. This model construction is very similar in motivation to Jensen s original argument for the consistency of NFU, but the construction is apparently due to Boffa in [2] (Thomas Forster references this paper as the source for this construction when he describes it in [6]). 10

11 Fix an M-rank r moved by the automorphism J. M = (!α.r = V α ). We fix α M witnessing this. Obviously the object α does not correspond to any actual ordinal, as the ordinals do not admit a nontrivial automorphism. The domain M 1 of the model we construct is {x M = x V α }. The membership relation E 1 of the model we construct is {(x, y) M = j(x) y y V j(α)+1 }. Satisfaction M 1 = φ of sentences φ of the language of set theory in the model (M 1, E 1 ) is defined in a manner precisely analogous to the definition of M = φ above, with one technical clause: empty set: M 1 = φ[ /x] iff M = φ[ /x]. This is needed because is a primitive notion in the language of NFU, whereas it is a defined notion in the language of Mac Lane set theory (and so eliminable from M = φ[ /x]). We stipulate that the empty set in our model of NFU is the empty set of M; we need to do this because the empty set of M is not the only elementless object in the model M 1 : all elements of V α V j(α)+1 in the sense of M are urelements in the sense of M 1. We call models (M 1, E 1 ) constructed in this way from models (M, E, J) Boffa models of NFU. It remains to prove that a Boffa model of NFU is actually a model of NFU. Theorem 4.1. If (M, E) is a model of Mac Lane set theory with Rank and without Infinity, and J is an automorphism of this model moving a rank downward, then the model (M 1, E 1 ) defined as above is a model of NFU: that is, for each axiom φ of NFU, M 1 = φ holds. Proof. Recall that r = V α is a fixed rank moved downward by the automorphism introduced in the definition of (M 1, E 1 ) above. The Axiom of Empty Set is obviously satisfied: M 1 = c because M = c, for any term c. We show that the Axiom of Extensionality is satisfied. M 1 = set(c) iff M 1 = c = ( x.x c), which will hold iff M = c = or M = ( x V α.j(x) c c V j(α)+1 ) (notice that we do not need to stipulate explicitly that M = c V α because a stronger condition in c is included). This is clearly equivalent to simply M = c V j(α)+1 : note that in the model M an element of V j(α)+1 has an element j(x) such that x V α precisely if it has an element in V j(α) and thus precisely if it has an element at all, so the first alternative is that c is the empty element of V j(α)+1 and the second alternative is that c is some nonempty element of V j(α)+1. 11

12 We fix c, A, B M 1, and assume M 1 = c A ( x.x A x B) and argue to M 1 = A = B. This shows that M 1 satisfies weak extensionality. Since M 1 = c A we have M = j(c) A A V j(α)+1. Since M 1 = ( x.x A x B), we have for any d M 1, M = j(d) A A V j(α)+1 iff M = j(d) B B V j(α)+1. In particular, letting d = c, we see that M = B V j(α)+1, so we can simplify this to M = j(d) A iff M = j(d) B for any d M 1. So M = ( x V α.j(x) A j(x) B), which is equivalent to M = ( x V j(α).x A x B), from which we can deduce M = A = B because we know that M = A V j(α)+1 B V j(α)+1, so in the sense of M, all elements of either set are in V j(α), and their equality follows from extensionality in M. We now show that for any stratified formula φ in which the variable A does not occur free, M 1 = ( A.( x.x A φ)). Fix such a stratified formula φ and a stratification σ of φ. Our aim is to show M = ( A.A V j(α)+1 ( x V α.(j(x) A φ 1 ))), where φ 1 is the translation into the language of (M, E, J) of φ (this is not precisely equivalent to the translation into the language of M of the original sentence, but clearly implies it). We recall that the translation of φ to φ 1 is to be effected by replacing each atomic membership formula x y in φ with j(x) y y V j(α)+1 and restricting each quantifier in φ to the set V α. We introduce the notation j n (x) for integers n with the obvious meaning. We also introduce the notation φ jn for the result of replacing each term in φ appearing as an argument in an atomic subformula with its image under j n (and write φ j instead of φ j1 ): evidently M = φ φ jn. Now observe that M = ( A.A V j(α)+1 ( x V α.(j(x) A φ 1 ))) is equivalent to M = ( A.A V j(α)+1 ( x V j(α).(x A φ 1 [j 1 (x)/x]))), which is equivalent to M = {x V j(α) φ 1 [j 1 (x)/x]} exists. The assertion M = {x V j(α) φ 1 [j 1 (x)/x]} exists appears to be an instance of Separation (note that all quantifiers in φ 1 are bounded), but is not. The problem is that it contains essential occurrences of the automorphism j. We complete the proof by showing that the formula φ 1 [j 1 (x)/x] is equivalent to a formula φ in which j occurs only in parameters (terms containing no bound variable): M = {x V j(α) φ } exists then holds because it is an instance of Separation after all, and is equivalent to the assertion we are trying to prove. We define a sequence of formulas equivalent to φ 1 [j 1 (x)/x] terminating with the desired formula φ. φ 2 we define as (φ 1 [j 1 (x)/x]) j. This formula has the merit that j 1 (x) 12

13 is replaced with x (while each term or free variable in φ is replaced with its image under j, which will not cause any serious difficulties as we are not committed to any details about these parameters). Notice that the constants V α and V j(α)+1 which we expect to occur in φ 1 are replaced by their images under j. We choose an integer constant N greater or equal to σ(u) for any variable u appearing in φ. We obtain φ 3 by replacing each atomic formula j(u) v in φ 3 with j N σ(v) (j(u)) j N σ(v) (v) [note that j N σ(v) (j(u)) is the same as j N σ(u) (u), because σ is a stratification], each atomic formula v V α+1 with j N σ(v) (v) j N σ(v) (V α+1 ), and each atomic formula u = v with j N σ(v) (u) = j N σ(v) (v) [note that j N σ(v) (u) is the same as j N σ(u) (u) because σ is a stratification]. Observe that each variable u occurring in φ now occurs with exactly N σ(u) applications of j (this only works because we know that φ is stratified: see the bracketed remarks in the previous paragraph). We can then eliminate all applications of j to bound variables by using the equivalence of ( u A.ψ[j n (u)/u]) to ( u j n (A).ψ). The formula which results from this transformation is φ 4. Finally, we define φ as (φ 4 ) j N+σ(x) : we need to convert all the occurrences of x with N σ(x) applications of j to occurrences of x with no application of j. φ is a formula in which all quantifiers are restricted and all occurrences of j are in parameters, so M = {x V α φ } exists is an instance of Separation in the model M and so holds. This completes the proof that (M 1, E 1 ) is a model of NFU. We now know that NFU is consistent on very reasonable assumptions, and we know that the Boffa model construction works to obtain a model of NFU from any model of Mac Lane set theory - Infinity + Rank with an external automorphism moving a rank [everything we say will apply if Infinity does hold, of course]. 5 The Axiom of Endomorphism and the type level membership relation In this section, we identify a function which exists in any Boffa model (M 1, E 1 ) of NFU which has a simple definition in terms of the underlying model (M, E, J) and whose behavior can be axiomatized in the language 13

14 of NFU, giving an additional statement in the language of NFU which holds in any Boffa model. We work in a fixed Boffa model (M 1, E 1 ) with underlying model (M, E, J) of Mac Lane set theory - Infinity + Rank (of course everything we do applies to a model in which Infinity holds). Whenever we have M = A V α, we have M = c A iff M = j(c) j(a) j(a) V j(α)+1 (the first clause being equivalent to the original statement and the second simply true), which is in turn true iff M 1 = c j(a). The set A of the model M is implemented by the set j(a) in M 1. A special case of this is that M 1 = x = {u, v} is equivalent for x, u, v M 1 to M = x = j({u, v}). This implies that M 1 = x = (u, v) is equivalent to M = x = j 2 (u, v). If M = A V α B V α f : A B, we observe that M = (x, y) f is equivalent to M = j 3 (x, y) j 3 (f) which is in turn equivalent to M = j 3 (x, y) j 3 (f) j 3 (f) V j(α)+1 (because the additional clause is simply true: f lives in V α+2 and two applications of j will serve to take f into V j(α)+1 ), which is in turn equivalent to M 1 = (x, y) j 3 (f). This shows that the function f in the model M is implemented by the function j 3 (f) in the model M 1 in a certain precise sense. We are now ready to define a function of interest which is found in every Boffa model of NFU. Definition 5.1. We define S M as the object such that M = S = j 3 ({({x}, x) x V j(α) }). M = ( x V α.j 3 (S)({j(x)}) = j(x)); from this it follows by considerations above that M 1 = ( x.s({x}) = j(x)). M 1 sees S as a function sending each singleton {x} to j(x). In other words, S codes information about the automorphism J (or at least its restriction to M 1 ). One should note here that we strictly speaking have to extend our structure from (M 1, E 1 ) to (M 1, E 1, J (M 1 ) 2 ) to support the addition of the notation j to the language of NFU. But note that we never have to use the notation j(x) again, since we have shown it to be equivalent to S({x}). We present some properties of S from the perspective of M 1. M 1 = S is a one-to-one function with domain the set of all singletons and codomain the collection of all sets. This should be evident from the discussion above and the fact that for any x V α we have j(x) V j(α) V j(α)+1. Note that M 1 does not see the range of S as the collection of all sets: there are sets not in its range. 14

15 If M 1 = set(x) we claim that M 1 = S({x}) = {S({y}) y x}, that is, M 1 = ( z.z S({x}) ( y.z = S({y}) y x)). There is no problem here with weak extensionality as opposed to strong extensionality because we know that S({x}) is a set in the eyes of M 1. This is equivalent to M = ( z V α.j(z) j(x) ( y V α.z = j(y) j(y) x)), and this is true (we do not need extra clauses providing that things are in V j(α)+1 because this is already granted in the prior assumptions: note that we know that M = x V j(α)+1, which is why we know that its element z has a preimage under j). Now we can frame an axiom to be adjoined to NFU which holds in all Boffa models. V is notation for the universal set. P 1 (A) is notation for the set of all one-element subsets of A. Axiom 5.1. (Endomorphism, for NFU) There is a one-to-one function S : P 1 (V ) P(V ) such that for any set A, S({A}) = {S({x}) x A} [and the range of S is downward closed under inclusion]. The additional condition that the range of S is downward closed under inclusion, not found in the original formulation in [10], holds because V j(α) contains all subsets of its elements. Once we have S at our disposal, we can recover not only the function J but the relation E. In the context of NFU, we assume the Axiom of Endomorphism and fix a specific function S witnessing the axiom. Definition 5.2. x E y is defined as x S({y}). Notice that this relation is a set relation in NFU : x and y would be assigned the same relative type by any stratification, so {(x, y) x E y} exists as a set. Further, we show the following very convenient result. Lemma 5.1. For all x, y M 1, M 1 = x E y iff M = x y. Proof. M 1 = x E y is equivalent to M 1 = x S({y}), which is in turn equivalent to M = j(x) j(y) j(y) V j(α)+1, which is equivalent to M = j(x) j(y) (the omitted clause is simply true), which is equivalent to M = x y as desired. 15

16 There is another important statement which is satisfied in all Boffa models which we will describe below (a native form of the Axiom of Rank in terms of the type level membership relation E). We prove a lemma that will be needed below. Definition 5.3. x E y is defined as x S({y}). J(x) is defined as S({x}). For any formula φ in the language of set theory with a function symbol j, φ E is the assertion in the language of NFU obtained by replacing with E and j with J everywhere. Lemma 5.2. There is a formula φ(x, y) of the language of set theory with j such that φ(x, y) E x y is a theorem of NFU + Endomorphism. φ(x, y) def j(x) y y {j(x) x = x}. Proof. (y = {j(x) x = x}) E ( z.z E y ( w.z = S({w}))), or equivalently ( z.z S({y}) ( w.z = S({w}))), or equivalently S({y}) = {S({w}) w = w}, and this is the case by the Axiom of Endomorphism iff y = {x x = x} = V, the universal set. So the symbol {j(x) x = x} in the language with type level membership and j refers to the universal set in the context of NFU. (x y) E ( z.z x z y) E which is in turn equivalent to ( z.z S({x}) z S({y})), which can in turn briefly be expressed as S({x}) S({y}). Now we verify the equivalence of φ(x, y) E with x y. (j(x) y y {j(x) x = x}) E S({x}) S({y}) S({y}) S({V }). We claim that x y implies S({x}) S({y}) S({y}) S({V }). If x y then y is a set, so S({y}) = {S({z}) z y}, and it follows that S({x}) S({y}). Further, S({V }) = {S({w}) w = w}, and it is then obvious that S({y}) = {S({z}) z y} {S({w}) w = w} = S({V }). We claim that S({x}) S({y}) S({y}) S({V }) implies x y. S({y}) S({V }) implies S({y}) {S({w}) w = w} so S({y}) = {S({w}) S({w}) S({y})}. Now x {w S({w}) S({y})}. Now observe that S({{w S({w}) S({y})}} = {S({w}) S({w}) S({y})} = S({y}) so {w S({w}) S({y})} = y (S is a bijection) so x y. We show that the theory of E and J satisfies certain familiar set theory axioms. We will only use these to support a brief remark later, but they have interest. The following assertions are provable in NFU + Endomorphism. We will have occasion to refer to these in a later section. This is the only place in the 16

17 paper where the strengthening of Endomorphism to include the condition that the range of S is downward closed under inclusion is used. (Extensionality) E : ( x.x A x B) E ( x.x S({A}) x S({B})), which is in turn equivalent to S({A}) = S({B}), which is in turn equivalent to A = B. (Separation) E : For any formula φ in membership and equality alone (not involving j), and any object a, {x S({a}) φ E } exists (because E is a set relation) and is equal to S({b}) for some b because the range of S is downward closed under inclusion. Now for any x, x E b is equivalent to x S({b}), and so to x S({a}) φ E, and thus to x E a φ E as desired. (j is an endomorphism) E : ( xy.j(x) = j(y) x = y) E ( xy.s({x}) = S({y}) x = y), which is true because S is injective. ( x.j(x) = {j(y) y x}) E is equivalent to ( y.y S({x}) S({y}) S({S({x})})). The latter statement holds because S({x}) is a set, so S({S({x})}) = {S{y} y S({x})}. From this the weaker assertion that ( xy.x y j(x) j(y)) E follows directly. ({j(x) x = x} exists) E : It has already been proved that (y = {j(x) x = x) E is equivalent to y = V. (P(j(x)) exists) E : S({x}) is a set, and every subset of S({x}) is of the form S({v}) for some v (because the range of S is downward closed under inclusion) so P(S({x})) is equal to {S({v}) S({v}) S({x})} = S({{v S({v}) S({x})}}). It follows that for any z, z E {v S({v}) S({x})} iff z {S({v}) S({v}) S({x})} iff z S({x}). Finally, z S({x}) is equivalent to (z j(x)) E, because in general z y is equivalent to (z y) E if y is a set. This establishes that (y = P(j(x))) E is equivalent to y = {v S({v}) S({x})}. 6 Recovering the lost information In the previous section we showed that if we have a Boffa model (M 1, E 1 ) and a name for the function S of the underlying model (M, E, J), we can define the restrictions of E and J to M 1 using the language of NFU augmented with a name for S. 17

18 In this section we show that if there is a function S witnessing the Axiom of Endomorphism in a Boffa model then there is only one such function. It follows from this result that S is definable in the language of NFU (in principle in terms of equality, membership and alone) as the unique object (if any) witnessing the truth of Endomorphism. It follows then that the restrictions to M 1 of the relations E and J of the underlying model (M, E, J) are definable in the language of NFU in the Boffa model (M 1, E 1 ), and in a way which does not depend on which Boffa model is considered, which is the main result of this paper. Theorem 6.1. In any Boffa model of NFU, there is at most one relation S witnessing the truth of the Axiom of Endomorphism. Proof. We observe that there is at least one such relation S, defined above. We suppose for the sake of a contradiction that there is a second such relation S. We argue entirely in terms of the model M, rather than attempting to prove anything in the theory NFU. It is possible to prove uniqueness of S entirely inside NFU (using the formulation of the Axiom of Rank native to Boffa models of NFU which we omitted to give above) but we do not take this approach here. In order to reason in terms of M, we need to express the defining property of S (and S ) entirely in terms of the model M. M 1 = S is a function with domain the set of singletons and codomain the set of all sets is true iff M = j 3 (S ) is a function from the set of singletons of elements of V j(α) to V j(α)+1. M 1 = ( x.set(x) S ({x}) = {S ({y}) y x}) is equivalent to M = ( x V α.x V j(α)+1 j 3 (S )({j(x)}) = j({j 3 (S )({j(y)}) j(y) x})). This is easier to sort out if we let Σ = j 3 (S ) (and Σ = j 3 (S)). Further, we can replace j(y) harmlessly with y. We then have that the model M says that Σ is a function from P 1 (V j(α) ) to V j(α)+1 with the property that for any x V j(α)+1 we have Σ ({j(x)}) = j({σ ({y}) y x}). Note that this is obviously true for the function Σ: j(x) = Σ({j(x)}) = j({σ({y}) y x}) = j({y y x}). Now if Σ Σ there must be an object x of least rank such that Σ({x}) Σ ({x}). Let β be minimal such that x V β. We have β j(α) by considerations of the domain of Σ and Σ. It follows that {x} V j(α)+1, so Σ ({{j(x)}}) = j({σ ({y}) y {x}}) = j({σ ({x})}), whence it follows that Σ({{j(x)}}) = {j(x)} = j({σ({x})}) j({σ ({x})}) = Σ {{j(x)}}), 18

19 so if x is a counterexample, so is {j(x)}. The minimal γ such that {j(x)} V γ is j(β) + 1, so we have j(β) + 1 β by choice of x. It is important to note that j(β) + 1 β actually implies j(β) + 1 > β. j(β) + 1 = β is impossible, because the parity of the finite parts of the ordinals β and j(β) must be the same. Thus j(β) + 1 β implies j(β) β as well. We now argue that x = j(y) for some y V j(α)+1. It is sufficient to show that β j 2 (α) + 1. To see this, observe that j(α) < α, so j n+1 (α) < j n (α) for each n, so for any γ with j n+1 (α) < γ j n (α) we would have j n+2 < j(γ) j n+1 (α) < γ j n (α), whence j(γ) < γ, so such a γ cannot be β. This is enough to see that we cannot have j 2 (α) < β j(α). Since β < j 2 (α) + 1, it follows that x = j(y) for some y V j(α)+1. It follows that Σ ({x}) = Σ ({j(y)}) = j({σ ({z}) z y}) = {j(σ ({z})) z y} = {j(σ ({z})) j(z) x}. Now since Σ ({x}) Σ({x}) = x, and since all members of x are of the form j(z) for some z, there must be a z such that j(σ ({z})) j(z), and so Σ ({z}) z. Let γ be minimal such that z V γ. We have β γ because z is a counterexample. We have j(γ) < β because j(z) x. From this we can deduce j(β) j(γ) < β, but this contradicts j(β) β. From this contradiction we conclude that the original assumption that S S and thus Σ Σ was false, completing the proof of the theorem. Theorem 6.2. (our main result) There are formulas φ(x, y) and ψ(x, y) of the language of NFU such that in any Boffa model (M 1, E 1 ) obtained from a model (M, E, J) of Mac Lane set theory with Rank as above, M 1 = φ(x, y) iff M = x y and M 1 = ψ(x, y) iff M = y = j(x). A less formal statement of this is In any Boffa model of NFU, the membership relation and automorphism of the underlying model of Mac Lane set theory with Rank are definable in the language of NFU [in a way which does not depend on the particular Boffa model considered]. Proof. The import of φ(x, y) is there is a witness S to the Axiom of Endomorphism and x S({y}). The import of ψ(x, y) is there is a witness S to the Axiom of Endomorphism and y = S({x}). It would be routine but space-consuming and not particularly helpful to develop these into formulas of the language of NFU in the most precise sense, with equality, membership and as the only non-logical notions used. The fact that these formulas work follows from the previous theorem and the fact that we already know how to represent the membership relation 19

20 and automorphism of the underlying model if we are given a name for the function S in a Boffa model. It is worth noting that it is not asserted here that the class of Boffa models is first-order definable (i.e., coincides with the class of models of some first order theory) though we will see below that this is the case. Remark 6.1. In this paper we have defined Boffa models using the description given by Forster in [6] (which Boffa also gave to us in person); however, the description in the original paper [2] is more general. Boffa introduces the theory ZFJ, ZF with an external automorphism j (nothing in his discussion would fail if we weakened ZF to Mac Lane set theory with Rank), and remarks that a model of NFU is obtained if one has a set X in a model of ZFJ such that P(j(X)) X. The domain of the model of NFU will be the set X. The definition of membership in a general Boffa model is x NF U y def j(x) y y P(j(X)). The construction we have given is the special case of this in which X is a rank. The proof that the more general Boffa models are models of NFU is essentially the same as the proof given above. The Axiom of Endomorphism holds in the general Boffa models for the same reasons that it holds in the ones we have described. Finally, the argument just given for the uniqueness of the function S witnessing the Axiom of Endomorphism can also be seen to succeed for the generalized Boffa construction with an additional condition on X somewhat weaker than X is a rank. Let V α be the smallest rank which contains X as a subset. The crucial step is the analogue of the one in which we argue from the rank of the minimal rank counterexample x being less than j 2 (α) + 1 that x must be j(y) for some y of rank less than j(α)+1. What we need in the more general context is that fact that the rank of the minimal rank counterexample x is less than j 2 (α) + 1 (which we can show essentially as above) must imply that x = j(y) for some y P(j(X)). What seems to be required for this to work is that X include the rank V j(α) as a subset (it is sufficient for X to include j 1 (X V j 2 (α)), but this does not seem to be a very natural condition). An example of a condition on X expressible without reference to j which would imply this is ( γ.v γ X V γ+1 ); what is required of such a condition is that it imply that X includes as a subset some rank perhaps below but not far below its own rank. 20

21 7 The theory of a rank with an endomorphism and the theory of a Boffa model Let (M, E, J) be a model of Mac Lane set theory with an automorphism. Let V α be a name for a rank in the model which is moved downward by the automorphism. Let M 1 be defined as {x M M = x V α }. We describe a theory of which (M 1, E (M 1 ) 2, J (M 1 ) 2 ) will be a model. Moreover, any model of this theory is isomorphic to a model (M 1, E (M 1 ) 2, J (M 1 ) 2 ) obtained as above. In other words, we isolate the theory of a rank in Mac Lane set theory with a nontrivial endomorphism. We described a similar but simpler theory in [10] and related it to models of NFU. The theory is the first order theory with equality, membership and a function j as primitive notions and with the following axioms. We call this theory truncated Mac Lane set theory with Endomorphism. Axiom 7.1. (Extensionality) ( AB.A = B ( x.x A x B)) Axiom 7.2. (Separation) For any formula φ in which the variable A does not occur free, in which the symbol j does not occur, and in which each quantifier is restricted to a set, the universal closure of ( a.( A.( x.x A x a φ))) is an axiom: in other words, for any such formula φ and any set a, {x a φ} exists. Axiom 7.3. (Endomorphism. for truncated Mac Lane) ( xy.x = y j(x) = j(y); ( x.j(x) = {j(y) y x}). Definition 7.1. (modified definition of rank) A hierarchy is a set H such that H is well-ordered by inclusion, H contains as an element the union of any subset of H, and the immediate successor of any h H other than H in the inclusion order is P(h). A pre-rank is a set which is an element of some hierarchy. A rank is a set r which is either the power set of a prerank or has the property that every element of r is an element of a pre-rank which is a subset of r (i.e, r is the union of a class of pre-ranks which is not necessarily a set). It should be evident that any pre-rank is a rank. Axiom 7.4. (Rank) Each set is a subset of a rank. Axiom 7.5. (Nontriviality) {j(x) x = x} is a set and a rank. Axiom 7.6. (Restricted Power Set) For any set x, P(j(x)) is a set. 21

22 This is not a minimal set of axioms: the last one can be proved from the others, but showing this is not useful here. Theorem 7.1. If (M, E, J) is a model of Mac Lane set theory with Rank with an automorphism and M 1 is the extension of an M-rank moved by the automorphism, then (M 1, E (M 1 ) 2, J (M 1 ) 2 ) is a model of truncated Mac Lane set theory with endomorphism. Proof. That Extensionality and Separation are satisfied in (M 1, E (M 1 ) 2, J (M 1 ) 2 ) is obvious. That Endomorphism is satisfied follows from the facts that J is an automorphism of the original model (M, E, J) and that J M 1 is downward closed under E. The axiom of Nontriviality is witnessed by Vj(α), whose E-preimage has as its elements exactly the images under J of the elements of M 1, and which is an element of M 1. Restricted Power Set holds because M sees any set j(x) as belonging to V j(α) and P(j(x)) as belonging to V j(α)+1 V α, so (P(j(x))) M 1, and M 1 sees j(x) as having a power set. The modification of the definition of rank (and so of the axiom of Rank) and the modification of the axiom of Power Set are both motivated by the fact that M might see V α as a successor rank. M 1 will then see V α 1 (and any other set of rank α 1) as having no power set. In addition, M 1 will not see V α 1 as a pre-rank, because M will see any hierarchy containing V α 1 as an element as having rank at least α and so M 1 will see no hierarchy containing V α 1 as an element at all. If V α 1 is itself a successor rank in M (and so also in M 1 ), then M 1 will see V α 2 as a pre-rank (a hierarchy containing it will live in rank α 1) and V α 1 as the power set of V α 2. If V α 1 is a limit rank in M (and so also in M 1 ) then M 1 will see every element of V α 1 as an element of some pre-rank which is a subset of V α 1. Every other M-rank belonging to M 1 will also be an M 1 -(pre-rank) and so an M 1 -rank (and indeed if M 1 is the extension of an M-limit rank the usual axioms of Rank and Power Set will both hold: V α 1, if it exists, is the only problem). It is further useful to observe that the modified definition of rank and the original definition are equivalent if the Power Set axiom holds. Theorem 7.2. Let (N, E 0, J 0 ) be any model of truncated Mac Lane set theory with endomorphism. There is a model (M, E, J) of Mac Lane set theory with automorphism with the property that (N, E 0, J 0 ) will be isomorphic to a model (M 1, E (M 1 ) 2, J (M 1 ) 2 ) obtained from (M, E, J) by restriction to the extension M 1 of an M-rank moved by the automorphism. 22

23 Proof. The basic idea is that we will augment the model N with iterated formal inverses j n (x) under the endomorphism j, thus converting the endomorphism into an automorphism. The intention that the extension of j be an automorphism determines exactly how equality between such formal terms is to be defined and how the membership relation is to be extended. We define an intermediate structure (N N,, E 1, J 1 ) whose equality relation is a nontrivial equivalence relation we will define shortly. The intention is that (x, n) will be the referent of j n (x) for each term x (and so each x M) and for each natural number n. (x, m) (y, n) is defined as holding iff J n (x) = J m (y). (x, m)e 1 (y, n) is defined as holding iff J n (x)e 0 J m (y). J 1 (x, n) is defined as (J 0 (x), n). M is defined as the collection of all -equivalence classes. [x]e[y] is defined as xe 1 y; J([x]) is defined as [J 1 (x)]; coherence of these definitions is straightforward to verify. For any natural number i, define D i as {[(x, i)] x M}. It is readily seen that (D i, E Di 2, J Di 2 ) is isomorphic to the original model N for each i via the map x [(x, i)] and further that D i is downward closed under E. From this it is easy to see that Extensionality, Separation, modified Rank and restricted Power Set hold in the new structure (M, E, J), since it is a union of suitably nested copies of structures in which these axioms hold. That full Power Set holds in (M, E, J) follows from the fact that for any x, j 1 (P(j(x))) will serve as the power set of x. More formally, for any [(x, n)] M, [((P(j(x))), n + 1)] can be seen to be the power set of x in the sense of the model M. Now the full form of the Axiom of Rank follows from the modified form and the axiom of Power Set. It is evident from the construction that J is an automorphism, and that J moves the element of M whose E-preimage is M 1 = {[(x, 0)] x N}, which is an M-rank moved downward by the automorphism, from which it is evident that (N, E 0, J 0 ) is isomorphic to a model (M 1, E (M 1 ) 2, J (M 1 ) 2 ) obtained from (M, E, J) as above. Now we connect this with what was done above. If (M, E, J) is a model of Mac Lane set theory with automorphism, M 1 is the extension of an M- rank moved downward by J, and (M 1, E 1 ) is the associated Boffa model, we showed in the previous section that the relations E (M 1 ) 2 and J (M 1 ) 2 are definable entirely in the language of the Boffa model (the converse fact that E 1 is definable in terms of the relations E (M 1 ) 2 and J (M 1 ) 2 is immediate from the definition of E 1 ). So we can recover a uniquely determined model 23

24 of truncated Mac Lane set theory with endomorphism from a Boffa model of NFU. Further, we can use the construction just given to obtain a uniquely determined model of Mac Lane set theory with an automorphism from the model of truncated Mac Lane set theory. This will be isomorphic to part of the original model (M, E, J) with which we started, namely the union of the sets J n M 1 for n N. This means that from every Boffa model of NFU we can obtain a uniquely determined model (M, E, J) of Mac Lane set theory with automorphism, with the property that the iterated inverse images of a particular M-rank under j are cofinal in the ranks of the model. This is a bit unsatisfactory as this class of models of Mac Lane is not describable in first-order terms. It is worth noting that the correspondence between Boffa models and models of Mac Lane is many-to-one: each Boffa model determines a unique model of Mac Lane as indicated above, but a model of Mac Lane with automorphism determines different Boffa models if different nonstandard ranks are taken as the domain. We show that the Boffa models of NFU are the models of a certain firstorder theory. We identify a theory of which any Boffa model is a model, and then demonstrate that any model of this theory is actually a Boffa model of NFU. The theory in question, to which we will give the nonce name Boffa set theory, has the following axioms: NFU: All axioms of NFU are axioms of this theory. Endomorphism for NFU with uniqueness: There is exactly one function S which witnesses the Axiom of Endomorphism for NFU. Definitions: x E y is defined as x S({y}). J(x) is defined as S({x}). For any formula φ in the language of Mac Lane set theory with endomorphism, φ E is the assertion in the language of NFU obtained by replacing with E and j with J everywhere. truncated Mac Lane: For each axiom φ of truncated Mac Lane set theory with endomorphism, φ E is an axiom of this theory. Theorem 7.3. The models of Boffa set theory are (up to isomorphism) exactly the Boffa models of NFU. Thus the class of Boffa models of NFU is first-order definable. 24