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5 Final remarks 40

6 Acknowledgements 42

1 Introduction

It is rather difficult to determine when functions were born, since mathematics itself significantly evolves along history. Specially nowadays we find different formal concepts associated to the label function. But an educated guess could point towards Sharaf al-D¯ın al-T¯us¯ı who, in the 12th century, not only introduced a ‘dynamical’ concept which could be interpreted as some notion of function, but also studied how to determine the maxima of such functions [13]. All usual mathematical approaches for physical theories are based on either differential equations or systems of differential equations whose solutions (when they exist) are either functions or classes of functions (see, e.g., [29]). In pure mathematics the situation is no different. Continuous functions, linear transformations, homomorphisms, and homeomorphisms, for example, play a fundamental role in topology, linear algebra, group theory, and differential geometry, respectively. Category theory [20] emphasizes such a role in a very clear, elegant, and comprehensive way. As remarked by Marquis [21], category theory allows us to distinguish between canonical and noncanonical maps, in a way which is not usually achieved within purely extensional set theories, like ZFC. And canonical maps “constitute the highway system of mathematical concepts”. Concepts of symmetry are essential in pure and applied mathematics, and they are stated by means of group transformations [32]. Some authors suggest that functions are supposed to play a strategic role into the foundations of mathematics [35] and even mathematics teaching [17], rather than sets. The irony of such perspective starts in the historical roots of set theories. Georg Cantor’s seminal works on sets were strongly motivated by Bernard Bolzano’s manuscripts on infinite multitudes called Menge [34]. Those collections were supposed to be conceived in a way such that the arrangement of their components is unimportant. However, Bolzano insisted on an Euclidian view that the whole should be greater than a part, while Cantor proposed a quite different approach: to compare infinite quantities we should consider a one-to-one correspondence between collections. Cantor’s concept of collection (in his famous Mengenlehre) was strongly committed to the idea of function. Subsequent formalizations of Cantor’s “theory” were developed in a way such that all strategic terms were associated to an intended interpretation of collection. The result of that effort is a strange phenomenon from the point of view of theories of definition: Padoa’s principle allows us to show domains of functions are definable from functions themselves [4] [5] [27]. Thus, even in extensional set theories functions seem to play a more fundamental role than sets. Nevertheless, there is another crucial role that functions (in a broader intu-

2 itive sense) play in the study of set theories and cannot be ignored, namely, those questions regarding the use of ∈-automorphisms [15] and non-trivial elementary embeddings of a universe into itself [16], among other examples. Some questions regarding set theories cannot be answered by means of the use of their axioms alone. That is where model theory provides metamathematical tools to answer to some of those questions [33]. Non-trivial elementary embeddings and ∈-automorphisms are examples of ‘functions’ which are not formalized into standard set theories but which are very useful to evaluate questions regarding independence of formulas like the Axiom of Choice, the Continuum Hypothesis and the existence of inaccessible cardinals. Within this context, the formal theory we introduce here is supposed to provide a framework where at least some of those ‘very large functions’ can be precisely defined. This paper was initially motivated by von Neumann’s original ideas [35] and a variation of them [27], and related papers as well ([4] [5]). In [27] it was provided a reformulation of von Neumann’s ‘functions’ theory (termed N theory). Nevertheless, we develop here a whole new approach called Flow. String diagrams [7] (whose edges and vertices can eventually be interpreted as morphisms in a monoidal category) introduce some ideas which seem to inter- sect with our own. A remarkable feature of string diagrams is that edges need not be connected to vertices at both ends. More than that, unconnected ends can be interpreted as inputs and outputs of a string diagram (with important applications in computer science). But within our framework no function has any domain whatsoever. Besides, the usual way to cope with string diagrams is by means of a discrete and finitary framework, while in Flow (our framework) we do not impose such restrictions. A more radical proposal where functions play a fundamental role is the Theory of Autocategories [10]. Autocategories are developed with the aid of autographs, where arrows (which work as morphisms) are drawn between arrows with no need of objects. Thus, once more we see a powerful idea concerning functions is naturally emerging in different places nowadays. When Ernst Zermelo introduced AC, his motivation was the Partition Prin- ciple [2], which is usually stated by means of functions. Indeed, AC entails PP. But since 1904 (when Zermelo introduced AC) it is unknown whether PP implies AC. For a review about the subject see, e.g., [2] [12] [14] [26]. Our answer is that AC is independent from PP, at least within ZFU if Flow is consistent. Our point is that ZF, ZFC, all their variants, and almost all their respective models (with a few exceptions like those in [1] and [6]) are somehow committed to a methodological and epistemological character which forces us to see sets as collections of some sort. Within Flow that does not happen, since our framework drives us to see sets as special cases of terms whose intended interpretation correspond to an intuitive notion of function. All sets in Flow are restrictions of 1, a function which works as some sort of ‘universal’ identity in the sense that for any x we have 1(x) = x. Nevertheless, functions who are no restrictions of 1 play an important role with consequences over sets. An analogous situation takes place with the well known Reflexion Principle in model theory [31]: some features of the von Neumann Universe motivate mathematicians to look for new

3 axioms which grant the existence of strongly inaccessible cardinals. Flow, however, provides a new way for coping with the metamathematics of ZF and its variants. As a final remark before we start, it is worth to observe we freely employ the term ‘Flow’ in this paper. Within a more general viewpoint, ‘Flow’ refers to any first order theory (i) whose objects are supposed to be interpreted as functions; (ii) which grants the existence of functions 0 and 1, in the sense that ∀x(1(x) = x ∧ 0(x) = 0) and 0 6= 1, as it follows in the next Section; and (iii) which allows a nontrivial composition among any functions at all, whether such a composition is associative or not. In this first paper about Flow, however, we explore one specific formal framework who is able to replicate ZF, ZFC, and ZFU. Our goal here is to suggest another way to construct models of set theories.

2 Flow theory

Our first Flow theory is a first-order theory F with identity [22], where x = y reads ‘x is equal to y’, and ¬(x = y) is abbreviated as x 6= y. From now on 2 we use Flow and F as synonyms. Flow has one functional letter f1 (termed 2 2 evaluation), where f1 (f, x) is a term, if f and x are terms. If y = f1 (f, x), we abbreviate this by f(x) = y, and read ‘y is the image of x by f’. All terms are called functions. Such a terminology seems to be adequate under the light of our intended interpretation: functions are supposed to be terms which ‘transform’ terms into other terms. Since we are assuming identity, our functions cannot play the role of non-trivial relations. Suppose, for example, f(x) = y ′ 2 2 ′ and f(x) = y , which are abbreviations for f1 (f, x) = y and f1 (f, x) = y , respectively. From transitivity of identity, we have y = y′. So, for any f and any x, there is one single y such that f(x)= y. Lowercase Latin and Greek letters denote functions, with the sole exception of two specific functions to appear in the next pages, namely, 0 and 1. Uppercase Latin letters are used to denote formulas or predicates (which are eventually defined). Any explicit definition in Flow is an abbreviative one, in the sense that for a given formula F , the definiendum is a metalinguistic abbreviation for the definiens given by F . Eventually we use bounded quantifiers. If P is a predicate defined by a formula F , we abbreviate ∀x(P (x) ⇒ G(x)) and ∃x(P (x) ∧ G(x)) as ∀P x(G(x)) and ∃P x(G(x)), respectively; where G is a formula. Finally, ∃!x(G(x)) is a metalinguistic abbreviation for ∃x∀y(G(y) ⇔ y = x). Postulates F1∼F11 of F are as follows.

2.1 The very basic F1 - Weak Extensionality ∀f∀g(((f(g)= f ∧g(f)= g)∨(f(g)= g∧g(f)= f)) ⇒ f = g)). If f(g)= f we say f is rigid with g. If f(g)= g we say f is ﬂexible with g. If both f and g are rigid (ﬂexible) with each other, then f = g.

4 F2 - Self-Reference ∀f(f(f)= f).

Any f is ﬂexible and rigid with itself. That may sound a strong limitation if we compare our framework, e.g., to lambda calculus [3]. But that feature is quite useful for our purposes. One task for the future is to admit some variations of Flow where F2 is no axiom.

Observation 1 Let y be a function such that ∀x(y(x)= r ⇔ x(x) 6= r). What about y(y)? If x = y, then y(y) = r ⇔ y(y) 6= r (Russell’s paradox). But that entails y(y) 6= y(y), another contradiction. However, our Self-Reference postulate does not allow us to deﬁne such an y, since x(x)= x for any x. That is not the only way to avoid such an inconsistency. Our Restriction axiom (some pages below) does the same work. That opens the possibility of variations of Flow where Self-Reference is not a theorem.

Our ﬁrst theorem states any f can be identiﬁed by its images f(x).

Theorem 1 ∀f∀g(f = g ⇔ ∀x(f(x)= g(x))).

Proof From substitutivity of identity in f(x) = f(x), proof of the ⇒ part is straightforward: if f = g, then f(x) = g(x), for any x. For the ⇐ part, suppose, for any x, f(x) = g(x). If x = f, f(f) = g(f); if x = g, f(g)= g(g). From F2, f(f)= f and g(g)= g. So, g(f)= f and f(g)= g. F1 entails f = g.

Axioms F1 and F2 could be rewritten as one single formula:

F1’ - Alternative Weak Extensionality ∀f∀g(((f(g) = f ∧ g(f) = g) ∨ (f(g)= g ∧ g(f)= f)) ⇔ f = g)).

F2 is a consequence from F1’. Ultimately, f = g entails f(g) = f (from F1’). And substitutivity of identity entails f(f) = f. But we prefer to keep axioms F1 and F2 (instead of F1’) to smooth away some discussions. From F1 and F2, we can analogously see that F1’ is a nontrivial theorem.

F3 - Identity ∃f∀x(f(x)= x).

There is at least one f such that, for any x, we have f(x)= x. Any function f which satisﬁes F3 is said to be an identity function.

Theorem 2 Identity function is unique.

Proof Suppose both f and g satisfy F3. Then, for any x, f(x) = x and g(x)= x. Thus, f(g)= g and g(f)= f. Hence, according to F1, f = g.

In other words, there is one single f which is flexible to every term. In that case we simply say f is flexible. That means “flexible” and “identity” are synonyms.

5 F4 - Rigidness ∃f∀x(f(x)= f).

There is at least one f which is rigid with any function. Observe the symmetry between F3 and F4! Any f which satisﬁes last postulate is said to be rigid.

Theorem 3 The rigid function is unique.

Proof Let f and g satisfy axiom F4. Then, for any x, f(x)= f and g(x)= g. Thus, f(g(x)) = f(g)= f and g(f(x)) = g(f)= g. From F1, f = g.

Now we are able to introduce new terminology. Term 1 is the identity (ﬂexi- ble) function, while 0 is the rigid function, since we proved they are both unique. So, ∀x(1(x)= x ∧ 0(x)= 0). 2 Since f(x) = y says f1 (f, x) = y, F3 states there is an f such that, for any x, 2 2 f1 (f, x) = x, while F4 says there is f where f1 (f, x) = f. If we do not grant the existence of other functions, it seems rather diﬃcult to prove 0 6= 1.

Theorem 4 0 is the only function which is rigid with 0.

Proof This theorem says ∀x(x 6= 0 ⇒ x(0) 6= x), i.e., ∀x(x(0) = x ⇒ x = 0). But 0(x)= 0. So, if x(0)= x ∧ 0(x)= 0, then x = 0 (F1).

Theorem 5 1 is the only function which is ﬂexible with 1.

Proof This theorem says ∀x(x 6= 1 ⇒ x(1) 6= 1), i.e., ∀x(x(1) = 1 ⇒ x = 1). But 1(x)= x. So, if x(1)= 1 ∧ 1(x)= x, then x = 1 (F1).

For more details about x(0) and x(1), for any x, see Theorems 10 and 13.

Deﬁnition 1 f[t] iﬀ t 6= f ∧ f(t) 6= 0. We read f[t] as ‘f acts on t’.

Whereas f(t) is a term, f[t] abbreviates a formula. No f acts on itself. The intuitive idea is to allow us to talk about what a function f does. For example, 0 does nothing whatsoever, since there is no t on which it acts. Next definition is a pedagogical move to write postulate F5 in a simple and direct way. If the reader is not comfortable with that, all you have to do is to rewrite h = f ◦ g in F5 (next postulate) as the conjunction of all five formulas in the next definition.

Deﬁnition 2 Given f and g, the F-composition h = f ◦ g, if it exists, must satisfy the next conditions: (i) h 6= 0; (ii) ∀x((x 6= f ∧ x 6= g ∧ x 6= h) ⇒ h(x)= f(g(x))); (iii) h 6= f ⇒ h(f)= 0; (iv) h 6= g ⇒ h(g)= 0; (v) (g 6= h ∧ f 6= h ∧ g 6= 1 ∧ f 6= 1) ⇒ (f(g(h)) = 0 ∨ g(h)= 0)

6 So, (i) No F-composition is 0. (ii) F-composition behaves like standard notions of composition between functions up to self-reference. Nevertheless, contrary to usual practice in standard set theories, we can deﬁne F-composition between any two functions. (iii and iv) No F-composition h = f ◦ g acts on any of its factors f or g. Finally, item (v) is a technical constraint to avoid ambiguities in the calculation of f ◦g. Either g does not act on the F-composition h or, if it acts, then f does not act on g(h).

F5 - F-Composition ∀f∀g(∃!h(h = f ◦ g)).

We never calculate (f ◦ g)(f ◦ g) as f(g(f ◦ g)), since (f ◦ g)(f ◦ g) is f ◦ g, according to F2. Given f and g, we can ‘build’ f ◦ g through a three-step process: (i) First we establish a label h for f ◦ g; (ii) Next we evaluate f(g(x)) for any x which is diﬀerent of f and g. By doing that we are assuming those x are diﬀerent of h. Thus, if such a choice of x entails f(g(x)) = y for a given y, then h(x) = y; (iii) Next we evaluate the following possibilities: is h equal to either f, g or something else? If h 6= g, then h(g) is supposed to be 0. If h(g) = 0 entails a contradiction, then h is simply g. An analogous method is used for assessing if h is f. Eventually, h is neither f nor g, as we can see in the next theorems. It is worth to remark that F-composition plays a crucial role in our main result (Theorems 67 and 68).

Theorem 6 There is a unique h such that h 6= 0 but h(x)= 0 for any x 6= h.

Proof From Deﬁnition 2 and F5, 0 ◦ 0 is a unique h 6= 0 such that, for any x where x 6= 0 and x 6= h, h(x)= 0(0(x)) = 0(0)= 0. Since h 6= 0, then F5 entails h(0)= 0. Thus, h(x) is 0 for any x 6= h, while h itself is diﬀerent of 0.

Such h of last theorem (which does not conﬂict neither with Theorem 3 nor with Theorem 4) is labeled with a special symbol, namely, ϕ0. So, 0 ◦ 0 = ϕ0, where ϕ0 6= 0. If the reader is intrigued by the subscript 0 in ϕ0, our answer is ‘yes, we intend to introduce ordinals, where ϕ0 is the ﬁrst one’. Let f, g, a, and c be pairwise distinct functions where: (i)f(a)= g, f(g)= c, f(c) = c, f(f) = f, and f(r) = 0 for the remaining values r; (ii) g(a) = a, g(g)= g, and g(r)= 0 for the remaining values r; (iii) a and c are arbitrary, as long they are neither 0 nor 1. Then, (f ◦ g) ◦ f = ϕ0, while f ◦ (g ◦ f)= l, where l(a)= c, l(l)= l, and l(r)= 0 for the remaining values r. Thus, F-composition among functions which act on each other (observe in this example f[g], since f 6= g and f(g)= c 6= 0), if they exist, is not necessarily associative.

Theorem 7 Let f, g, and h be terms such that neither one of them acts on the remaining ones. Then F-Composition is associative among f, g, and h.

Proof Both f ◦ (g ◦ h) = p and (f ◦ g) ◦ h = q correspond (Deﬁnition 2) to y = f(g(h(x))), if x is diﬀerent of f, g, h, p, and q. So, f ◦(g◦h) = (f ◦g)◦h

7 for those values of x. But f, g, and h do not act on f, g, or h; and no F-composition acts on any of its factors. So, f ◦ (g ◦ h) = (f ◦ g) ◦ h = f(g(h(x))).

2 Evaluation f1 is not associative as well. Consider, e.g., x(1(x)), for x different of 0 and diﬀerent of 1, and such that x(1) = 0 (functions like that do exist, as we can see later on). Thus, x(1(x)) = x(x) = x. If evaluation was associative, we would have x(1(x)) = x(1)(x)= 0(x)= 0; a contradiction, since we assumed x 6= 0! Of course this rationale works only if we prove the existence of a function like x and that 0 6= 1. Both claims are the subject of the next two theorems. It is good news that evaluation is not associative. According to F2, for all t, g(t) = (g(g))(t), since g(g)= g. If evaluation was associative, we would have g(t)= g(g(t)) and, thus, g = g ◦ g. So, F-composition would be idempotent, a useless feature for our purposes.

Theorem 8 0 6= 1.

Proof 0 ◦ 0 = ϕ0 6= 0 (Theorem 6). But 0(ϕ0) = 0, while 1(ϕ0)= ϕ0. Thus, from Theorem 1, 0 6= 1.

Theorem 9 For any x we have 0 ◦ x = x ◦ 0 = ϕ0

Proof First we prove 0◦ x = ϕ0. From Definition 2 and axiom F5, (0 ◦ x)(t)= 0(x(t)) = 0 for any t 6= 0 and t 6= x. But F5 demands 0 ◦ x is different of 0. Hence, (0 ◦ x)(0)= 0. Regarding x, there are three possibilities: (i) x = 0; (ii) x = ϕ0; (iii) x is neither 0 nor ϕ0. The first case corresponds to Theorem 6, which entails 0◦x = ϕ0. In the second case, if 0◦x is different of x = ϕ0, then (0 ◦ x)(ϕ0) = 0. But that would entail (0 ◦ x)(t) = 0 for any t 6= 0◦x, which corresponds exactly to function ϕ0 proven in Theorem 6, a contradiction. So, 0 ◦ x is indeed ϕ0, when x = ϕ0. Concerning last case, since x 6= 0 and x 6= ϕ0, then (Theorem 1) there is t 6= ϕ0 such that x(t) 6= 0 for x 6= t. From F5, (0 ◦ x)(t) = 0 for such value of t. But once again we have a function 0 ◦ x such that (0 ◦ x)(t)= 0 for any t 6= 0, which corresponds to ϕ0 from Theorem 6. Concerning the identity x ◦ 0 = ϕ0, the proof is analogous. If h = x ◦ 0, then h 6= 0 (from F5). Besides, for any x, x 6= h entails h(x)= 0. Hence, h = ϕ0.

Theorem 10 ∀x(x(0)= 0).

Proof x ◦ 0 = ϕ0 (Theorem 9). Hence, for any t, (t 6= x ∧ t 6= 0 ∧ t 6= ϕ0) ⇒ ϕ0(t)= x(0(t)) = x(0). But ϕ0(t)= 0 for any t 6= ϕ0. Thus, x(0)= 0.

Theorem 11 1 ◦ 1 = 1.

Proof From F5, h = 1 ◦ 1 entails h(x) = x for any x 6= 1. Since the F- composition is unique and 1 guarantees all demanded conditions, then h = 1.

8 Theorem 12 ∀x∀y(x ◦ y = 1 ⇒ (x = 1 ∧ y = 1)).

Proof Let x 6= 1. Then 1(x)= 0, according to F5. But that happens only for x = 0. And 0 ◦ y 6= 1. Analogous argument holds for y 6= 1.

There can be no functions diﬀerent of 1 such that their composition is 1. Next theorem is important for a better understanding about F1, although its proof does not demand the use of such a postulate.

Theorem 13 ∀x((x 6= 0 ∧ x(1)= 0) ⇔ (1 ◦ x = x ∧ x ◦ 1 = x ∧ x 6= 1)).

Proof The ⇒ part. If x(1)= 0, then x 6= 1, since 1(1)= 1. If x ◦ 1 = h, then, for any t diﬀerent of x, 1, and h, we have h(t)= x(1(t)) = x(t). If h = x, then h satisﬁes all conditions from F5, since x 6= 1, h(1) = x(1) = 0, and h(t) = x(t) for any t 6= h. Since F5 demands h to be unique, then h = x. If 1 ◦ x = h, we use an analogous argument. For the ⇐ part, 1 ◦ x = x ◦ 1 = x entails x 6= 0 (Theorem 9). Since x 6= 1, then F5 demands for the F-composition x that x(1)= 0.

Observation 2 If it wasn’t for the uniqueness of F-compositions in F5, Flow would be consistent with the existence of many functions, like x and h (from the proof of Theorem 13), which “do” the same thing. We refer to such functions as clones. For a brief investigation about clones see Definition 12 and its subsequent discussion. Clones are meant to be different functions x and h which share the same images x(t) and h(t) for any t different of both x and h, and such that x(h) = 0 and h(x) = 0. From Theorem 1, x 6= h (recall F2). We use this opportunity to prove Theorem 6, since 0 ◦ 0 = ϕ0, where ϕ0 and 0 are clones. But we cease using clones when we talk about other functions. That is why we refer to F1 as “weak extensionality”. A strong extensionality postulate would demand that other clones besides 0 and ϕ0 cannot exist. We do something like that in some remaining postulates where we use the quantifier ∃!.

Deﬁnition 3 Given a term f, g is the F-successor of f, and we denote this by Σ(f,g), iﬀ f(g)= 0 ∧ ∀x(x 6= g ⇒ g(x)= f(x)).

For example, let f be given by f(a)= b, f(b)= c, f(c)= c, f(f)= f, and f(r)= 0 for the remaining values r. Now let g be given by g(a)= b, g(b)= c, g(c) = c, g(f) = f, g(g) = g, and g(r) = 0 for the remaining values r, where g 6= f. In that case, Σ(f,g), if functions like those exist. Observe this has nothing to do with our previous discussion about clones.

F6 - F-Successor Function ∃!σ(σ 6= 0 ∧ ∀f((f 6= σ ∧ f 6= 0) ⇒ (∃g(Σ(f,g) ⇔ σ(f)= g) ∨ (∀h(¬Σ(f,h)) ⇔ σ(f)= 0)))).

This last axiom states the existence and uniqueness of a special function σ. From now on we write mostly σf for σ(f). Every time we use the symbol σ we are referring to the same term from F6: the only one such that σf = g 6= 0 is equivalent to Σ(f,g), and σf = 0 is equivalent to ∀h(¬Σ(f,h)), as long f is

9 neither σ nor 0. Thus, σ successfully ‘signals’ F-successors, when they exist, for all functions, except when f is either 0 or σ. In those cases, we have Σ(0, ϕ0) and σ0 = 0 (Theorem 10), while ∀h(¬Σ(σ, h)) and σσ = σ (F2). Next we want to grant the existence of F-successors for many other terms. As previously announced, this axiom states the existence of an hierarchy of functions, where ϕ0 is the ﬁrst one.

F7 - Infinity ∃i((∀t(i(t) = t ∨ i(t) = 0)) ∧ σi 6= 0 ∧ (i(ϕ0) = ϕ0 ∧ ∀x((x 6= 0 ∧ i(x)= x) ⇒ (i(σx)= σx ∧ σx 6= 0)))). Deﬁnition 4 Any i which satisﬁes F7 is said to be inductive.

Since the existence of ϕ0 is granted by F5, we can use σ and F7 to grant the existence of ϕ1 = σϕ0 such that ϕ1(ϕ1) = ϕ1, ϕ1(ϕ0) = ϕ0, and for the remaining values r (those who are neither ϕ0 nor ϕ1) we have ϕ1(r)= 0. That happens because F7 states the existence of at least one other function i and inﬁnitely many other functions. It says i(ϕ0) = ϕ0. Besides, there is a non-0

F-successor of ϕ0 such that i(σϕ0 )= σϕ0 . More than that, if x admits a non-0 F-successor σx (where i(x)= x), then i(σx)= σx, where σx 6= 0. Thus, ϕ0 6= 0 and σϕ0 = ϕ1, where ϕ1 6= ϕ0 and ϕ1 6= 0. Analogously we can get ϕ2, ϕ3, and so on. Along with those terms ϕn, F7 says any inductive function i admits its own non-0 F-successor σi. Subscripts 0, 1, 2, 3, etc., are metalinguistic symbols based on an alphabet of ten symbols (the usual decimal numeral system) which follows the lexicographic order ≺, where 0 ≺ 1 ≺ 2 ≺ ··· ≺ 8 ≺ 9. If n is a subscript, then n +1 corresponds to the next subscript, in accordance to the lexicographic order. In that case, we write n ≺ n + 1. n + m is an abbreviation for (...(...((n +1)+ 1)+ ...1)...) with m occurrences of + and m occurrences of pairs of parentheses. Again we have n ≺ n + m. Besides, ≺ is a strict total order. That fact allows us to talk about a minimum value between subscripts m and n: min{m,n} is m iff m ≺ n, it is n iff n ≺ m, and it is either one of them if m = n. Of course, m = n iff ¬(m ≺ n) ∧ ¬(n ≺ m). If m ≺ n ∨ m = n, we denote this by m n. Such a vocabulary of ten symbols endowed with ≺ is called here (meta) language L. F7 provides us some sort of “recursive definition” for functions ϕn, while it allows as well to guarantee the existence of inductive functions: (i) ϕ0 is such that ϕ0(x) is ϕ0 if x = ϕ0 and 0 otherwise; (ii) ϕn+1 is such that ϕn+1(ϕn+1)= ϕn+1, ϕn+1 6= ϕn, and ϕn+1(x)= ϕn(x) for any x different of ϕn+1. Observe that ϕn+1(ϕn) = ϕn(ϕn) = ϕn, while ϕn(ϕn+1) = 0. Moreover, ϕn+2(ϕn+1)= ϕn+1, and ϕn+2(ϕn)= ϕn+1(ϕn)= ϕn; while ϕn(ϕn+2)= 0. Figure 1 illustrates how to represent some functions f in a quite intuitive way. A diagram of f is formed by a rectangle. On the left top corner inside the rectangle we find label f. The remaining labels refer to terms x such that either f(x) 6= 0 (f acts on x) or there is t such that f(t)= x 6= 0. Term 0 never occurs in any diagram. For each x inside the rectangle there is a unique corresponding arrow which indicates the image of x by f, as long x is not f itself. Due to self-reference, label f at the left top corner inside the rectangle does not need to be attached to any arrow, to avoid redundancy.

10 From left to right, the ﬁrst diagram in Figure 1 refers to ϕ0. It says, for any x, ϕ0(x) is 0, except for ϕ0 itself. The second diagram says ϕ1(ϕ1) = ϕ1, and ϕ1(ϕ0) = ϕ0. The circular arrow associated to ϕ0 in the second diagram says ϕ1(ϕ0)= ϕ0 (ϕ1[ϕ0]). The third diagram says ϕ2(ϕ2)= ϕ2, ϕ2(ϕ1)= ϕ1, and ϕ2(ϕ0) = ϕ0. Finally, for the sake of illustration, the last diagram corresponds to an f such that f(a) = b, f(b) = c, and f(c)= 0. That is why there is no arrow ‘starting’ at c. In those cases we represent c outside the rectangle. The existence of functions like that is granted by F10α some pages below. The diagrams of 0 and 1 are, respectively, a blank rectangle and a ﬁlled in black rectangle. Other examples are provided in the next paragraphs.

ϕ0 ϕ1 ϕ2 f ··· x x x ab✲ ✲ c ϕ0 ϕ0 ϕ1

Figure 1: From left to right, diagrams of ϕ0, ϕ1, ϕ2, and an arbitrary f. Observe that ϕm+n(ϕn)= ϕn, for any m and n of L.

Theorem 14 0 and ϕ0 are the only functions who do not act on any t.

Proof If f does not act on any t, then ∀t(t = f ∨ f(t) = 0). Suppose f is neither 0 nor ϕ0. Then there is t such that t 6= f ∧ f(t) 6= 0. But that entails f[t] (Deﬁnition 1). So, the only functions which do not act on any t are 0 and ϕ0.

Theorem 15 σ1 = 0.

Proof σ1 6= 0 ⇒ 1(σ1)= 0 (Deﬁnition 3). That happens only if σ1 = 0.

Theorem 16 For any g, σg 6= 1

Proof Suppose there is g such that σg = 1. Since 1 6= 0, then g(1) = 0 and 1(g)= g, according to Deﬁnition 3. So, g 6= 1. Once again from Deﬁnition 3, g(x) = 1(x) for any x 6= 1. But those are the same conditions for 1 ◦ 1 = g, according to F5. Since any F-composition is unique, then g = 1 (Theorem 11), which contradicts the assumption g 6= 1. So, there is no such g.

In other words, the existence of some ‘functions’ in Flow is forbidden.

Definition 5 For any function f, a restriction g of f is defined as g ⊆ f iff g 6= 0 ∧ ∀x((g[x] ⇒ f[x]) ∧ ((g[x] ∧ f[x]) ⇒ f(x)= g(x))),

Proper restrictions are deﬁned as g ⊂ f iﬀ g ⊆ f ∧ g 6= f. We abbreviate ¬(g ⊆ f) and ¬(g ⊂ f) as, respectively, g 6⊆ f and g 6⊂ f. For example, ϕ1 ⊆ ϕ3, ϕ1 ⊂ ϕ3, and ϕ3 6⊆ ϕ1.

Theorem 17 ∀f∀g(g ⊂ f ⇒ g(f)= 0).

11 Proof Suppose g(f) 6= 0. Since g 6= f, then g[f]. But for any f we have ¬f[f]. In other words, ¬(g[f] ⇒ f[f]). Hence, g 6⊆ f.

Some of the most useful restrictions are obtained from a given formula F , in a way which resembles the well known Separation Scheme in ZF. That is achieved thanks to careful considerations regarding the F-successor function σ. But before that, we need more concepts, since we are interested on a vast number of situations.

Deﬁnition 6 A term f is comprehensive iﬀ there is g such that g 6= 0, g ⊆ f, and σg = 0; and we denote that by C(f). Otherwise, f is uncomprehensive.

Comprehensive functions are supposed to describe “huge” functions who act on “many terms”, and they are partially regulated by some of the the next axioms. If f itself is such that σf = 0, then f is comprehensive, except when f = 0. Besides, it is easy to see that 1 is comprehensive and 0 is uncomprehensive. On the other hand, there are other comprehensive functions which, by the way, play an important role within our proposal for building a model of ZFU. But for now we are mostly interested on uncomprehensive functions, as it follows in the next Subsection.

Theorem 18 If f ⊆ g and f is comprehensive, then g is comprehensive.

Proof If f is comprehensive, then there is h such that h 6= 0, h ⊆ f, and σh = 0. Since h ⊆ f ⊆ g, then h ⊆ g. Thus, g is comprehensive.

2.2 Emergent functions

Deﬁnition 7 E(f) iﬀ (i) f 6= σ; (ii) σf 6= 0; (iii) ∀x(f[x] ⇒ σx 6= 0); (iv) ∀y((∃x(f(x)= y ∧ y 6= 0)) ⇒ σy 6= 0).

We read E(f) as ‘f is emergent’. ϕ0 is vacuously emergent. That entails ϕ1 is emergent. Emergent functions are supposed to be uncomprehensive terms who act only on uncomprehensive terms. The behavior of emergent functions is regulated by the next axiom, F8.

Definition 8 g E f iff g 6= 0 ∧ ∀x(g[x] ⇒ ((f[x] ∨ ∃a(f(a) = x)) ∧ (f[g(x)] ∨ ∃b(f(b) = g(x))))). If g E f, we say g lurks f. Besides, g⊳f iff g E f and g 6= f. In that case we say g properly lurks f. Finally, ¬(g E f) and ¬(g⊳f) are abbreviated as g 5 f and g ⋪ f, respectively.

Last deﬁnition is a generalization of restriction, as it follows in the next theorem. Besides, if f is a function such that f(f)= f, f(ϕ0)= ϕ1, f(ϕ1)= ϕ0 and f(r)= 0 for the remaining values, then f E ϕ2, ϕ2 E f, f E ϕ3, ϕ3 5 f.

Theorem 19 If g ⊆ f and g 6= 0, then g E f.

The proof is straightforward. The converse is obviously not a theorem.

12 Deﬁnition 9 h = p(f) iﬀ ∀x(h[x] ⇔ x E f). h is the full power of f. The full power p(f) of f acts on all terms x who lurk f.

Theorem 20 If f ⊂ 1 acts on n terms, then p(f) acts on (n + 1)n terms.

Proof If f acts on n terms xi and g lurks f, each xi may correspond to any xj (where eventually xj = xi) in the sense we may have g(xi)= xj . On the other hand, we may have g(xi)= 0 as well. Thus, for each xi (n possible values) there are n + 1 possible images. So, there are (n + 1)n terms who lurk f.

F8 - Cohesion ∀f(E(f) ⇒ (∀g((g E f ⇒ σg 6= 0) ∧ ∀h((h[g] ⇔ g E f) ⇒ σh 6= 0)) ∧ ∀i(∀t(i[t] ⇒ ∃x(x[t] ∧ f[x]))) ⇒ σi 6= 0) ∧ E(σf )). If f is emergent, then any g (if it exists) who lurks f has a non-0 F-successor. Function h at F8 refers to the full power of f, if it exists. Term i is useful for dealing with arbitrary union, if we are able to define it. Finally, if f is emergent, so it is σf . Observe F8 is an existence postulate. For understanding this, recall F6, which provides necessary and sufficient conditions for knowing if σf 6= 0: there must be a g different of f such that certain conditions are met. The point here is that F8 grants the existence of certain terms which are F-successors of others, as long some conditions are met. When we say, as above, that σg 6= 0, we state there is a function z 6= 0 such that σg = z. The same happens to σh and σi.

2.3 More about restrictions Deﬁnition 10 Let f be a function and F (t) be a formula where all occurrences of t are free. We say g is restriction of f under F (t), and denote this by g = f , iﬀ: (i) g 6= 0; (ii) f(σg)= 0 ∨ ¬F (g); (iii) g 6= f ⇒ g(f)= 0; (iv) F (t) ∀t((t 6= f ∧ t 6= g) ⇒ ((g(t)= f(t) ∧ F (t) ∧ f[t]) ∨ (g(t)= 0 ∧ (¬F (t) ∨ ¬f[t])))).

Formula g = f is somehow equivalent to g ⊆ f, as we see in the next F (t) two theorems.

Theorem 21 If g ⊆ f, then it is possible to state a formula F where g = f . F

Proof If g ⊆ f, assume as formula F (t) the next one: g[t]. Item (i) of Defini- tion 10 is a consequence from Definition 5. Item (ii) of the same definition is granted thanks to the fact that ¬g[g] (¬F (g)) for any g. Item (iii) is due to Theorem 17. Finally, item (iv) is granted from Definition 5. Theorem 22 If g = f , then g ⊆ f. F

Proof If g = f , then either g = ϕ or g 6= ϕ . In the ﬁrst case, the proof F 0 0 is immediate by vacuity (ϕ0 does not act on any term). If g 6= ϕ0, then item (iv) of Deﬁnition 10 demands, for any t, g[t] ⇒ (f[t] ∧ g(t)= f(t)). Thus, g ⊆ f.

13 A natural way of getting some restrictions g of f is through F9E below. Observe as well item (iv) of Deﬁnition 10 takes into account the self-reference postulate, since we demand t 6= f ∧t 6= g. Now, if F (t) is a formula (abbreviated by F ) where all occurrences of t are free, then the following is an axiom.

F9E - E-Restriction ∀f∀x((F (x) ⇒ E(x)) ⇒ ∃!g(g = f )). F

Subscript E highlights a strong commitment to emergent functions, although there is no need of f to be emergent. Many restrictions due to this last postulate grant the existence of emergent functions. Next theorem, for example, shows we do not need F-composition to prove there is ϕ0. Theorem 23 0 = ϕ if ∀x(F (x) ⇒ E(x)). F (x) 0

Proof Immediate, since 0 does not act on any term and no restriction can be 0. Theorem 24 For any emergent f we have: (i) f = ϕ ; (ii) f = f; x=6 x 0 x=x (iii) f = f; (iv) f = ϕ . x=6 f x=f 0

Proof Item (i) is proven by vacuity. (ii) takes into account all terms where f acts. Observe F8 grants any emergent function who acts on any term, acts on emergent functions. About (iii) and (iv), recall f plays no role into the calculation of its restriction. All that matters are the terms where f acts.

From F9E, there are four possible restrictions g of ϕ2. If F (x) is, for example, “x = ϕ0”, then g = ϕ1. Accordingly, assume f = ϕ2 in F9E. So, consider, e.g., x = ϕ0. Such a value for x is different of ϕ2. Besides, F (ϕ0). That implies g(ϕ0) = ϕ2(ϕ0) = ϕ0. For all remaining values x 6= ϕ0, we know g does not act on x. That means g acts solely on ϕ0. And according to F7, that function is supposed to be ϕ1. Observe ϕ1 is allowed to have a free occurrence in F (x), according to our Restriction Axiom. Nevertheless g does not act on ϕ1 in our first example. That means either g(ϕ1) = 0 or g = ϕ1. In this case, we have g = ϕ1. Later on we define a membership relationship ∈ (Definition 24) where x ∈ f iff f[x] and some conditions are imposed over f. That entails we can guarantee that in a translation of ZF’s Separation Scheme into Flow’s language, any free occurrence of g in F (x) will have no impact (in a precise sense). After all, in this first example F (x) is x = ϕ0, while g is ϕ1. For more details see Section 3. Resuming the discussion about restrictions of ϕ2, if F (x) is the formula “x = ϕ0 ∨ x = ϕ1”, then g = ϕ2. If F (x) is “x = x”, then again g = ϕ2. If F (x) is “x 6= x”, then g = ϕ0. The novelty here, however, happens with the formula F (x) given by “x = ϕ1”. In that case we have a proper restriction γ such that γ 6= ϕ1, γ(γ) = γ, γ(ϕ1) = ϕ1, and γ(r) = 0 for any remaining r different of ϕ1 and γ itself. Thus, γ is a new function whose existence is granted thanks to F9E and no other previous postulate. Besides, F8 grants γ is emergent. Hence, σγ 6= 0.

14 Deﬁnition 11 z is the restricted power of f 6= 0 iﬀ ∀x((x 6= z ∧ x 6= 0) ⇒ ((z(x)= x ⇔ x ⊆ f) ∧ (z(x)= 0 ⇔ x 6⊆ f))). We denote z as ℘(f). We adopt the convention ℘(0)= ϕ0.

For example, ℘(ϕ0) = ϕ1, ℘(ϕ1) = ϕ2, and ℘(ϕ2) = f, where f(ϕ0) = ϕ0, f(ϕ1) = ϕ1, f(ϕ2) = ϕ2, f(γ) = γ, f(f) = f, and f(x) = 0 for the remaining values x. Recall γ acts only on ϕ1 and γ(ϕ1)= ϕ1. Observe z = ℘(f) is a restriction of 1, even if f is not. Besides, ℘ is not a function, but a metalinguistic symbol which helps us to abbreviate the formula z = ℘(f) given by the deﬁnition above. Observe as well ℘(f) ⊆ p(f), for any f. While p(f) refers to a function who acts on all terms that lurk f, ℘(f) acts on all terms that lurk f as long they are restrictions of f.

Observation 3 In a sense, F9E is similar to the Separation Scheme in ZFC, since it states the existence of a unique g obtained from a given f and a formula F (x). Nevertheless, the role of Separation Scheme in ZFC is not limited to grant the existence of subsets of a given set. Thanks to that postulate, ZFC avoids antinomies like Russell’s paradox. In our case those antinomies are avoided by means of the simple use of Self-Reference (Observation 1). That is one of the reasons why we do not prohibit free occurrences of g in F (x) (like what happens in ZFC). Actually, if we demanded no free occurrences of g in F (x), we would be unable to obtain some useful restrictions, as we can see in the examples below. Nevertheless, we demand ∀x((x 6= g ∧ x 6= f) ⇒ ((f(x) = g(x) ∧ F (x)) ∨ (g(x)= 0 ∧ ¬F (x))) (Definition 10), which is a weaker condition than the prohibition of occurrences of g in F (x). If we recall Observation 1, we can easily see that, for any y and any r, neither y nor y allow us to x(x)=6 y x(x)=6 r get any contradiction in the style of Russell’s paradox. Finally, if anyone tries to “define” a function f from a formula F (x) without using F9E, it is perfectly possible to get a contradiction. For example, we can “define” a function f as it follows: ∀x(f(x)= ϕ0). Since no f acts on 0, we have a contradiction, namely, f(0)= ϕ0. Nevertheless, that would not be a definition at all, but simply a new postulate which is inconsistent with our axioms. Definitions are supposed to be conservative, in the sense of not allowing new theorems [4].

Deﬁnition 12 f ∼ g iﬀ ∀t((f[t] ⇔ g[t]) ∧ ((t 6= f ∧ t 6= g) ⇒ f(t)= g(t))).

If f ∼ g but f 6= g, we say f and g are clones, as discussed in Observation 2. Regarding their images, clones f and g diﬀer solely on f and g: f(g)= 0, while g(g)= g, and g(f)= 0, while f(f)= f. It is easy to see that ∼ is reﬂexive.

Theorem 25 Let f be emergent. Then, for any h 6= 0, h ∼ f entails h = f.

Proof From Theorem 24, f = f . Since h ∼ f, h = f as well. But F9E t=t t=t says any restriction is unique. Thus, h = f.

This last theorem grants 0 and ϕ0 are the only clones in Flow, at least among emergent functions.

15 Theorem 26 ∀f(ϕ0 ⊆ f).

Proof Theorem 14 states ϕ0 is the only function diﬀerent of 0 who do not act on any term. Thus, Deﬁnition 5 grants formula above is proven by vacuity. Theorem 27 For any x and y, x ⊂ y entails y 6⊂ x.

Proof If x and y are, respectively, ϕ0 and 0, the proof is straightforward, since 0 is never any restriction, according to F9E and Deﬁnition 5. For the remaining cases, x ⊂ y entails x 6= y and, for any t, x[t] ⇒ (y[t]∧x(t)= y(t)). So, there is t′ where y[t′] ∧ ¬x[t′] or y[t′] ∧ x[t′] ∧ x(t′) 6= y(t′). In anyone of those cases we have y 6⊂ x.

Theorem 28 σ and 0 are the only functions f such that σf = f.

Proof We already know σσ = σ (F2) and σ0 = 0 (Theorem 10). If f 6= 0 and σf = 0, then σf 6= f. If f 6= σ, f 6= 0 and σf 6= 0, then F6 and Deﬁnition 3 demand σf 6= f. Observe as well there is no clone of σ, according to F6, although σ is not emergent.

Theorem 29 ∀f(E(f) ⇒ f = σf ). x=6 f

Proof E(f) entails σf 6= 0. So, σf acts on all terms where f acts. But σf acts on just one more term: f itself. Since formula F (x) in the restriction above is “x 6= f” (observe we follow all demands for F (x) in F9E), then σf acts exactly on all terms where f acts. Besides, they share all x=6 f∧ their images, according to F9E. So, σf = f, since f[x] ⇒ x 6= f. x=6 f

Definition 13 Let f be a restriction of 1 where ∀g∀t((f[g] ∧ g[t]) ⇒ E(t)). The arbitrary union of all terms g where f acts (or arbitrary union of f, for short) is defined as u = g = 1 . f[g] ∃g(f[g]∧g[t]) S The idea of arbitrary union is as it follows. If f acts on any g which acts on any x, then u acts on that very same x; and if no g acts on a given x, then u does not act on that x. Particularly, we write u = g ∪ h for the case where f acts at most on g and h. Definition 14 Let f be a restriction of 1 where ∀g∀t((f[g] ∧ g[t]) ⇒ E(t)). Then the arbitrary intersection of f is given by g = 1 . Tf[g] ∀g(f[g]⇒g[t])

Observe the deﬁnition above is equivalent to g = g . Tf[g] Sf[g] ∀g(f[g]⇒g[t]) If f acts only on g and h, the arbitrary intersection may be written as g ∩ h. In particular, for any m and n from language L, ϕm ∩ ϕn = ϕm ◦ ϕn.

Theorem 30 0 = 0 = 1 = ϕ0. S [g] Sϕ [g] Sϕ [g] Proof is straightforward. Last identity from last theorem illustrates our previous claim that it is possible the union u be one of the terms where f does act: ϕ1 acts on ϕ0, and = ϕ0. Sϕ1[g]

16 2.4 Ordinals Deﬁnition 15 Let F (t) be the formula t ⊆ 1 ∧ E(t) ∧ ∀r(t[r] ⇒ r ⊆ t) ∧ ∀r∀s((t[r] ∧ t[s]) ⇒ (r = s ∨ r[s] ∨ s[r])) ∧ ∀r((r ⊆ t ∧ ∃s(r[s])) ⇒ ∃m(r[m] ∧ ∀x(r[x] ⇒ ¬m[x]))). Then, ̟ = 1 is called the ordinal function. Moreover, F (t) if ̟[t], we say t is an ordinal.

The ordinal function ̟ is defined from a conjunction of five formulas: any ordinal is supposed to be a restriction of 1 and emergent and transitive and totaly ordered by actions and well-ordered by actions (term m above is the least element with respect to actions). The first two formulas are obviously satisfied by ϕ0. The remaining ones are satisfied by vacuity. Thus, ϕ0 is an ordinal. That fact entails any ϕn is an ordinal. In order to grant we are talking about ordinals in an usual sense, we prove the next theorems.

Theorem 31 Every ordinal acts only on ordinals.

Proof All we have to prove is ∀r(∀t(̟[t] ∧ t[r]) ⇒ ̟[r]). Formula F (t) from Definition 15 is a conjunction of five formulas. So, this proof is divided into five parts, where we assume t is an ordinal: (i) we know (̟[t]∧t[r]) ⇒ r ⊆ t. Since t ⊆ 1, then r ⊆ 1. That means the first formula in the five factors conjunction F (t) is satisfied when we replace t by r. (ii) According to F8, any restriction r of an emergent function t is emergent. Thus, since t[r] ⇒ r ⊆ t, then E(r). That settles the second condition from F (t). (iii) We know (̟[t] ∧ t[r]) ⇒ r ⊆ t. Thus, if r[s], then t[s]. That entails s ⊆ t. Now, suppose s 6⊆ r. Then, ∃y(s[y] ∧ ¬r[y]). Since t[r] ∧ t[y], then r = y ∨ y[r] ∨ r[y]. Nevertheless, the possibilities r[y] and r = y cannot take place. So, we must have y[r]. Therefore, r[s]∧s[y]∧y[r], where t acts on all of them: r, s, and y. Now, let m be a restriction of t who acts only on r, s, and y. Therefore, last formula of the five factor conjunction F (t) cannot be satisfied for m (there is no least term with respect to actions of m on r, s, and y). In other words, our hypothesis s 6⊆ r leads to a contradiction. Thus, s ⊆ r. (iv) We know (̟[t] ∧ t[r]) ⇒ r ⊆ t. Thus, (r[u]∧r[v]) ⇒ (t[u]∧t[v]). So, from the definition of ̟, u = v ∨u[v]∨v[u]. (v) We know (̟[t] ∧ t[r]) ⇒ r ⊆ t. If u ⊆ r, then u ⊆ t. So, if u acts on at least one term, then last condition of F (t) is satisfied when we replace t by r.

Theorem 32 If t is an ordinal, then σt is an ordinal.

Proof Analogously to last theorem, we split this proof into ﬁve parts. (i) If t is an ordinal, then it is emergent. According to F8, the F-successor of any emergent function is emergent. That entails σt 6= 0. (ii) If t ⊆ 1, then σt ⊆ 1. (iii) Since σt 6= 0, then σt[t] and ∀r(σt[r] ⇒ r ⊆ σt). Last two parts are straightforward.

Theorem 33 The ordinal function ̟ is comprehensive.

17 Proof Suppose σ̟ 6= 0. If F (t) is the formula used in the definition of ̟, then F (̟). But according to item (ii) of Definition 10, that entails 1(σ̟)= 0. Such a condition is satisfied only for σ̟ = 0, a contradiction. Therefore, ̟ cannot be emergent, although it acts only on emergent terms.

Deﬁnition 16 An ordinal t is a limit ordinal iﬀ ∄r(σr = t).

In particular, ϕ0 is a limit ordinal.

Theorem 34 Any ordinal is well-ordered with respect to actions.

Proof We adopt the next convention: if r and s are ordinals, r is lesser than s iﬀ s[r]. Trichotomy, transitivity, and the existence of a least element (with respect to actions) are straightforward consequences from Deﬁnition 15.

Definition 17 The least limit ordinal who acts on ϕ0 is denoted by ω. We say r is a finite ordinal iff ω[r].

Theorem 35 If r is an ordinal, then there is a limit ordinal s such that s[r].

Proof Analogous to standard literature [15].

2.5 ZF-sets It is convenient to introduce the next concepts.

Deﬁnition 18 If E(f) and f acts solely on emergent functions, DomF = 1 f f[t] and ImF = 1 are, respectively, the F-domain and the F-image of f ∃x(f[x]∧f(x)=t) F F f. We denote f as f : Domf → Imf .

F F When we write f : x → y, that means x = Domf and y = Imf , as long f is F F emergent. We write Domf and Imf instead of Domf and Imf due to the fact we want to compare F-domain and F-image with the usual notions of domain and image (range) of a function in a theory like ZFU, for example. Details in Subsection 2.8. Now, let α(x, y) be a formula where there is at least one occurrence of x, one occurrence of y, and all of them are free. Then, next formula is an axiom.

F10α - Creation ∀x∃!y(α(x, y)) ⇒ ∀f(E(f) ⇒ ∃!g(∀x∀y((α(x, y) ∧ x 6= g) ⇒ F ((f[x] ⇒ g(x) = y) ∧ (¬f[x] ⇒ g(x) = 0))) ∧ σ(g) 6= 0 ∧ σ(Img ) 6= 0 ∧ (f[g] ⇒ ¬α(g,f(g))))).

We employed the notation σ(g) instead of σg due to limitations in the LATEX editor of this text, particularly for the case of F-successor of the F-image of g. Last postulate is supposed to deﬁne functions g from certain formulas α and any f, as long f is emergent. We say g is created by f and α(x, y), and α(x,y) α denote this by g = f or, simply, g = f . Besides, g is emergent, as

18 well as its F-image (and consequently its F-domain). Last conjunction in F10α avoids ambiguities in the calculation of g. Postulate F10α motivates us to employ an alternative notation for functions. When we see fit to do so, we f may eventually write a 7−→ b meaning f(a) = b. One obvious advantage of x this notation a 7−→ b is that it can be used to easily write down strings like f f f a1 7−→ a2 7−→ a3 7−→ ···, meaning f(a1) = a2, f(a2) = a3, and so on. We adopt the next convention: every time we state a finite sequence like either f f f f f f f f f f a1 7−→ a2 7−→ a3 7−→ ··· 7−→ an 7−→ 0 or a1 7−→ a2 7−→ a3 7−→ ··· 7−→ an 7−→ ai (i is any value among terms from 1 to n), we assume f(f)= f and f(r) = 0 if r is different of all previous terms already stated. F10α allows us to get restrictions by other means besides F9E. For example, g g g from F10α there is a g such that ϕ1 7−→ ϕ2 7−→ ϕ3 7−→ 0. In that case g ⊂ σ. Function g could be created from, e.g., ϕ4 and an appropriate formula α. With the aid of F10α it is quite easy to show F-composition is not commu- f f tative. As an example, consider f as a function given by ϕ1 7−→ ϕ2 7−→ ϕ2 and g g g◦f g◦f g given by ϕ2 7−→ ϕ3 7−→ ϕ3. In that case g ◦ f is given by ϕ1 7−→ ϕ3 7−→ 0 and g◦f g◦f ϕ2 7−→ ϕ3 7−→ 0, while f ◦ g is simply ϕ0 (which is obviously different of g ◦ f). It is worth to remark there are some functions which cannot exist (besides those already discussed), as we can see in the next theorem.

x x Theorem 36 There is no x where, for a given t 6= 1, t 7−→ 1 7−→ t.

Proof Suppose there is such a function x. Then, from F5, x ◦ x is given by x◦x x◦x t 7−→ t and 1 7−→ 1. But from Theorem 5 we have that any function h such that h(1)= 1 entails h = 1. Thus, x ◦ x is supposed to be 1. But since x is diﬀerent of 1, then x ◦ x cannot be 1, according to Theorem 12. That is a contradiction!

Theorem 37 ∀f(E(f) ⇒ ∃!g(g = f )). F

Proof Since f is emergent, we can use F10α. Let α(x, y) be the next formula: α (F (x) ⇒ f(x) = y) ∧ (¬F (x) ⇒ y = 0). In that case, f = f = g. F Concerning the uniqueness of g, that is granted by F10α.

Next we use the axioms of Creation and Restriction to deﬁne ZF-sets, who are supposed to play the same role sets play in ZF set theory.

Deﬁnition 19 Let α(u, v) be a formula recursively deﬁned as it follows: i α(ϕ0, ϕ0); ii ∀u∀v(̟[u] ⇒ (α(u, v) ⇒ α(σu,℘(v)))); iii ∀u((̟[u] ∧ ∄x(σx = u)) ⇒ α(u, 1 )); ∃r∃v(u[r]∧α(r,v)∧v[t]) iv ∀u((¬̟[u]) ⇔ α(u, 0)). v ∀u∀v∀w((α(u, v) ∧ α(u, w)) ⇒ v = w).

19 α If η = r , where r is an ordinal, we say η is a von Neumann function with rank r. For the sake of abbreviation, any von Neumann function with rank r is denoted by ηr.

ηϕ4 ηϕ4 As an example consider ηϕ4 . That is a function given by ϕ0 7−→ ϕ0, ϕ1 7−→ ϕ1, ηϕ4 ηϕ4 ϕ2 7−→ ϕ2, ϕ3 7−→ ℘(ϕ2), where ℘(ϕ0) = ϕ1, ℘(ϕ1) = ϕ2, and ℘(ϕ2) is a restriction of 1 who acts only on ϕ0, ϕ1, ϕ2 and γ, where γ acts only on ϕ1.

Theorem 38 For any ordinal r and any t such that r[t], ηr(t) is emergent and a restriction of 1.

Proof Observe we are not saying that any ηr is a restriction of 1, but only their F-images. We prove this by transfinite induction. Let α be the formula from Definition 19. The first ordinal r is ϕ0. F10α trivially α says ϕ = ϕ . That entails ηϕ0 = ϕ . Since ϕ does not act on any 0 0 0 0 term, then it satisfies the theorem by vacuity. For the sake of argument, next ordinal ϕ1 is the first one who acts on some term. In that case,

ηϕ1 = ϕ1. The only term where ϕ1 acts is ϕ0. Besides, ϕ1(ϕ0) = ϕ0. And ϕ0 is emergent and a restriction of 1. Now, suppose all non-0 images of a given ηr are emergent and restrictions of 1. As we could see, that takes place with the first two ordinals. Since r is an ordinal and any ordinal acts only on ordinals (Theorem 31), then, ηr(σt) = ℘(ηr(t)), for any ordinal t where r acts, as long r acts on σt besides t. From F8 (see function h in that axiom), ℘(ηr(t)) is emergent. Besides, from the very definition of restricted power (Definition 11), ℘(ηr(t)) is a restriction of

1. An analogous result holds for ησr , since σr acts on all terms where r acts and on r itself. In the case where u is a limit ordinal (item iii of Deﬁnition 19), then ησ (u)= 1 . But, according to F8 u ∃r∃v(u[r]∧α(r,v)∧v[t]) (see function i in that postulate), that last term is emergent. Besides, it is trivially a restriction of 1.

Deﬁnition 20 (i) ν = 1 ∃r ̟ r ∧η r t is the von Neumann universe; (ii) ( [ ] σr ( )= ) ∀u(Z(u) ⇔ ∃t(ν[t] ∧ t[u])); we read Z(u) as ‘u is a ZF-set’.

Theorem 39 Every ordinal is a ZF-set.

Proof For any ordinal r, ησσr (σr) acts on r (although it may act on other terms as well), according to Deﬁnition 19. So, from Deﬁnition 20, Z(r).

Theorem 40 If ν is the von Neumann universe, then σν = 0.

Proof From Theorem 39, ̟ ⊂ ν. From Theorem 33, σ̟ = 0. So, from Theorem 18, ν is comprehensive. Suppose σν 6= 0. Then, ν is emergent, since it acts only on terms with non-0 F-successor. Thus, from F8, every restriction of ν has a non-0 F-successor. But ̟ is a restriction of ν and σ̟ = 0, a contradiction. Therefore, σν = 0.

Theorem 41 ∀u(Z(u) ⇔ ∃r(̟[r] ∧ u ⊆ ησr (r))).

20 Proof From Deﬁnition 20, Z(u) ⇔ ∃r∃t(̟[r]∧ησr (r)[t]∧t[u]). From Deﬁnition

19, item (ii), ησσr (σr)= ℘(ησr (r)). Hence, for any x, ησσr (σr)[x] ⇔ x ⊆

ησr (r).

Deﬁnition 21 The rank of a ZF-set u is the least ordinal r such that u ⊆ rank ησr (r). We denote this by r = (u).

a b As an example, consider a function a such that b 7−→ b, where ϕ1 7−→ ϕ1. The rank of a is ϕ3, while the rank of b is ϕ2, and the rank of ϕ1 is ϕ1.

Theorem 42 ∀u(Z(u) ⇒ ∀x(u[x] ⇒ rank(u)[rank(x)])).

Proof Straightforward from Theorem 41 and the deﬁnition of rank of a ZF-set.

Last theorem, in a sense, resembles the deﬁnition of rank of a set in [23].

Theorem 43 ∀f(Z(f) ⇔ (σf 6= 0 ∧ f ⊆ 1 ∧ ∀x(f[x] ⇒ Z(x))).

Proof The ⇒ part is a direct consequence from Theorem 38 and Deﬁnition 20. For the ⇐ part, we already know f is emergent, since σf 6= 0 and f acts only on ZF-sets. That entails we can use F10α to create a function g from f, if an adequate formula β is used. For that purpose, consider formula β(x, y) given by “(f[x] ⇒ y = rank(x)) ∧ (¬f[x] ⇒ y = 0)”. According F F to F10α, σ(Img ) 6= 0. On the other hand, Img is a restriction of ̟. F rank F Therefore, Img is a ZF-set (Theorem 39). Now, let r = (Img ). That entails r acts on the rank of any term x where f acts, i.e., ∀x(f[x] ⇒ rank r[ (x)]). Now, suppose f is no restriction of ησr (r) (see Theorem

41). That entails there is u such that f[u] ∧ ¬ησr (r)[u] (remember u is rank supposed to be a ZF-set). Since ¬ησr (r)[u], then (u)[r], which entails ¬r[rank(u)]. That contradicts the fact that ∀x(f[x] ⇒ r[rank(x)]). Thus,

indeed f ⊆ ησr (r). From Theorem 41, f is a ZF-set.

Theorem 44 u = 1 is comprehensive. Z(t)

Proof Suppose σu 6= 0. Then, according to Theorem 43, u is a ZF-set. Thus, σu = u, a contradiction.

Definition 22 f is an ordered pair (a,b), with both values a and b different of 0, iff there are α and β such that α 6= f, β 6= f, α 6= a, β 6= b and

α if x = α  f(x)=  β if x = β 0 if x 6= f ∧ x 6= α ∧ x 6= β  where α(a)= a, α(x)= 0 if x is neither a nor α, β(a)= a, β(b)= b, β(x)= 0 if x is neither a nor b or β.

21 We have two kinds of ordered pairs: those where α 6= b (ﬁrst kind) and those where α = b (second kind). Their respective diagrams are as it follows: g x f x x β x α β α = b x x x a a b xa

Figure 2: Diagrams of f = (a,b) and g = (a,b) of the first and second kind. Left diagram above concerns the case where f acts only on α and β, while α acts only on a, and β acts only on a and b. In the particular case where a = b, we have α = β, and the ordered pair f is (a,a). So, (a,a) is a function f which acts solely on α, while α acts solely on a. Observe that f(a) = f(b) = 0 (f never acts neither on a nor on b) if f is an ordered pair of the first kind. Thus, f is (a,b) iff f acts only on α and β, which act, respectively, only on a and only on a and b. To get (b,a), all we have to do is to exchange α by a function α′ which acts only on b. Our definition is obviously inspired on the standard notion due to Kuratowski. In standard set theory (a,b) is a set {{a}, {a,b}} such that neither a nor b belong to (a,b). In Flow, on the other hand, an ordered pair (a,b) of the first kind is a function which does not act neither on a nor on b. Nevertheless, the second kind of ordered pair shows our approach is not equivalent to Kuratowki’s. In the case where α = b, we have the diagram to the right of Figure 2. This non-Kuratowskian ordered pair g = (a,b) acts on b, although it does not act on a. And no Kuratowskian ordered pair (a,b) ever acts on either a or b. So, in the general case, no ordered pair (a,b) ever acts on a. That means:

Theorem 45 Any ordered pair (a,a) is Kuratowskian.

The proof is straightforward. Since any ordered pair (a,b) is a function, for the sake of abbreviation we write x(a,b) for x((a,b)), for a given function x. The reader can observe, from Figure 2, that in a non-Kuratowskian ordered pair f, β is the non-0 F-successor of α. In other words:

Theorem 46 f = (a,b) is a non-Kuratowskian ordered pair iﬀ f acts only on a function α - which, in its turn, acts on one single term a - and on its F-successor σα, where σα 6= 0.

Proof If f = (a,b), then f acts at most on α and β, where α acts only on a, and β acts only on a and b. Suppose f = (a,b) is non-Kuratowskian. Then, a 6= b (Theorem 45); and f = (a, α), since α = b. But a 6= b entails a 6= α. And since α acts on a, then β acts on two terms: α and a. But that is the condition given for granting β = σα, where σα 6= 0. Finally, if f acts only on α and σα, where α acts only on a and σα 6= 0, then f = (a, α) is non-Kuratowskian.

22 Suppose f = 1 . In that case f acts on ϕ3 and on its F-successor x=ϕ3∨x=ϕ4 ϕ4. Nevertheless, ϕ3 does not act on just one single term. Thus, such an f is not an ordered pair, let alone a non-Kuratowskian ordered pair.

Theorem 47 (a,b) = (c, d) iﬀ a = c and b = d.

Proof Straightforward from Deﬁnition 22 and F9E.

Theorem 48 If a and b are both emergent, then there is f = (a,b).

Proof From F9E we deﬁne the proper restriction β of 1 for “t = a ∨ t = b” as formula F (t). So, β(a) = a, β(b) = b, β(β) = β, and β(x) = 0 for all remaining values of x. Analogously, the proper restriction α of 1 for “t = a”gives us α(a) = a, α(α) = α, and α(x) = 0 for the remaining values of x. And, from F9E, we have α 6= a. Analogously, we have β 6= a and β 6= b. Finally, the proper restriction f of 1 for “t = α∨t = β gives us f(α)= α, f(β)= β, f(f)= f, and f(x)= 0 for all the remaining values of x. Besides, f 6= α and f 6= β. But function f is exactly that one in Deﬁnition 22. Hence, f = (a,b).

To get an ordered pair (0, 0), all we have to do is to consider ϕ1, which acts only on ϕ0. So, ϕ1 =def (0, 0). If f acts only on α and ϕ0, where α acts only on a 6= 0, then we adopt the convention f = (a, 0). For example, ϕ2 is the ordered pair (ϕ0, 0). There are no ordered pairs of the form (0,b), where b 6= 0.

Theorem 49 If f is a ZF-set, then ℘(f) is a ZF-set.

Proof If f is a ZF-set, then it is emergent. So, f acts only on emergent functions and F8 says any t who lurks f is emergent. Now, let h = 1 . tEf Then, h[t] ⇔ t E f. So, h = p(f) (Deﬁnition 9). But F8 also says such an h is emergent. From Deﬁnition 11, p = ℘(f) = 1 . But p lurks h. t⊆f So, once again from F8, σp 6= 0.

Deﬁnition 23 f is injective iﬀ ∀r∀s((f[r] ∧ f[s] ∧ r 6= s) ⇒ f(r) 6= f(s)). We denote this by I(f).

For example, any ϕn is injective.

Deﬁnition 24 x ∈Z f iﬀ f ⊂ 1 ∧ Z(x) ∧ f[x].

The negation of formula x ∈Z f is abbreviated as x 6∈Z f. We read x ∈Z f as ‘x Z-belongs to f’ or ‘x is a Z-member of f’. The symbol ∈Z is called Z- membership relation.

Theorem 50 (i) ∀x(x 6∈Z x); (ii) ∀x(0 6∈Z x); (iii) ∀x(1 6∈Z x).

Proof Item (i) is consequence from F2, since no x acts on itself. Item (ii) is consequence from Theorem 10: no x acts on 0. Item (iii) is immediate, since 1 is not a ZF-set.

23 At ﬁrst glance, item (i) looks like an evidence that Flow is well-founded. Nevertheless, the issue of regularity is a little more subtle than that, as we can see later on. {f,g} =def 1 , where E(f) ∧ E(g). So, a pair {f,g} is a restriction x=f∨x=g of 1 which acts only on f and g, as long they are both emergent.

Theorem 51 Let E(f) ∧ E(g). If u = {f,g} is a pair, then E(u).

α(x,y) Proof Let h = ϕ be deﬁned from F10α by formula α(x, y) given by 2 (x = ϕ0 ⇒ y = f) ∧ (x = ϕ1 ⇒ y = g) ∧ ((x 6= ϕ0 ∧ x 6= ϕ1) ⇒ y = 0). According to F10α, σh 6= 0. Since h acts on emergent functions, then h is emergent. Now, let u = 1 . That entails u lurks h. And t=g(ϕ0)∨t=g(ϕ1) axiom F8 grants σu 6= 0. Hence, u is emergent.

In particular, any pair of ZF-sets is a ZF-set. On the left side of the image below we find a Venn diagram of a set f (in the sense of ZF), with its elements g, h, and i. On the right side there is a diagram of a ZF-set f (in the sense of Flow) which acts on g, h, and i. One of the main differences between both pictorial representations is the absence of arrows in the Venn diagram. Venn diagrams do not emphasize the full role of the Extensionality Axiom from ZF: a set is defined either by its elements or by those terms who do not belong to it as well. Within Flow, however, it is explicitly emphasized that anything which is not inside the rectangle of a diagram has an image 0. Thus, in a sense, Flow diagrams were always somehow implicit within Venn diagrams. At the end, standard extensional set theories, like ZFC, are simply particular cases of a general theory of functions, as we can formally check in Subsection 3.2. f ✬✩g h f x x x i g h i Figure 3: Comparison between Venn diagrams and Flow diagrams. ✫✪ Theorem 52 Any inductive function which acts only on ZF-sets is a ZF-set.

Proof Immediate from the deﬁnitions of inductive function and ZF-set.

Deﬁnition 25 f is a proper class iﬀ ∀x(f[x] ⇒ Z(x)) ∧ C(f).

In other words, no proper class is a ZF-set.

Theorem 53 There is one single ZF-set f such that for any x, we have x 6∈Z f.

Proof f = ϕ0. From F6, ϕ0 is unique. From Theorem 14, ϕ0 is the only term diﬀerent of 0 (recall 0 is not a ZF-set) which satisﬁes the condition given above. In other words, ϕ0 is the empty ZF-set, which can be denoted by ∅.

24 2.6 Atoms Next we introduce a generalization of ZF-sets which is quite helpful for us.

Definition 26 i Zϕ0 (f) iff Z(f); ii Zϕ1 (f) iff f 6⊆ 1 ∧ E(f) ∧ ∀x(f[x] ⇒ (Zϕ0 (x) ∧ Zϕ0 (f(x))));

iii Zσr (f) iff ̟[r] ∧ r[ϕ0] ∧ E(f) ∧ ∃x(f[x] ∧ (Zr(x) ∨ Zr(f(x)))) ∧ ∀y((f[y] ∧ ¬Zr(y)) ⇒ ∃s∃t(r[s] ∧ r[t] ∧ Zs(y) ∧ (Zt(f(y)) ∨ Zr(f(y))))). iv Let s be a limit ordinal different of ϕ0. Then, Zs(f) iff E(f) ∧ ∀x(f[x] ⇒ ∃r∃t(s[r] ∧ s[t] ∧ Zr (x) ∧ Zt(f(x)) ∧ ∃u∃y(u[r] ∧ u[t] ∧ s[u] ∧ f[y] ∧ Zu(y)))). If r is an ordinal, we read Zr(f) as ‘f is a ZF-emergent function of degree r’ or, simply, ‘ZF-emergent’, if the degree r is irrelevant.

The only ZF-emergent functions which are required to be no restrictions of 1 are those with degree ϕ1. Nevertheless, for our purposes in this Subsection, we are mostly interested on ZF-emergent functions which are no restrictions of 1, regardless of their degree, according to next deﬁnition.

Deﬁnition 27 A(f) iﬀ ∃r(̟[r] ∧ Zr(f) ∧ f 6⊆ 1). We read A(f) as ‘f is an atom’.

In other words, atoms are ZF-emergent functions who are no restriction of 1.

Deﬁnition 28 Let a 6= ϕ0 be an emergent function who acts only on atoms and such that a ⊂ 1. In that case we say a is a brick of atoms, and denote this by B(a). If we change item i of Deﬁnition 19 to α(ϕ0,a), while keeping a α all remaining items, then ηr = r is an atomic von Neumann function, where a r is an ordinal. Besides, ν = 1 ∃r(̟[r]∧ηa (r)=t) is an atomic von Neumann σr a universe. Furthermore, ∀u(Ma(u) ⇔ ∃t(ν [t] ∧ t[u])); we read Ma(u) as ‘u is a Menge in νa’ or simply ‘u is a Menge’ if there is no risk of confusion.

Obviously, the name Menge was borrowed from Cantor’s Mengenlehre. Ob- serve no Menge is an atom and no atom is a Menge. Besides, function a of last deﬁnition works, for all practical purposes, as the set of all atoms within the context of our atomic von Neumann universe.

a Theorem 54 For any brick of atoms a, σ(ν )= σ 1 M = 0. a(t)

Proof Analogous to the proofs of Theorems 40 and 44.

Theorem 55 ∀a∀f((B(a) ∧ Z(f)) ⇒ Ma(f)).

25 The proof of last theorem is trivial. It means every ZF-set is a Menge in any atomic von Neumann universe. Obviously there may be a Menge who is not a ZF-set (recall every ZF-set is a ZF-emergent of degree ϕ0). Our atoms are supposed to mimic Urelemente from a theory in the style of ZFU (ZF with Urelemente), while certain ZF-emergent functions f such that f ⊂ 1 are supposed to play the role of ‘collections’ or ‘sets’. In order to do that we need to introduce a specialized concept of ‘membership’ which grants all atoms are empty. Thus:

Definition 29 (i) x ∈ y iff Ma(y) ∧ y[x], if a is a brick of atoms; (ii) x 6∈ y iff ¬(x ∈ y). We read x ∈ y as ‘x is an element of y’ or ‘x belongs to y’.

In other words, x ∈ y iff y is a Menge and x is either an atom or a ZF-set such that y[x] or a Menge such that y[x] and so on. After all, any Menge who acts on some term, acts either on an atom or on a ZF-set or on another Menge. Since no atom is a Menge, then every atom y is empty, in the sense that, for any x, we have x 6∈ y. That means our ZF-sets correspond to pure sets. From an intuitive point of view, although our atoms are empty (with respect to ∈), they are not ontologically equivalent to the classical notion of Ure- lemente. Urelemente, from German, are supposed to be primordial elements, i.e., ‘unbreakable into minor parts’. However, our atoms have some sort of ‘inner structure’, since they are functions who act on other functions. For ex- x x x ample, if x is a function such that ϕ0 7−→ ϕ1 7−→ ϕ2 7−→ 0, and y is given by y y y ϕ2 7−→ ϕ1 7−→ ϕ0 7−→ 0, they are both atoms. But x ◦ y is a ZF-set whose only elements are the ZF-sets ϕ1 and ϕ2. No version of ZFU in literature suggests the possibility of compositions among atoms who are able to produce ‘pure sets’. Besides, our atoms may be ‘ranked’ by means of degrees: the degree of an atom a is the least ordinal r such that Zr(a). That is why we prefer to avoid the terms ‘Urelement’ and ‘Urelemente’ for describing empty terms who are not Menge. On the other hand, the term ‘atom’ is not appropriate either, since its etymol- ogy is attached to the notion of indivisibility. Nevertheless, for our purposes in this paper, we refer to our empty terms as atoms, as long they are not Menge but can be elements of a Menge. If x is a ZF-emergent of degree ϕ0, then it is a Menge in any von Neumann universe (Theorem 55). If x is a ZF-emergent of degree ϕ1, then it is an atom. Therefore, it is an atom in any atomic von Neumann universe νa where a acts on x. If x is a ZF-emergent of degree greater than ϕ1, then either x 6⊂ 1 or x ⊂ 1. In the first case, x is an atom in νa, for any a which acts on x. Finally, if x ⊂ 1, then ∀t(x[t] ⇒ (t ⊂ 1∨t 6⊂ 1)). Thus, either b = x t6⊆1 is a brick of atoms c or b = ϕ0. In the first case b is a Menge in ν for any c such that b ⊆ c. In the second case Theorem 55 once again grants ϕ0 is a Menge in any atomic von Neumann universe. All of this points to an interesting phenomenon, namely, ZF-emergent functions of degree ϕ1 constitute some sort of frontier between collections with atoms and collections without atoms.

26 2.7 Interpreting ZFU Our main goal here is to introduce a minimum set of ingredients to build a model of ZF set theory with atoms (ZFU) where we have PP but not AC. It is well known that permutation models of ZFU are easily capable of violating the Axiom of Choice thanks to one single fact: the axioms of ZFU cannot distinguish among atoms, in the sense that there are non-trivial ∈-automorphisms on the universe of ZFU. Although Definition 27 allows the existence of a vast number of atoms, we are interested only on a specific range of them. But first we need a new concept.

Deﬁnition 30 Let x be any function. The kernel of x is given by K(x) = 1 . Z(t)∧(x[t]∨∃r(x(r)=t))

Last deﬁnition is sound, since every ZF-set is emergent. Thus, F9F is applicable.

Theorem 56 Let νa be an atomic von Neumann universe. Then, K(νa) = ν, where ν is the von Neumann universe.

Proof Straightforward from Theorem 55.

Thus, we have here a generalization of the usual concept of kernel [15] which is applicable in a nontrivial way to atoms as well, as we can see in the next Observation.

Observation 4 According to our previous discussions we have a lot of options for choosing a brick a of atoms for defining any atomic von Neumann universe νa. That is why we now introduce some simple examples for the sake of illustration. Let µ be a function obtained from ̟ and F10α as it follows, t=ω∨t=σω where ω is the limit ordinal who acts only on finite ordinals (Definition 17): µ µ ′ ω 7−→ σω 7−→ ω. Let µ be obtained from ̟ t σ ∨t σ and F10α as it follows: = ω = σω µ′ µ′ ′ σω 7−→ σσω 7−→ σω. Obviously, µ 6= µ . Let p be a function obtained from ϕ3 and p p p F10α as it follows: ϕ0 7−→ ϕ1 7−→ ϕ2 7−→ ϕ0. Using once again F10α over p we g g g g ′ g can obtain the next function g given by ϕ1 7−→ ϕ0 7−→ µ 7−→ 0 and ϕ2 7−→ µ 7−→ 0. h h h ′ h h Analogously there is an h such that µ 7−→ ϕ0 7−→ ϕ1 7−→ 0 and µ 7−→ ϕ2 7−→ 0. aϕ0 aϕ0 ′ aϕ0 Hence, there is a function aϕ0 where µ 7−→ ϕ0 7−→ µ 7−→ µ. By using analogous aϕ0 ′ aϕ0 aϕ0 arguments we introduce the next atom aϕ0 : µ 7−→ µ 7−→ ϕ0 7−→ µ. Observe aϕ0 6= aϕ0 . Besides, aϕ0 ◦ aϕ0 = aϕ0 , aϕ0 ◦ aϕ0 = aϕ0 , but K(aϕ0 )= K(aϕ0 )=

K(aϕ0 ◦ aϕ0 ) = K(aϕ0 ◦ aϕ0 ) = K(aϕ0 ◦ aϕ0 ) = K(aϕ0 ◦ aϕ0 ) = ϕ1. Although both atoms aϕ0 and aϕ0 are empty (there is no t such that t belongs to either one of them), their kernels are ϕ1, a non-empty ZF-set. We say aϕ0 and aϕ0 are conjugates of each other, in the sense that they cannot be distinguished from each other neither by means of their kernels nor by means of the kernels of their F-compositions. So, yes, we are inspired on the original ideas by Abraham A. Fraenkel [8], despite the fact that his seminal paper has some well known mistakes [24]. Nevertheless we use an analogous technique to introduce atoms

27 which can be distinguished from aϕ0 and aϕ0 . That is a key point for our main result.

Deﬁnition 31 The rank of a Menge u in an atomic von Neumann universe νa a rank is the least ordinal r such that u ⊆ ησr (r). We denote this by r = (u). The rank of any atom in an atomic von Neumann universe is ϕ0.

Observe this last concept of rank of a Menge in νa is independent from the notion of degree of any atom where a acts. Next we introduce the concept of transitive closure, which is a copy from the usual notion in set theory.

Definition 32 Let Ma(x), i.e., x is a Menge from an atomic von Neumann a universe ν . Then: (i) xϕ0 = x; (ii) xσ = t, where r is a finite ordinal; r Sxr[t] (iii) TC(x)= xr. We read TC(x) as the ‘transitive closure’ of x. Sω[r] Definition 33 Let a be a brick of atoms such that ∀x(a[x] ⇒ ∃!y(a[y] ∧ x 6= y ∧ K(x)= K(y) ∧ K(x ◦ y)= K(y ◦ x))). Then, in the corresponding atomic von Neumann universe νa, we have: i If Z(x) and y is any term, then x ≡ y iff x = y; ii If a[x] ∧ a[y], then x ≡ y iff K(x)= K(y) ∧ K(x ◦ y)= K(y ◦ x); iii Let x = 1 , where a[r] ∧ a[s] ∧ r 6= s ∧ r ≡ s; then ∀y(y ≡ x ⇔ y = x); t=r∨t=s iv Let f = {r}x be an abbreviation for the formula x ⊆ a ∧ a[r] ∧ ∀s(f[s] ⇔ (x[s] ∧ s ≡ r)); now, let x ⊆ a ∧ y ⊆ a, then x ≡ y iff ∀u(x[u] ⇒ ∃!v(v = {u}x ∧ v ≡{u}y)); v Let Ma(x) ∧ Ma(y), then x ≡ y iff rank(x)= rank(y) ∧ ∀r∀s((̟[r] ∧ ̟[s]) ⇒ TC(x ) ≡ TC(y ) ), where all rank(t)=r∨rank(t)=s a[m] rank(t)=r∨rank(t)=s a[m] occurrences of m in formula a[m] are free; vi ∀x∀y(x ≡ y ⇔ y ≡ x); vii If a[x] ∧ Ma(y), then ¬(x ≡ y). We read x ≡ y as ‘x is indistinguishable of y’ or ‘x is indiscernible of y’. x 6≡ y abbreviates ¬(x ≡ y), and we read it as ‘x is distinguishable of y’ or ‘x is discernible of y’.

a Obviously, any x of ν (i.e., either a[x] or Ma(x)) is indistinguishable of itself. The same criteria used for indiscernible atoms (item ii) could be used to ZF-sets. After all, if both x and y are ZF-sets, then x ◦ y = y ◦ x is simply the intersection between x and y. In that case indistinguishability ≡ is simply identity =, since K(x) = x, K(y) = y, K(x ◦ y) = x ◦ y, and K(y ◦ x) = y ◦ x. Nevertheless, there may exist diﬀerent atoms which are indistinguishable, like aϕ0 and aϕ0 in Observation 4. That is a key feature we intend to use in our proof of independence of AC from the Partition Principle within ZFU.

28 Item iv of last deﬁnition introduces classes of indiscernible atoms. For in- aϕ0 aϕ0 ′ aϕ0 stance, let aϕ0 be the atom µ 7−→ ϕ0 7−→ µ 7−→ µ. Let aϕ0 be the atom aϕ0 ′ aϕ0 aϕ0 µ 7−→ µ 7−→ ϕ0 7−→ µ. Those are the same ones from Observation 4. Now, let aϕ1 aϕ1 ′ aϕ1 aϕ1 ′ aϕ1 aϕ1 aϕ1 be the atom µ 7−→ ϕ1 7−→ µ 7−→ µ, and aϕ1 be the atom µ 7−→ µ 7−→ ϕ1 7−→ µ.

Observe aϕ0 6≡ aϕ1 . If x is a Menge who acts only on aϕ0 , aϕ0 , and aϕ1 , then

{aϕ0 }x = {aϕ0 }x, namely, a Menge who acts only on aϕ0 and aϕ0 . On the other hand, {aϕ1 }x = {aϕ1 }x, namely, a Menge who acts solely on aϕ1 . If aϕ2 is defined in a similar fashion, then {aϕ2 }x = ϕ0. Bricks of atoms in the sense of last Definition provide necessary but not sufficient conditions for achieving our goals. That is why we need the next concept. We introduce below our Appropriate Brick of Atoms which is used in our main result of this paper.

α α ′ α Definition 34 For any finite ordinal r, let αr be the atom r 7−→r µ 7−→r µ 7−→r r α ′ α α ′ and αr be the atom r 7−→r µ 7−→r µ 7−→r r, where µ and µ are the same functions introduced in Observation 4. Then, a = 1 is the Appro- ∃r(ω[r]⇒(t=αr∨t=αr)) priate Brick of Atoms, where ω is the limit ordinal who acts only on all finite ordinals (Definition 17), i.e., all ϕn. Any t where a acts is an appropriate atom. Within this context, νa is the Appropriate Atomic von Neumann Universe, or Appropriate Universe, for short.

That means the Appropriate Brick of Atoms a is a Menge in νa. From now on we use the same notation of last deﬁnition for appropriate atoms, i.e., anytime we write either αr or αr we mean both are appropriate atoms. Next theorem is quite important, although its proof is a simple exercise.

Theorem 57 For any ﬁnite ordinals r and s, such that r 6= s, we have: (i) K(αr)= K(αr)= K(αr ◦αr)= K(αr ◦αr)= K(αr ◦αr)= K(αr ◦αr)= γ, where γ ′ γ is given by r 7−→ r; (ii) K(αr ◦αs)= K(αr ◦αs)= K(αr ◦αs)= K(αr ◦αs)= γ , ′ γ′ γ′ where γ is given by r 7−→ r and s 7−→ s. Proof Just a bunch of boring calculations.

Theorem 58 For any ﬁnite ordinals r and s, such that r 6= s, we have: (i) αr ≡ αr; (ii) αr 6≡ αs; (iii) αr 6≡ αs; (iv) αr 6≡ αs; (v) αr 6≡ αs.

Proof For any ﬁnite ordinal r we have K(αr)= K(αr)= K(αr ◦ αr)= K(αr ◦ γ αr)= γ, where γ is given by r 7−→ r. That settles item i. For the remaining ′ γ′ γ′ items observe that K(αs)= K(αs)= γ , where r 7−→ r and s 7−→ s. Since r 6= s, then γ 6= γ′.

Theorem 59 ∀f(Ma(f) ⇔ (σf 6= 0 ∧ f ⊆ 1 ∧ ∀x(f[x] ⇒ (a[x] ∨ Ma(x))))). Proof Analogous to the proof of Theorem 43, since any Menge in νa can be ranked.

Next we follow a maneuver somehow inspired on Cantor-Schr¨oder-Bernstein theorem [18].

29 Deﬁnition 35 g and h are equipotent, denoted by g =∼ h, iﬀ there is τ where, for any t: (i) τ[t] ⇒ τ(τ(t)) = t; (ii) g[t] ⇒ (τ[t] ∧ h[τ(t)]); (iii) h[t] ⇒ (τ[t] ∧ g[τ(t)]); (iv) τ[t] ⇒ (g[t] ∨ h[t]). We call function τ a connector of g/h.

For example, any ϕn is equipotent to itself. And a connector of ϕn/ϕn is ϕn.

Definition 36 g is Dedekind-finite iff ∃h(g =∼ h∧τ =6∼ g), where τ is a connector of g/h (Definition 35); and g is Dedekind-infinite iff it is not Dedekind-finite.

As an example, h = ϕ5 acts only on ϕ3 and ϕ4. So, if we prove the t=ϕ3∨t=ϕ4 existence of τ such that τ( ϕ0)= ϕ3, τ(ϕ3)= ϕ0, τ(ϕ1)= ϕ4, and τ(ϕ4)= ϕ1, then h =∼ ϕ2 with connector τ. Besides, τ =6∼ ϕ2 ∧ τ =6∼ h, since τ acts on four terms, while ϕ2 and h act on two terms each. To prove the existence of such a τ we use F10α.

Theorem 60 =∼ is reﬂexive among emergent functions.

Proof If f is ϕ0, assume 0 as connector, according to Deﬁnition 35. For the remaining cases, make τ = 1 as a connector of f/f. f[x]

Theorem 61 ∀f∀g((Z(f) ∧ Z(g) ∧ f =∼ g) ⇒ σf =∼ σg).

Proof If f and g are ZF-sets, then f ∪ g is a ZF-set. That means σf 6= 0, σg 6= 0, and σf∪g 6= 0. If f =∼ g, then there is a connector τ of f/g. Since f ∪ g is uncomprehensive, we can apply axiom F10α over f ∪ g, where formula α(x, y) is τ(x)= y. That entails, from F10α, that στ 6= 0. Besides, σf , σg, and σf∪g are uncomprehensive and ZF-sets. That entails ′ we can deﬁne a new connector τ for σf /σg. All we have to do is to apply ′ F10α over σf ∪ σg with formula α (x, y) given by α(x, y) when x 6= f and ′ ′ ′ x 6= g, and such that α (f,g) and α (g,f). Hence, σf =∼ σg, where τ is the connector of σf /σg.

Definition 37 g is infinite iff there is no finite ordinal r such that r =∼ g. Otherwise, we say g is finite.

If we recall that E (emergent), I (injective), and B (brick of atoms) are given, respectively, in Deﬁnitions 7, 23, and 28, next formula is our last axiom of this ﬁrst version of Flow.

F F F F11 - F-Partition ∀f∀a((B(a)∧E(f)∧Ma(Domf )∧Ma(Imf )) ⇒ ∃c(Domc = F F F Imf ∧ Imc ⊆ Domf ∧ I(c))).

2.8 The model Next we prove that if a = a, then we are able to introduce a model of ZFU where the Partition Principle holds but the Axiom of Choice fails.

30 Definition 38 Let a be a brick of atoms. A function f is a ZFU-function in an atomic von Neumann universe νa iff: i Ma(f); ii ∀t(f[t] ⇒ ∃u∃v((a[u] ∨ a[v] ∨ Ma(u) ∨ Ma(v)) ∧ t = (u, v))); iii ∀u∀v∀w((f[(u, v)] ∧ f[(u, w)]) ⇒ v = w). Observe the concept of ZFU-function corresponds to the usual notion of function in a theory like ZFU: a specific set of ordered pairs (u, v) where u and v are either atoms or sets. Theorem 62 Let a be a brick of atoms and f : x → y (Definition 18) be a function such that both x and y are Mengen of νa. Then there is one single ZFU- function f ′ such that ∀u∀v(f[u] ∧ f(u)= v) ⇔ (f ′[(u, v)] ∧ f ′((u, v)) = (u, v)). Conversely, if f ′ is a ZFU-function in νa, then there is a single f : x → y such ′ ′ that Ma(x)∧Ma(y) and ∀u∀v(f[u]∧f(u)= v) ⇔ (f [(u, v)]∧f ((u, v)) = (u, v)).

Proof First part. Let f : x → y be a function such that Ma(x) and Ma(y). Then, ∀r(f[r] ⇒ ∃s(y[s]∧f(r)= s)). Now, let α(r, t) be the next formula: α (f[r] ⇒ t = (r, f(r)))∧(¬f[r] ⇒ t = 0). Then, h = f is a function whose

non-0 images h(r) are just ordered pairs (r, s) such that f(r)= s. Besides, F ′ F F10α entails σ(Imh ) 6= 0. Another way to write this is f = Imh = ′ 1 ∀a∀b(t=(a,b)⇔(f[a]∧f(a)=b)). So, f is a ZFU-function, whose uniqueness ′ a is granted by F9E. Theorem 59 grants f is a Menge in ν , since any ordered pair (r, s) of either atoms or Mengen has a non-0 F-successor. Second part. Let f ′ be a ZFU-function in νa. Then ∀u∀v∀w((f ′[(u, v)] ∧ f ′[(u, w)]) ⇒ v = w). Now, let x = 1 . By analogous ∀u(t=u⇔∃v(f ′[(u,v)])) arguments used in the ﬁrst part, σx 6= 0. Next, let α(r, s) be the formula (x[r] ⇒ ∀t(f ′[(r, t)] ⇔ t = s)) ∧ (¬x[r] ⇒ s = 0). The deﬁnition of ZFU-function grants ∀r∃!s(α(r, s)). That means we can use F10α. Thus, α f = x is a function f : x → y such that ∀r(f[r] ⇒ ∃s(y[s] ∧ f(r)= s)).

Besides, σy 6= 0. The uniqueness of f is granted by F10α. Theorem 59 grants x and y are Mengen in νa. Last theorem could be rewritten in terms of an injective function f from the ‘space’ of all ZFU-functions of νa to the ‘space’ of all functions f : x → y where x and y are Mengen of νa. Deﬁnition 39 We say f : x → y is in correspondence to a ZFU-function f ′ in a given νa iﬀ ∀u∀v(f[u] ∧ f(u) = v) ⇔ (f ′[(u, v)] ∧ f ′((u, v)) = (u, v)), where x, y, and f ′ are Mengen in νa.

′ ′ Deﬁnition 40 Let f be a ZFU-function. Then, the domain of f is Domf ′ = ′ 1 , and the range of f is Imf ′ = 1 . ∀u(t=u⇔∃v(f ′[(u,v)])) ∀v(t=v⇔∃u(f ′[(u,v)]))

Theorem 63 If f : x → y is in correspondence to a ZFU-function f ′, then F F ′ Domf = Domf ′ = x and Imf = Imf ′ = y. Besides, f 6= f .

31 Proof See the proof of Theorem 62.

Next deﬁnition provides all necessary and suﬃcient conditions to introduce the universe of our main model.

Definition 41 Let F (t) be the formula ∃v(νa[v] ∧ v[t] ∧ ∀r(TC(t) [r] ⇒ a[m] a a ∃s(r ≡ s ∧ r 6= s ∧ TC(t) a[m][s]))). Then, ν† = ν F (t) is the Azriel Lévy a universe or Lévy universe, for short. If x is a Menge in ν† , we denote this by † Ma(x).

The Lévy universe is a restricted appropriate atomic von Neumann universe a a a in the sense that ν† ⊂ ν . Last definition grants there is no Menge f in ν† which acts on a single u if there is any v in νa where v ≡ u and v 6= u. For a f example, Flow allows us to define a Menge f in ν such that αr 7−→ αr, where a αr is an appropriate atom. Nevertheless, ν† does not act on any t such that t a acts on that f. If g is a Menge in ν† who acts on any αr, then g acts on αr as well, since αr is indiscernible from αr (although αr 6= αr).

† † Theorem 64 ∀x∀y((Ma(x) ∧ Ma(y) ∧ x ≡ y) ⇒ x = y).

† a Proof For any x, Z(x) ⇒ Ma(x) (from Theorem 55 and the fact that ν† ⊂ νa). But, from item i of Definition 33, for any ZF-set x and any y, x ≡ y ⇔ x = y. Now, let x be a Menge who is not a ZF-set, recalling a satisfies all conditions imposed in Definition 33. First we consider a the case where x acts only on atoms. Since no Menge x in ν† acts on any atom r without acting on s when r ≡ s, then item iii of Defini- tion 33 grants x ≡ y ⇔ x = y. Finally, if x and y are any Mengen in νa, then TC(x ) ≡ TC(y ) † rank(t)=r∨rank(t)=s a[m] rank(t)=r∨rank(t)=s a[m] iff TC(x ) = TC( y ) , where all rank(t)=r∨rank(t)=s a[m] rank(t)=r∨rank(t)=s a[m] occurrences of m in formula a[m] are free. So, item v from Definition 33 grants x ≡ y ⇔ x = y.

† Theorem 65 ∀f(Ma(f) ⇔ (σf 6= 0 ∧ f ⊆ 1 ∧ ∀x(f[x] ⇒ (∀t(t ≡ x ⇒ f[t]) ∧ † (a[x] ∨ Ma(x)))))).

Proof Analogous to the proof of Theorem 43 and a special case of Theorem 59.

Next theorem is a key result.

† † Theorem 66 Any f : x → y, such that Ma(x) and Ma(y), corresponds to a ′ † ′ ZFU-function f where Ma(f ) iﬀ neither x nor y acts on any atom.

Proof From Theorem 62 and Deﬁnition 39 we know any f : x → y, such † † F that Ma(x) and Ma(y), corresponds to a ZFU-function where Domf = F ′ Domf ′ = x and Imf = Imf ′ = y (Theorem 63) and Ma(f ). The question

32 † ′ is whether Ma(f ) or not. For the ⇐ part, suppose x acts on an atom u from a. That entails f(u)= v for some v 6= 0. But f is in correspondence to a ZFU-function f ′ iﬀ ∀u∀v(f[u] ∧ f(u)= v) ⇔ (f ′[(u, v)] ∧ f ′((u, v)) = ′ a (u, v)), where x, y, and f are Mengen in ν† . Thus, any ordered pair (u, v) a ′ is supposed to be a Menge in ν† , where f acts. Nevertheless, according to (u,v) (u,v) Deﬁnition 22, (u, v) is a function such that α 7−→ α and β 7−→ β, where α β β a u 7−→ u, and u 7−→ u and v 7−→ v. Although α is a Menge in ν , it is no a a Menge in ν† . After all, any Menge from ν† who acts on any atom u is supposed to act on u′ as well, where u ≡ u′. Since any atom u in a admits ′ ′ ′ † † ′ an u such that u 6= u and u ≡ u , then ¬Ma(α). Therefore, ¬Ma(f ). An analogous rationale holds for the case where y acts on any atom. For the ⇒ part, suppose f : x → y corresponds to a ZFU-function f ′ such that † ′ ′ Ma(f ). That entails f acts on ordered pairs (u, v) where x[u] and y[v]. If either u or v is an atom, we get the same contradiction we got in the ⇐ part.

a Theorem 67 The Partition Principle holds in ν† .

′ † ′ Proof Let f be a ZFU-function such that Ma(f ) and whose domain and range are, respectively, x and y (Deﬁnition 40). Then, there is a unique f : x → y in correspondence to f ′ (Theorems 62 and 63). Axiom F11 grants an injection c : y → x′ such that x′ ⊆ x. But since f : x → y is ′ † ′ in correspondence to f where Ma(f ), that entails neither x nor y act on any atom (Theorem 66). Therefore, c : y → x′ is in correspondence to a ′ † ′ ′ ZFU-function c where Ma(c ) (recall x ⊆ x). Consequently, for any ZFU- ′ a ′ function f in ν† such that f is correspondence to f , Partition Principle ′ a ′ a holds for f in ν† , in the sense that c is a Menge in ν† . Observe as well that c′ is injective in the usual sense concerning functions as collections of ordered pairs (labeled here as ZFU-functions).

a Theorem 68 Axiom of Choice fails in ν† .

Proof Let f : a → r be a function where r is an ordinal (Deﬁnition 15). † † Observe we have Ma(a) and Ma(r). According to Theorem 66, there is ′ † ′ ′ no ZFU-function f such that Ma(f ) and f is in correspondence to f . Therefore, even if there is an injection f from a to r, there is no ZFU- ′ a function f in ν† which corresponds to such an injection. Therefore, a a ′ cannot be well ordered in ν† by means of any ZFU-function f . The trick in the last two theorems is the fact that PP is a conditional: if there is a function from x to y, then there is an injection from y to x′ ⊆ x. Nevertheless, certain ZFU-functions (those whose domains or ranges act on a atoms) do not exist in ν† (Theorem 66). Obviously, theorems 67 and 68 do not grant we have succeeded into the introduction of a model of ZFU where PP holds but AC fails. After all, we still need to prove the remaining axioms of ZFU are, in some sense, true in our L´evy

33 universe. That is why we need the next Section. But before we go any further, we need to state our model, based on our previous discussions. † † Deﬁnition 42 x ∈a y iﬀ y[x] ∧ Ma(y) ∧ (∃t(t ≡ x ∧ (a[t] ∨ Ma(t))) ⇒ y[t]). If ¬(x ∈a y), we abbreviate that as x 6∈a y.

For example, formula αr ∈a y is equivalent to αr ∈a y despite the fact that αr 6= αr for any ﬁnite ordinal r. Observe Theorem 64 grants we do not need to worry about any indiscernible of x when x ∈a y and x is a Menge. In that case † x ∈a y is equivalent to y[x] ∧ Ma(y). Thus, † Theorem 69 Let z be a function such that Ma(z), and x and y be both atoms a (or both Mengen) of ν† . If x ≡ y, then x ∈a z ⇔ y ∈a z. Proof Straightforward from Deﬁnition 42.

a x x If, e.g., x is a Menge of ν† , given by αr 7−→ αr and αr 7−→ αr, then the only a restrictions of x who are Menge of ν† are x itself and ϕ0. In other words, atoms in a L´evy universe always ‘come in pairs’. That is feasible thanks to the fact that the Axiom of Extensionality of ZFU cannot decide who is who between two indiscernible atoms. So, if we need any atom to form a Menge, we always take its ‘twin’ with it. That is a diﬀerent approach if we we compare it with permutation models [15]. Finally,

Deﬁnition 43 Let u = 1 M† , F (t) be the formula ∃a∃b(a ∈a b) ⇔ t = (a,b), a (t) and ǫ = 1 . Then, M = (u, a, ϕ ,ǫ) is the L´evy model. F (t) 0

Concerning last deﬁnition, it is worth to remark that ǫ acts on some terms where u does not, namely, ordered pairs (a,b) where a is an atom and b is a Menge, among others (see Theorem 66). In the next Section (speciﬁcally in Subsection 3.2), it will be clear the meaning of ∃x(P ), where P is a formula a from ZFU’s language. It means either x is an atom (a[x]) or x is a Menge in ν† † (Ma(x), which is equivalent to u[x]).

3 ZFU is immersed in Flow

We prove here vast portions of standard mathematics can be replicated within Flow.

3.1 ZFU

2 ZFU is a first-order theory with identity and one binary predicate letter A1, 2 such that the formula A1(x, y) is abbreviated as x ∈ y, if x and y are terms, and is read as ‘x belongs to y’ or ‘x is an element of y’. The negation ¬(x ∈ y) is abbreviated as x 6∈ y. Besides, there are two constants, namely, ∅ and A, as primitive concepts. From this we are able to define a predicate S as it follows: S(a) iff a 6∈ A. If a ∈ A we say a is an atom. Otherwise, we say a is a set. The postulates of ZFU are the following:

34 ZFU1 - Atoms ∀z(z ∈ A ⇔ (z 6= ∅ ∧ ∄x(x ∈ z))).

ZFU2 - Extensionality ∀Sx∀S y(∀z(z ∈ x ⇔ z ∈ y) ⇒ x = y). ZFU3 - Empty set ∀y(¬(y ∈∅)).

ZFU4 - Pair ∀x∀y∃Sz∀t(t ∈ z ⇔ t = x ∨ t = y).

x ⊆ y =def S(x) ∧ S(y) ∧ ∀z(z ∈ x ⇒ z ∈ y).

ZFU5 - Power set ∀S x∃Sy∀Sz(z ∈ y ⇔ z ⊆ x).

If F (x) is a formula where there is no free occurrences of y, then:

ZFU6F - Separation ∀Sz∃Sy∀x(x ∈ y ⇔ x ∈ z ∧ F (x)).

The set y is denoted by {x ∈ z | F (x)}. If α(x, y) is a formula where all occurrences of x and y are free, then:

ZFU7α - Replacement ∀x∃!yα(x, y) ⇒ ∀S z∃Sw∀t(t ∈ w ⇔ ∃s(s ∈ z ∧ α(s,t))).

ZFU8 - Union set ∀Sx∃Sy∀z(z ∈ y ⇔ ∃t(z ∈ t ∧ t ∈ x)).

The set y from ZF7 is abbreviated as y = t∈x t. Intersection is deﬁned from union in the usual way. We adopt the conventionS that x∩y T= ∅ if either x or y is anS atom.

ZFU9 - Infinity ∃Sx(∅ ∈ x ∧ ∀S y(y ∈ x ⇒ y ∪{y} ∈ x)).

ZFU10 - Choice ∀Sx(∀S y∀Sz((y ∈ x ∧ z ∈ x ∧ y 6= z) ⇒ (y 6= ∅ ∧ y ∩ z = ∅)) ⇒ ∃Sy∀Sz(z ∈ x ⇒ ∃w(y ∩ z = {w}))).

ZFU11 - Regularity ∀Sx(x 6= ∅ ⇒ ∃y(y ∈ x ∧ x ∩ y = ∅)).

3.2 L´evy model Deﬁnition 44 Consider the next translation table,

Translating ZFU into M ZFU M

† ∀x ∀Aa ∨Ma x † ∃x ∃Aa ∨Ma x † S(x) Ma(x) A a ∅ ϕ0 x ∈ y x ∈a y x ⊆ y ∀t(t ∈a x ⇒ t ∈a y) x = y x = y

35 where M is the L´evy model and Aa is a monadic predicate where Aa(x) says x is an atom (A(x)) and a[x]. If Ξ is a formula from ZFU, then its translation by means of the table above is denoted by Translated(Ξ). Besides,

M Ξ iﬀ ⊢F Translated(Ξ), where F is Flow theory. We read M Ξ as ‘Ξ is true in M’.

Proposition 1 All axioms of ZFU are true in the L´evy model, with the only exception of the Axiom of Choice.

The proof of Proposition 1 is made through the following lemmas. For the

† † sake of readability, translated quantiﬁers ∀Aa ∨Ma x and ∃Aa ∨Ma x will be simply written as ∀x and ∃x, respectively. There is no risk of confusion since the proofs of all lemmas refer to translated formulas. Observe as well that ∀Sx and ∃S x

† † from ZFU language are translated as ∀Ma x and ∃Ma x, respectively. Lemma 1 M ZFU1.

Proof Translated(ZFU1) is ∀z(z ∈a a ⇔ (z 6= ϕ0 ∧ ∄x(x ∈a z))). For the ⇒ part, observe z ∈a a is equivalent to say z is an atom (Definitions 27 and 34). Since no atom is a restriction of 1, then any atom z is different of ϕ0. Besides, no term belongs to (we are talking about ∈a) any atom z, since no atom is a Menge. For the ⇐ part, observe z is either an atom or a Menge. We already know there is no x such that x ∈a ϕ0, besides † the fact that Ma(ϕ0). Suppose z 6= ϕ0 is an empty Menge in the Lévy universe, i.e., ∄x(x ∈a z). That entails z does not act on any Menge nor on any atom of the Lévy universe. If z acts on any other term, then z is a a no Menge in ν† . Thus, ϕ0 is the only empty Menge in ν† . Therefore, z is an atom.

Lemma 2 M ZFU2.

† † Proof Translated(ZFU2) is ∀Ma x∀Ma y(∀z(z ∈a x ⇔ z ∈a y) ⇒ x = y). If x and y are Mengen and z ∈a x and z ∈a y, then x(z) = z and y(z) = z. Besides, if z′ ≡ z, then x(z′) = z′ and y(z′) = z′ (Theorem 69). If either z 6∈a x or z 6∈a x, then either x(z) = 0 or y(z) = 0 or z = x or z = y (where we may replace all occurrences of z by z′ if z ≡ z′). So, Translated(ZFU2) considers the case where both x and y share all their images x(z) and y(z) for any z, except perhaps for z = x or z = y. That means x ∼ y (Deﬁnition 12). But Theorem 25 says no emergent function has any clone which is emergent. Thus, Translated(ZFU2) is a theorem of Flow thanks to Theorem 1.

Lemma 3 M ZFU3.

† Proof Translated(ZFU3) is ∀y(¬(y ∈a ϕ0)). But Ma(ϕ0), which means ϕ0 is a Menge in the L´evy universe. Since ϕ0 does not act on any term, then there is no y such that y ∈a ϕ0.

36 Lemma 4 M ZFU4.

† Proof Translated(ZFU4) is ∀x∀y∃Ma z∀t(t ∈a z ⇔ t = x ∨ t = y). Let α(t,r) h = ϕ4 , where α(t, r) is the next formula: (t = ϕ0 ⇒ r = x) ∧ ((t = ′ ′ ′ ′ ′ ′ ′ ϕ1 ∧ ∃x (x 6= x ∧ x ≡ x)) ⇒ r = x ) ∧ ((t = ϕ1 ∧ ∄x (x 6= x ∧ x ≡ x)) ⇒ ′ ′ ′ ′ r = 0)∧(t = ϕ2 ⇒ r = y)∧((t = ϕ3∧∃y (y 6= y∧y ≡ y)) ⇒ r = y )∧((t = ′ ′ ′ ϕ3 ∧ ∄y (y 6= y ∧ y ≡ y)) ⇒ r = 0) ∧ ((t 6= ϕ0 ∧ t 6= ϕ1 ∧ t 6= ϕ2 ∧ t 6= ϕ3) ⇒ r = 0). Observe the quantifier ∃ in formula α(t, r) is supposed to be read as in the table from Definition 3.2. According to F10α, σh 6= 0 and σ(ImF) 6= 0. Now, let z = ImF = 1 . h h t=h(ϕ0)∨t=h(ϕ1)∨t=h(ϕ2)∨t=h(ϕ3) † From Definition 41 and Theorem 65, Ma(z). Thus, t ∈a z ⇔ t = x∨t = y.

If the reader wants a more intuitive view about this last lemma, observe the only terms x and y in a Levy universe who are indiscernible and diﬀerent from each other are atoms, according to Theorem 64. Thus, any pair z of Mengen has either one or two elements with respect to ∈a, as it is expected. Now, let x be an atom. Hence, the pair z such that t ∈a z ⇔ t = x is a Menge who acts on x and its conjugate x′. We say x′ is a conjugate of x iﬀ x ≡ x′ and x 6= x′

(see Theorem 69). As a ﬁnal example, if x = αϕ0 and y = αϕ1 , then the pair z acts on four terms, namely, αϕ0 , αϕ0 , αϕ1 , and αϕ1 .

Lemma 5 M ZFU5.

Proof Translated(ZFU5) is ∀M† x∃M† y∀M† t(t ∈a y ⇔ t ⊆ x). Let y = 1 a a a t⊆x (see the translation of ⊆ in Definition 44). According to F8, every term where y acts is emergent. Besides, y itself is emergent (which entails † † σy 6= 0). More than that, since Ma(x), then Ma(y) (Definitions 41 and 28 grant every restriction of a Menge is a Menge). Recall Definition 28 makes use exactly of the notion of restricted power ℘. Finally, observe † t ∈a y ⇔ t ⊆ x, in the sense that y[t] ∧ Ma(y) is equivalent to t ∈a y according to Definition 42 and its subsequent paragraph.

Lemma 6 For any formula F with the syntactic conditions in ZFU6F we have M ZFU6F .

† † Proof Translated(ZFU6F ) is ∀Ma z∃Ma y∀t(t ∈a y ⇔ t ∈a z ∧ F (t)), if F (t) is a formula with no free occurrences of y. Let y = z ∃r∃s((r≡s∧F (s))⇔t=r) (remember every term is indiscernible from itself and, eventually, it can be † † indiscernible from something else). Since Ma(z) and y ⊆ z, then Ma(y) (recall that if y acts on any t, then it acts on any t′ such that t ≡ t′). Observe t ∈a y is equivalent to t ∈a z ∧ F (t), since Theorem 69 grants t ∈a y is equivalent to s ∈a y when t ≡ s. Recalling Observation 3, the calculation of y = z does not take into account any ∃r∃s((r≡s∧F (s))⇔t=r) t = y. So, any occurrence of y in F turns out to be irrelevant.

37 From an intuitive point of view, the only atypical situation in t ∈a z ∧ F (t) takes place when t is an atom. In that case, formula t ∈a z ∧F (t) can be read as ‘t and its conjugate belong to z and, at least for one of them, formula F holds’.

Lemma 7 For any formula α with the syntactic conditions in ZFU7α we have M ZFU7α.

† † Proof Translated(ZFU7α) is ∀x∃!yα(x, y) ⇒ ∀Ma z∃Ma w∀t(t ∈a w ⇔ ∃s(s ∈a z∧α(s,t))), where all occurrences of x and y in α(x, y) are free. Due to the use of quantifier ∃!, we are supposed to rewrite ZFU7 by means of ∀ and ∃ (as defined at the end of the second paragraph of Section 2), and only α(s,t) then translate it. Let g = z be defined from F10α by formula α(s,t) F provided by Translated(ZFU7 α). From F10α, σg 6= 0 and σ(Img ) 6= 0. That makes g emergent. Now, let w′ = ImF = 1 . But w′ is g ∃s(z[s]∧t=g(s)) a restriction of 1, with σw′ 6= 0, who acts only on either atoms or Mengen. Therefore, from Theorem 59, w′ is a Menge in the appropriate atomic von Neumann universe νa. Now, let w = 1 . Observe ∃s(w′[s]⇔(t≡s∧(a[t]∨Ma(t)))) ′ ′ w is a restriction of w. Since Ma(w ), so is w, where, by the way, both † share the same rank (Definitions 28 and 31). But, more than that, Ma(w) (Definition 41), since for any s where w acts, the latter acts on any indis- a cernible of s as well. Thus, for any t (either an atom or a Menge in ν† ), t ∈a w ⇔ ∃s(s ∈a z ∧ α(s,t)).

Nothing prevents formula α(s,t) from last lemma be deﬁned in a way such that t is an atom (for a given s) but there is no s′ where α(s′,t′) and t′ ≡ t and ′ a a t 6= t. That is the reason why we had to use ν before we get to ν† . Lemma 8 M ZFU8.

† † Proof Translated(ZFU8) is ∀Ma x∃Ma y∀t(t ∈a y ⇔ ∃z(t ∈a z ∧ z ∈a x)). Let y = z =def 1 . Axiom F8 entails σy 6= 0. Theorem Sz∈a x ∃z(z∈a x∧t∈a z) † 65 entails Ma(y).

Lemma 9 M ZFU9.

† † Proof Translated(ZFU9) is ∃Ma x(ϕ0 ∈a x ∧ ∀Ma y(y ∈a x ⇒ σy ∈a x)). Let x = ω, from Deﬁnition 17. Recall any ZF-set is a Menge in the L´evy universe (Theorem 55).

Lemma 10 M ZFU11.

† Proof Translated(ZFU11) is ∀Ma x(x 6= ϕ0 ⇒ ∃y(y ∈a x ∧ x ∩ y = ϕ0)). Any Menge x in the L´evy universe has a rank (Deﬁnition 31). Let y ∈a x be any term such that there is no z where rank(y)[rank(z)] and z ∈a x (see Theorem 42, which can be extended to any atomic von Neumann universe), i.e., y has the least rank among the elements of x (relative to ∈a). Then, there is not t such that t ∈a x and t ∈a y.

38 This concludes the proof of Proposition 1. Hence, Proposition 1 and Theo- rems 67 and 68 guarantee M is a model of ZFU where the Partition Principle holds but the Axiom of Choice fails. After all, M (Deﬁnition 43) is deﬁned from a ν† .

3.3 Other models

Definition 45 Let u = 1 , F (t) be the formula ∃a∃b(a ∈Z b) ⇔ t = (a,b), Z(t) and ǫ = 1 . Then, N = (u,ǫ) is the von Neumann model, where Z is the F (t) predicate “to be a ZF-set” from Definition 20 and ∈Z is given in Definition 24. Definition 46 Consider the next translation table. Translating ZF into N ZF N ∀ ∀Z ∃ ∃Z x ∈ y x ∈Z y x ⊆ y x ⊆ y If Ξ is a formula from ZF, then its translation by means of the table above is denoted by Translated(Ξ). Besides,

N Ξ iﬀ ⊢F Translated(Ξ), where F is Flow theory. We read N Ξ as ‘Ξ is true in N’.

Proposition 2 All axioms of ZF are true in the von Neumann model. Proof Analogous to the proof of Proposition 1, but much simpler.

a a If the reader is still suspicious about last proof, observe ν ⊂ ν† ⊂ ν . To obtain a model of ZFC, all we have to do is to replace F11 by Translated(AC) according to Definition 46. Obviously, both AC and PP are true in such a model. † If we replace ∈a by ∈ (Definition 29) and all occurrences of Ma by Ma in Definition 44 (keeping F11), we have a model of ZFU where we are unable to answer if AC is independent from PP.

4 On the consistency of Flow

It is far from us to answer if Flow is consistent, due to G¨odel’s Second Incom- pleteness Theorem [9]. But we can talk about the consistency of F relative to ZF, ZFC, and ZFU. In last Section we proved that, if Flow is consistent, then ZF, ZFC, and ZFU are consistent. We did that by means of a metatheory which provided a translation from the language of ZF (and some of its variations) into Flow’s language. Thus, there seems to be two possibilities, if we want to assess the consistency of Flow relative to ZF:

39 1. We introduce a translation from the language of Flow into the language of ZF and, then, prove every translated axiom of Flow is a theorem in ZF. 2. We use some other metatheory (quite diﬀerent from a simple translation) to decide if Flow has a greater (or equal) consistency strength than ZF.

The first possibility seems to bump into many obstacles. How to interpret 0 and 1 into the language of ZF? If 0 does not translate as the empty set, what is it? If 0 is to be interpreted as the empty set, who is ϕ0? Recall 0 and ϕ0 are the only clones in the present formulation of Flow. Hence, to avoid inconsistency with extensionality in ZF, we cannot interpret both 0 and ϕ0 as the empty set. On the other hand, if 1 acts on every single term (except itself and 0), how to interpret it within ZF? It is worth to remark that 1 is not a proper class, in the sense of a theory like, e.g., NBG [22]. After all, 1 acts on many universes (like a a ν, ν for any brick of atoms a, ν† , and so on) where those universes behave like proper classes. Thus, if we associate actions to membership relations in the sense of Definition 42, that would entail many classes belonging to 1. Besides, our main results show the usual correspondence between sets and identity functions is not that trivial. What Theorem 66 (and its consequences in Theorems 67 and 68) reveals is that a set x is not necessarily associated to a function f : x → x (in the ordinary set-theoretic sense, where a function is a set of ordered pairs) such that ∀t(f(t) = t). That non-correspondence takes place in our Lévy universe when x has atoms as elements. Concerning the second possibility, could we try and use forcing [15] over Flow? Obviously, no, at least for now. Forcing requires the previous existence of at least one model in order to conceive other models for the sake of investigation about relative consistency among formal theories. As remarked by Saharon Shelah [28], “forcing can be used only to make the universe ‘fatter’, not ‘taller’. In technical terms: if we use forcing starting from models of ZFC to prove that ‘ZFC neither proves nor refutes statement A’ (or equivalently, each of ‘ZFC + A’ and ‘ZFC + non-A’ is consistent), then the ‘consistency strengths’ of ‘ZFC’, ‘ZFC+A’, ‘ZFC + non-A’ are all equal”. The ‘universe’ mentioned here refers to a previously known model. The ‘taller’ independence results mentioned by Shelah refer to situations where the consistency strength of ‘ZFC + statement A’ is strictly higher than the consistency strength of ZFC alone. That ‘statement A’ could be, for example, ‘ZFC is consistent’. Such ‘statement A’ is far beyond the reach of forcing techniques. But our concern here is a little less demanding. The point here is the fact that we have no metalinguistic interpretation which could serve as a model of Flow. That is a task to be carried on in the future. Besides, we admit the possibility that Flow demands some quite new techniques for investigating its consistency strength relatively to other theories like ZF.

5 Final remarks

As a reference to Heraclitus’s ﬂux doctrine, we are inclined to refer to all terms of Flow as ﬂuents, rather than functions. In this sense Flow can be roughly

40 regarded as a theory of fluents. That is also an auspicious homage to the Method of Fluxions by Isaac Newton [25]. The famous ‘natural philosopher’ referred to functions as fluents, and their derivatives as fluxions. Whether New- ton was inspired or not by Heraclitus, that is historically uncertain ([30], page 38). Notwithstanding, we find such a coincidence quite inspiring and utterly opportune. From the mathematical point of view, our framework was strongly motivated by von Neumann’s set theory [35], as we briefly discussed in the Introduction. But what are the main differences between Flow and von Neumann’s ideas? We refer to von Neumann’s set theory as N. The first important difference lurks in the way how von Neumann seemed to understand functions: “a function can be regarded as a set of pairs, and a set as a function that can take two values... the two notions are completely equivalent”. That philosophical viewpoint seems to be committed to a set-theoretic framework. Here we follow a quite different path, since our main result suggests those notions are not necessarily equivalent: i) In N there are two privileged objects termed A and B which resemble our terms 0 and 1. Nevertheless, in N there is no further information about A and B. Besides, those constants play a different role if we compare to 0 and 1 in F, as we can see in the next item. ii) In N there are two sorts of objects, namely, arguments and functions. Eventually some of those objects are both of them. And when an object f is an argument and a function, which takes only values A and B, then f is a set. Our terms 0 and 1 have no similar role. Besides, we do not need to distinguish functions from any other kind of term. All objects of F are functions. iii) Von Neumann believed a distinction between arguments and functions was necessary to avoid the well known antinomies from naive set theory. Nevertheless, we proved a simple axiom of Self-Reference (F2) is enough to avoid such a problem. iv) The distinction between sets and other collections which are ‘too big’ to be sets depends on considerations if a specific term in N is both an argument and a function. Within F that distinction depends on considerations regarding F-successor. v) Axiom I4 of N ([35], page 399) says any function can be identified by its images. Our weak extensionality F1 allows us to derive a similar result as a non-trivial theorem (Theorem 1). vi) One primitive concept in N is a binary functional letter (using modern terminology) which allows to define ordered pairs. In F that assumption is unnecessary. vii) In N there are many constant functions ([35], page 399, axiom II2). In F there are only two constant functions, namely, 0 and ϕ0. viii) In N the well-ordering theorem and the axiom of choice are consequences of its postulates. In F such a phenomenon does not take place. Actually, it makes no sense at all to state something like AC within Flow. Some open problems refer to how precisely Flow is related to Lambda Cal- culus [3], Category Theory [11] [19], String Diagrams [7], and Autocategories [10]. Our technique for building a model of ZFU is supposed to be compared with permutation models as well [15]. But, for now, the most important open problem is whether AC can be proven to be independent from PP within ZF.

41 6 Acknowledgements

We thank Aline Zanardini, Bruno Victor, and Cléber Barreto for insightful discussions which inspired this work. We acknowledge with thanks as well Asaf Karagila, Hanul Jeon, Samuel Gomes da Silva, Newton da Costa, Jean-Pierre Marquis, Edélcio Gon¸calves de Souza, Décio Krause, Jonas Arenhart, Kherian Gracher, Bryan Leal Andrade, and Bárbara Guerreira for valuable remarks and criticisms concerning earlier and problematic versions of this paper. AK was the most patient, critic, and thoughtful reader of a very poor earlier version of this work. We were also benefitted by comments and criticisms made by several participants of seminars delivered at Federal University of Paraná and Fed- eral University of Santa Catarina. Finally we acknowledge Pedro D. Damázio’s provocation, more than 20 years ago, which worked as the first motivation towards the development of Flow.

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