Follow the Flow: sets, relations, and categories as special cases of functions with no domain

We introduce, develop, and apply a new approach for dealing with the intuitive notion of function, called Flow Theory. Within our framework all functions are monadic and none of them has any domain. Sets, proper classes, categories, functors, and even relations are special cases of functions. In this sense, functions in Flow are not equivalent to functions in ZFC. Nevertheless, we prove both ZFC and Category Theory are naturally immersed within Flow. Besides, our framework provides major advantages as a language for axiomatization of standard mathematical and physical theories. Russell's paradox is avoided without any equivalent to the Separation Scheme. Hierarchies of sets are obtained without any equivalent to the Power Set Axiom. And a clear principle of duality emerges from Flow, in a way which was not anticipated neither by Category Theory nor by standard set theories. Besides, there seems to be within Flow an identification not only with the common practice of doing mathematics (which is usually quite different from the ways proposed by logicians), but even with the common practice of teaching this formal science.

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1 Introduction

Throughout the ages mathematicians have considered their objects, such as numbers, points, etc., as substantial things in themselves. Since these entities had always defied attempts at an adequate description, it slowly dawned on the mathematicians of the nineteenth century that the question of the meaning of these objects as substantial things does not make sense within mathematics, if at all. The only relevant assertions concerning them do not refer to substantial reality; they state only the interrelations between mathematically “undefined objects” and the rules governing operations with them. What points, lines, numbers “actually” are cannot and need not be discussed in mathematical science. What matters and what corresponds to “verifiable” fact is structure and relationship, that two points determine a line, that numbers combine according to certain rules to form other numbers, etc. A clear insight into the necessity of a dissubstantiation of elementary mathematical concepts has been one of the most important and fruitful results of the modern postulational development.
Richard Courant, What is Mathematics, 1941.

All usual mathematical approaches for well-known physical theories can be easily associated to either differential equations or systems of differential equations. Newton’s second law, Schrödinger’s equation, Maxwell’s equations, and Einstein field equations are all differential equations which ground classical mechanics, quantum mechanics, classical electromagnetism, and general relativity, respectively. Other similar examples may be found in thermodynamics, gauge theories, the Dirac electron, etc. Solutions for those differential equations (when they exist) are either functions or classes of functions. So, the concept of function plays a major role in theoretical physics. Actually, functions are more relevant than sets, in a very precise sense [4] [5].

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. And category theory emphasizes such a role in a very clear, elegant, and comprehensive way.

Functions allow us to talk about the dynamics of the world, in the case of physical theories. Regarding mathematics, functions allow us to talk about invariant properties, whether those properties refer to either algebraic operations or order relations.

From a historical point of view, some authors have advocated the idea that functions are supposed to play a strategic role into the foundations of mathematics [17] and even mathematics teaching [9], rather than sets. Notwithstanding, the irony of such discussions lies in a closer look at Georg Cantor’s seminal works about the concept of set. Cantor - the celebrated father of set theory - was strongly motivated by Bernard Bolzano’s work on infinite multitudes called Menge [25]. 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. According to the latter, in order to compare infinite quantities we should consider a one-to-one correspondence between collections. That means 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. And the result of that effort is a strange phenomenon which we describe in the next paragraphs, based on [21].

Let be an axiomatic system whose primitive concepts are , , …, . One of these concepts, say , is independent (undefinable) from the remaining if and only if there are two models of in which , …, , , …, have the same interpretation, but the interpretations of in such models are different. (Of course, a model of is a set-theoretic structure in which all axioms of are true, according to the interpretation of its primitive terms [15].)

As an example, consider a very simple axiomatic system, namely, a Minimalist Space , whose axioms are:

MS1

is a non-empty set.

MS2

is a function whose domain and codomain are both .

By using Padoa’s method [1] [18] [24] we can easily prove that is undefinable, since we can exhibit two models of a minimalist system such that has the same interpretation in both models but has two different interpretations within these models. Consider, for this: the Model A, where is interpreted as the set of real numbers and is the identity function defined on ; and the Model B, where is interpreted again as the set , but is the function given by , with the same domain . This means that the interpretation of does not fix the interpretation of . In other words, cannot be defined (or fixed) from . On the other hand, is definable, since any two models with two different interpretations for would unavoidably entail different interpretations for . The reason for this is grounded on the fact that the domain and the codomain of a function are ingredients for the definition of the function itself, at least within the scope of a standard set theory like Zermelo-Fraenkel’s. Different domains imply different functions.

So, at least two questions remain:

  1. How to define ?

  2. What does it mean to say that is eliminable?

The answers are:

  1. ( is the domain and the codomain of ).

  2. We do not need to explicitly mention . We could rephrase the definition of a minimalist system by saying that a minimalist system is just a function whose domain is equal to its codomain.

In a similar way, it is possible to prove that in usual axiomatic frameworks for physical theories, time and spacetime are concepts that are definable, and so, eliminable. That happens because time and spacetime are usually considered as domains of functions that describe forces, fields, currents, and so on. For example, according to Padoa’s principle, the primitive concept time (described as an interval of real numbers) in a physical theory is independent from the remaining primitive concepts (mass, position, force, speed, magnetic field etc.) if, and only if, there are two models of the physical theory such that time has two interpretations and the remaining primitive symbols have the same interpretation. But usually these two interpretations are not possible, since mass, position, force, speed, magnetic field and other physical concepts are in general described as functions whose domains are time. If we change the interpretation of time, we change the interpretation of the other primitive concepts. So, time is not independent and hence can be defined. Since time is definable, it is eliminable. Time is eliminable in the sense that many physical theories can be rewritten without any explicit mention of time. A similar argument can be used to dispense with spacetime. (Details about this approach can be found in [4, 5].)

Results of this kind suggest the idea that functions are indispensable, but the explicit presence of sets as the domains of these functions is questionable. After all, it’s not clear that the notion of a set is playing any strategic explanatory role in the context of physical theories, since these sets are definable by means of functions. Sets seem to be carried along as just a surplus structure of the mathematical framework in which these theories are formulated. Moreover, in the context of standard set theories, such as Zermelo-Fraenkel’s, to reformulate a physical theory without any explicit mention of either time or spacetime is not an easy task—after all, the latter notions are typically expressed in terms of sets. Such reformulations of physical theories in Zermelo-Fraenkel are also unnatural, given that usually the functions that are invoked in the theories demand an explicit mention of their domains, and in this way, sets are brought back. (For an example of a mathematical description of thermodynamics without any explicit mention of time, see [5].)

Sets can be viewed as the result of a process of collecting objects. An object is collected if it is assigned to a given set. But the fundamental mechanism here is to attribute something to a certain collection. And that notion of attributing something to a given collection resembles a function. From another point of view, we should recall that sets and functions are meant to correspond to an intuitive notion of properties. Usually properties allow to define either classes or sets (like the Separation Schema in ZFC). But another possibility is that properties correspond to functions. Talking about objects that have a given property corresponds to associate certain objects to a label which represents ; and any other remaining objects are supposed to be associated to a different label. The correspondence itself between and a given label does have a functional, rather than a set-theoretical, appeal. And usually, those labels are called sets. So, why do we need sets? Why can’t we deal only with functions? In other words, why can’t we label those intended properties with functions instead of sets?

What would happen if we could avoid any explicit mention of domains of functions? Could we obtain better axiomatic formulations of physical theories? Could we avoid the presence of time and spacetime structures in a natural way? Could we go more directly to the point, i.e., to the functions that usually describe fields and forces, tensors and metrics, speeds and accelerations?

It could be thought that category theory provides a framework to develop this sort of approach. After all, category theory deals primarily with “functions”, called morphisms (see [12]). However, even morphisms have domains, which are other morphisms, and so we still wouldn’t have the appropriate framework to develop the approach we have in mind. So, even Category Theory is somehow committed to set-theoretic view about what a function is supposed to be.

What we are looking for is a mathematical theory where functions have no domains at all. In this way, we would immediately avoid the introduction of superfluous primitive notions, such as sets or domains, when we use this theory as the mathematical basis for the formulation of physical theories. Sets work as the stage where functions, the actor, play. So, we advocate a way of doing mathematics where the stage itself is unimportant. The relevant agents of mathematics are functions, and functions alone.

In 1925, John von Neumann introduced his axiomatization of set theory [17]

. There are two major assumptions in his approach, namely, the use of two kinds of collections, sets and classes, and the use of functions as the intuitive basic notion, instead of sets or classes. More specifically, von Neumann deals with three kinds of terms: I-objects (arguments), II-objects (characteristic functions of classes), and I-II-objects (characteristic functions of sets). The axiomatic system originally proposed was further developed by R. M. Robinson, P. Bernays, and Kurt Gödel, and it came to be known as the von Neumann-Bernays-Gödel (NBG) set theory. However, NBG is not faithful to the idea of the priority of functions instead of collections. In the end, NBG is a standard approach to set theory, where the novelty is the use of classes (mainly proper classes: those classes which are not sets), besides sets.

Intuitively speaking, a function is supposed to be a term which allows us to uniquely associate certain terms to other terms. In standard set theories, for example, a function is a special case of set, namely, a specific set of ordered pairs of sets. That means standard set-theoretic functions do not actually

act on terms in the sense of transforming them into other terms. In contrast, in category theory morphisms have an intended interpretation which is somehow associated to functions. But even in that case we show morphisms can always be treated as restrictions of an identity function. Besides, in both cases functions are somehow attached to domains and codomains which are sets in set theories and identity morphisms in category theories. In this paper we develop a new approach - Flow Theory - for dealing with the intuitive notion of function. In a precise sense, in Flow Theory functions have no domain at all. Within our approach, a set is a special case of function. Russell’s paradox is avoided without any equivalent to the Separation Scheme. We provide a comprehensive discussion of Flow Theory as a new foundation for mathematics, where functions explicitly play a more fundamental role.

The name Flow is a reference to Heraclitean flux doctrine, according to which things are constantly changing. Accordingly, in Flow theory all terms are “active objects” under the action of other “active objects”.

So, this paper is strongly motivated by [17] and [21] and related papers as well ([4] [5]). In [21] it was provided a reformulation of von Neumann’s original ideas (termed theory) which allowed the authors to reformulate standard physical and mathematical theories with much less primitive concepts in a very natural way. Nevertheless, in theory there are two fundamental constants which are not clarified in any way. Those constants, namely, and , allow us to define sets as particular cases of functions, in a way which is somehow analogous to the usual sense of characteristic functions in standard set theories.

In this paper Flow theory is introduced as a generalized formulation of concepts derived from theory. Constants and are still necessary. Notwithstanding, we are able to define them from our proposed axioms and some related theorems. And that fact entails an algebra defined over functions. Such an algebra shows us that both category theory and ZFC set theory are naturally present within our framework.

Besides the presentation and discussion of Flow axioms, we introduce several applications and foundational issues by comparing Flow with ZFC set theories and Category Theory.

Our punch line may be summarized by something like this: (i) the concept of set (as a collection of objects) is somehow implicitly assumed through ZF axioms; (ii) nevertheless, sets play a secondary role in mathematics and applied mathematics, since the true actors are always functions, while sets work as just a stage (setting) for such actors; (iii) so, why cannot we explicitly assume the notion of function right at the start on the foundations of mathematical theories?

2 Flow theory

Flow is a first-order theory with identity, where the formula should be read as “ is equal to ”. The formula is abbreviated as . Flow has one functional letter , where and are terms. If , we abbreviate this by , and we say is the image of by . We call evaluation. All terms of Flow are called functions. We use lowercase Latin and Greek letters to denote functions. Uppercase letters are used to denote predicates (which are eventually defined). The axioms of Flow follow in the next subsections. But first we need to make a remark. Any explicit definition in Flow is an abbreviative one, in the sense that for a given formula , the definiendum is just a metalinguistic abbreviation for the definiens given by .

2.1 Functions

P1 - Weak Extensionality

.

This first axiom is tricky. Any function such that is said to be rigid with . And any function such that is said to be flexible with . So, if both and are rigid with each other, then we are talking about the very same function (). Another possibility to identify a function is by checking if and are both flexible with each other. If that is the case, then again .

P2 - Self-Reference

.

Our first theorem has a very intuitive meaning.

Theorem 1

.

Proof:

By using the substitutivity of identity in the formula (which is a theorem in any first-order theory with identity), proof of the part is quite straightforward. After all, if , then , for any . In particular, we have . Concerning the part, suppose for any we have . In particular, for , we have . And for , we have . Nevertheless, according to P2, and . So, and . And from P1, that entails .

Axiom P2 says every function is rigid and flexible with itself. That fact deserves a more detailed discussion. Our main purpose here is to avoid any Flow-theoretic version of Russell’s paradox. Consider, for example, the next statement: is a function such that

In the formula above we are explicitly trying to define a function . On the left side of we have the definiendum and on the right side we have the definiens. If we ignore P2, what about ? If , then we are considering where is . Hence, according to the formula above we entail . Analogously, if , we are considering where is again . And according to the formula above we have . Consequently, we have . That is Russell’s paradox! To avoid such an embarrassment (which could explode Flow theory, since we are grounding our axiomatic system within classical logic) all we need to do is to introduce axiom P2. According to P2, any function defined by the formula above guarantees that cannot be equal to . Since for any we have and the definiens above demands that , that entails . But the definiendum states . Hence, . Therefore, Theorem 1 guarantees , since and do not share all their images. Hence, there is no paradox! After all, the paradox was entailed from the possibility that . Axiom P2 prohibits the definition of a function like . Otherwise, a formula like the one proposed above would be creative, allowing us to derive contradictions. That is a much simpler solution to Russell’s paradox than any equivalent to the Separation Scheme in Zermelo-Fraenkel-like set theories. Besides, as we shall see below, Flow theory allows us to talk about sets and proper classes in the usual sense of standard set theories, like ZFC with classes, NBG and their variations.

It is worth to observe that axioms P1 and P2 could be rewritten as one single axiom as it follows:

P1’ - Alternative Weak Extensionality

.

If that was the case, then P2 would be a consequence from P1’. Ultimately, would entail that (from P1’). And substitutivity of identity entails . On the other hand, we prefer to keep axioms P1 and P2 (instead of P1’) for pedagogical purposes. From P1 and P2, we can analogously see that P1’ is a theorem.

One philosophical remark concerning axiom P2 refers to Richard Courant’s quote presented in the Introduction. Functions, by themselves, are irrelevant. What matters is what they do. That point is gradually clearer thanks to the next postulates.

P3 - Identity

.

This is the first axiom which guarantees the existence of a specific function. Any function which satisfies P3 is said to be an identity function.

Theorem 2

The identity function is unique.

Proof:

Suppose both and satisfy axiom P3. Then, for any we have and . Thus, and . Hence, according to P1, .

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

P4 - Rigidness

.

In other words, there is at least one function which is rigid with any function. Observe the symmetry between axioms P3 and P4! Any function which satisfies this last postulate is simply said to be rigid.

Theorem 3

The rigid function is unique.

Proof:

Suppose both and satisfy axiom P4. Then, for any we have and . Thus, for any we have and . Thus, according to P1, .

Now we are able to justify the extensionality axiom P1. Our purpose here is to define constants and , in order to accommodate our view about von Neumann’s ideas. So, is the identity (flexible) function and is the rigid function, since we proved they are both unique. In other words

If we recall that is an abbreviation for , we can read axioms P3 and P4 as statements regarding the existence of two “spurs”. Axiom P3 states there is a function such that for any we have , while P4 says there is such that for any we have .

Theorem 4

is the only function which is rigid with .

Proof:

The statement above is equivalent to say that . In other words, . But we already know that . Therefore, if we have , according to P1, we have .

Theorem 5

is the only function which is flexible with .

Proof:

The statement above is equivalent to say that . In other words, . But we already know that . Therefore, if we have , according to P1, we have .

The last two theorems do not say what are the images or (when , in the last case). Nevertheless, such values prove to be rather important for future applications of Flow Theory. But before discussing that, we introduce another axiom.

P5 - Composition

.

P5 allows us to define unique functions from other functions and . That means this last postulate allows us to define a “binary operation” over functions. To be clearer about that, we state the next definition, based on P5.

Definition 1

For any functions and we may define the composition of with . Function is the one stated in axiom P5.

Beware! We never calculate as , since is always according to P2.

The idea of composition is quite simple, although it is not a constructive process. Given functions and , we can build the composition through a three-step process as it follows:

  1. First we establish a label for .

  2. Next we calculate for any which is different of and . By doing that we are assuming those are different of . Thus, if such a choice of entails for a given , then .

  3. Next we evaluate the following possibilities: is equal to either , or something else? How can we answer to that question? If is different of , then is supposed to be . If entails a contradiction, then is simply . And an analogous method is used for assessing if is . Eventually, is neither nor , as we can see in the next theorems.

Remarkable examples of how to calculate compositions can be found in the proofs of Theorems 18 and 19.

Theorem 6

Composition is associative.

Proof:

Here is a sketch for the proof. Both situations and correspond, according to Definition 1, to the formula , as long we are talking about values which are different of , , , , and . That means , if . Now, all we have to do is to consider four situations which contemplate all possible relations among , , , , and , in order to evaluate the images and when is either one of those remaining terms: (i) ; (ii) ; (iii) ; (iv) . If we have situation (i), then . That means has all the features of composition . But, according to P5, any composition is unique. Thus, . Regarding situation (ii), which states , we should consider two possibilities: either or . If , that means (according to P5). That entails (an idempotent function); and associativity among idempotent functions is a trivial result, when and . On the other hand, if and , then we have , and once again has all the features of . Hence, from uniqueness of composition, . If , while , then from we have , which entails for the case , since in that case is idempotent. And the case was already discarded within the first possibility of situation (ii). And if , while , then , and once again has the same features of ; hence, . Going back to the second possibility of situation (ii), when while , the last identity implies , since no composition can ever be (the only function such that ). So, neither nor can be , since . That implies and . Hence, . The proof of situations (iii) and (iv) is analogous to that one for situation (ii).

This last theorem is somehow interesting, since evaluation is not associative and composition is defined from evaluation. Consider, for example, , for different of and different of , and such that (functions like this will be available afterwards). Thus, . If evaluation was associative, we would have . A contradiction! Of course this rationale works only if we prove the existence of other functions besides and . That happens thanks to the last axiom, as we discuss below. So, although evaluation is not associative, we are still able to define a binary functional letter from such that is associative. That happens because and are not necessarily the same thing. So, contrary to the usual slogan from category theory [11], evaluation is not a special case of composition.

Actually, it is good news that evaluation is not associative. According to P2, we have, for all , , since for any we have . If evaluation was associative, we would have for any , and thus, . So, composition would be an idempotent operation. That would be an undesirable result for anyone who intends to develop, e.g., category theory within Flow.

Theorem 7

There is a unique function such that but for any .

Proof:

According to Definition 1 and axiom P5, is a unique function such that for any , we have . But since , then P5 guarantees that . Thus, is for any , while itself is different of .

Such a function of last theorem is rather important for future applications. So, we label it with a special symbol, namely, . That means , where .

Theorem 8

For any we have

Proof:

According to Definition 1 and axiom P5, for any and . But P5 demands is different of . Hence, . Now, regarding , there are three possibilities: (i) ; (ii) ; (iii) is neither nor . The first case corresponds to Theorem 7, which entails . In the second case, if is different of , then . But that would entail for any , which corresponds exactly to function proven in Theorem 7, a contradiction. So, is indeed , when . Concerning the last case, since and , then (according to Theorem 1) there is such that for . Thus, according to P5, for such value of . But once again we have a function such that for any , which corresponds exactly to function proven in Theorem 7

Concerning , that value is supposed to be discussed later, due to Theorem 4.

Theorem 9

Proof:

This proof is similar to the previous one in the last theorem.

Theorem 10

There is a unique function such that and for any such that .

Proof:

According to Definition 1 and axiom P5, is a unique function such that for any , we have . But since , then P5 guarantees that . Thus, is for any , while itself is different of .

Such a function of last theorem is rather important for future applications. So, we label it with a special symbol, namely, .

The next theorem is rather important for a better understanding about the weak extensionality axiom P1 (which entails Theorem 1), although its proof does not demand the use of such a postulate.

Theorem 11

.

Proof:

Suppose , for the sake of abbreviation. That means for any different of and different of , we have . Suppose now is different of . According to P5, that would entail . But for any such that and as well (as long , of course). And according to P5 is supposed to be unique, which entails . The proof of is analogous to the proof of .

Observation 1

This last result is quite subtle. If it wasn’t for the uniqueness requirement of compositions (axiom P5), Flow Theory would be consistent with the existence of many functions, like and , which “do” the same thing. By multiple functions “doing the same thing” we mean different functions and which share the same images and for any different of both and . Since P2 demands and , that would allow and . And according to Theorem 1, that fact would guarantee . We use this ambiguity for functions “doing the same thing” in Flow to prove Theorems 7 and 10, since and , where and . Function “does the same things” “does”, and “does the same things” does. Notwithstanding, we stop using that opportunity when we are talking about functions which are neither nor . Observe, however, the uniqueness of both and is not imposed. Their uniqueness is granted by Theorems 2 and 3. That is why we refer to P1 as “weak extensionality”. A strong extensionality postulate would demand that functions which “do the same thing” are necessarily the same. But that assumption is inconvenient for us, since it does not allow us to guarantee the existence of other functions besides and without considering other primitive concepts besides evaluation . In order to guarantee a strong concept of extensionality we use, along several axioms of Flow, the quantifier . The pragmatic impact of axioms of Flow Theory is that all of them work together into the direction of a strong concept of extensionality, in the sense that functions and who “do the same thing” are the very same, with the sole exceptions of and , and and . On the other hand, in Section 8 we discuss about some possible variations of Flow. And one those variations considers the possibility of replacing all occurrences of within our axioms by . In this sense, we consider the possibility of grounding such a variation of Flow with an intuitionistic logic rather than a classical predicate calculus, as the one used in this article.

One of the major advantages of our concept of composition is that it allows us to mimic many-variables functions, although all functions in Flow are monadic. That feature allows us to talk even about non-associative binary operations, despite the fact that composition is associative. For details, see Section 7.

Theorem 12

Suppose is idempotent with respect to composition, and there are some and such that . Then, is flexible with .

Proof:

If , then . But for . Since is idempotent with respect to composition, then . Thus, . Hence, .

This last theorem is not important for further developments of Flow Theory. We just proved it to show that our framework is able to mimic well known results regarding the usual way composition is defined within standard set theories. Similar results about idempotent functions can be generated.

P6 - Expansion

.

This last axiom guarantees the existence and uniqueness of a special function . Differently from other functions in Flow, this one deserves a special notation. From now on we write instead of , where is any term. If we take a look at the right hand of above, we see that in the case where is different of , we have , since is different of . Within this context, may be defined as it follows:

Definition 2

is a function such that for any , iff .

The intuitive idea of a term like is that of successor of a given function . If the successor of is a non- term , then and share the same images for any different of and , although and are different. Besides, . That is why and are different, since while (remember we are considering the case where is a non- term). In the case where there is no which satisfies such demands, then is simply , and once again and are different (if, of course, we guarantee the existence of any function like , as it is done in axiom P6).

Axioms P1-P5 work as “a soil prep to enhance the germination of functions”. Axiom P6, on the other hand, states the existence of another function . And that fact (together with the next axiom) entails the existence of infinitely many other functions. Besides and , there is a unique function whose images are either or itself, where is different of . In other words, P6 is consistent with the existence of a such that for any different of , both and share the same images and . Such a function is simply .

Theorem 13

.

The proof of this last theorem was already done in the previous paragraph.

Function is quite handy here. Actually, is ubiquitous within our discussions, since we prove latter can be associated to the empty set within ZFC. Since we intend to introduce further axioms regarding the existence of multiple functions (specially those functions which capture the everyday needs of standard mathematics), it is perfectly possible that some compositions correspond to certain functions whose images are always , except for , of course. In view of the fact that axiom P5 demands the composition can never be , function proves to be quite valuable to cope with such situations. In other words, if is always for any different of , then is simply .

Observation 2

A word of caution is necessary here. Rigorously speaking, the label “Definition 2” by itself does not necessarily refer to a definition. Consider, for example, there is a function such that and , where has the same properties of in Definition 2. In that case, we have and . That is a result which confirms

, according to the axiom of weak extensionality. On the other hand, something odd is happening here, since there seems to be

two successors for the same function , despite the fact that is unique. From an intuitive point of view, we cannot actually see or decide which is which. It does not matter which function is a successor of , if there is more than one successor which satisfies the allegedly definition 2. All that matters is how this successor does work. An analogous remark can be done about the successor of any function which admits a non- successor (as we intend to pursue in the next paragraphs). Nevertheless, if and , that entails , which conflicts with the assumption that . That means, from a rigorous point of view, “Definition 2” may somehow be a creative statement. After all, if “Flow without Definition 2” is consistent, then “Flow with Definition 2” may allow us to entail a contradiction. That means our choice above for stating Definition 2 and axiom P6 has a pedagogical rationale. That is why we used the quantifier in P6. In the next postulate, we intend to talk about the successor of some other functions, in the sense that the successor of the successor of does exist and so on. But from now on we don’t have to worry with the quantifier, since the uniqueness of guarantees the uniqueness of for any . Our pedagogical solution to cope with Flow is based on the convenience of how to easily read our axioms.

P7 - Infinity

.

Definition 3

Any function which satisfies axiom P7 is said to be inductive.

Since the existence of is granted by P6 and P7 (and independently by P5), we can now apply again to get a function such that , , and for the remaining functions (functions which are neither nor ) we have . Concerning P7, this postulate states the existence of another function . It says if a function admits a non- successor (in a way such that ), then . Analogously we can get (from P6) functions , , and so on. Besides, according to P7, any inductive function admits its own non- successor .

Subscripts , , , , etc., are simply metalinguistic symbols based on an alphabet of ten symbols (the usual decimal numeral system) which follows the lexicographic order. The lexicographic order is denoted here by , where . If is a subscript, then corresponds to the next subscript, in accordance to the lexicographic order. In that case, we write . is an abbreviation for with occurrences of and occurrences of pairs of parentheses. And again we have . As it is well known for any finite alphabet, is a strict total order. That fact allows us to talk about a minimum value between two subscripts and . Within that context, is iff , it is iff , and it is either one of them if . Of course, iff . If , we denote this by .

Such a vocabulary of ten symbols endowed with is called here (meta) language .

Thus, P6 provides us some sort of “recursive definition” for functions , while P7 allows us to guarantee the existence of inductive functions:

  • is such that is if and otherwise.

  • is such that , , and for any different from .

Observe that , while . Moreover, , and ; while . For a generalization of such results, see Theorems 15, 16, and 17.

Notwithstanding, P7 says much more, since it states function itself has its own non- successor .

The diagrams below (Figure 1) help us to illustrate how can we represent any function in a quite straightforward way. Each diagram is formed by a rectangle. The left top corner of any rectangle introduces the label of the function which is represented by the diagram. The remaining labels refer to functions such that . For each label there is a unique corresponding arrow which indicates the image of by . Since for any function we have , then the function represented at the left top corner of the rectangle does not need to be attached to any arrow. So, our first three examples below refer to functions , , and .

From left to right, the first diagram refers to . It says, for any , is , except for itself. The second diagram says , and . Observe the circular arrow attached to label in the second diagram is not a reference to the fact that . Circular arrows referring to axiom P2 are simply omitted. So, the circular arrow associated to in the second diagram says solely that . Finally, the third diagram says , , and . The diagram representations for and are, respectively, a blank rectangle and a filled in black rectangle. More sophisticated examples of functions are represented by diagrams in the next Section.

\begin{picture}(60.0,60.0)\par\put(3.0,3.0){\framebox(50.0,50.0){}} \par\par\put(10.0,43.0){$\phi_{0}$} \par\par\put(63.0,3.0){\framebox(50.0,50.0){}} \par\par\put(70.0,43.0){$\phi_{1}$} \par\put(70.0,18.0){$\boldmath\curvearrowleft$} \par\put(70.0,10.0){$\phi_{0}$} \par\par\put(123.0,3.0){\framebox(50.0,50.0){}} \par\par\put(130.0,43.0){$\phi_{2}$} \par\put(130.0,18.0){$\boldmath\curvearrowleft$} \par\put(130.0,10.0){$\phi_{0}$} \par\put(160.0,18.0){$\boldmath\curvearrowleft$} \par\put(160.0,10.0){$\phi_{1}$} \par\put(190.0,25.0){$\cdots$} \par\end{picture}

Figure 1: From left to right, diagram representations for functions , , and .

Observe those diagrams above may be easily identified with reflexive graphs, from Graph Theory. Since objects and morphisms of a category (in the sense of Category Theory) may be viewed as, respectively, the vertices and edges of a graph, that fact seems to ease our discussion in Section 4 concerning Category Theory. Nevertheless, we show latter that is not the case.

Theorem 14

If is inductive, then for any of language we have

The proof is straightforward.

The next theorems are provable by induction.

Theorem 15

For any and of the vocabulary given above, and .

Theorem 16

For any and of the vocabulary given above, if at least one of them is different of , then and .

Recall our previous argument for the non-associativity of evaluation holds, since we can now guarantee the existence of other functions besides and .

Definition 4

iff .

While is a term for any and , is a metalinguistic abbreviation for a formula. We read as “ acts on ”. And acts on iff is not itself and . The intuitive idea of this last definition is to allow us to talk about what effectively a function does. For example, both and do nothing at all, since there is no on which they act. On the other hand, there is a term on which acts, namely, .

Theorem 17

For any and of the vocabulary of language , .

Proof:

We present here a sketch for the proof. Without loss of generality, suppose first . That is equivalent to say there is some such that . So, we can use the previous propositions regarding functions . According to Definition 1, if , then the images of are given by . But according to the last two propositions, those are exactly the same images of . Since those functions of kind are generated by axiom P6, then is exactly . An analogous argument shows that . For the case where , the proof is straightforward.

This last proposition proves all functions are idempotent with respect to composition. Besides, composition is commutative among functions , although a given does not necessarily commute with any arbitrary function , as we can see in the next two theorems.

Theorem 18

For any from language , is a function such that: (i) ; (ii) for any (if there is any); (iii) ; and (iv) for the remaining values of .

Proof:

Item (i) is a direct consequence from axiom P2. If , then , for , , and . If , then . So, item (ii) is satisfied. That means is different of , which entails , according to P5. On the other hand, if, e.g., (which is different of , of and of ), then , which entails . That means is not either. Therefore, item (iii) is satisfied. For the remaining terms (those which are different of for , different of , different of and different of ), we have . Therefore, item (iv) is satisfied.

The proof of last theorem helps us to understand the non-constructive character of the calculation of compositions. In standard differential and integral calculus, for example, the definition of limit of a real function on a given point does not allow us to calculate limits, even when they do exist. Theorems about limits are the usual tools which allow us to calculate limits. A similar situation happens regarding composition in Flow. Axiom P5 does not provide any methodology for calculating compositions in a constructive fashion. But all theorems about composition can provide useful tools for calculations. Next theorem together with the previous one, e.g., show us that is never equal to , for any .

Theorem 19

For any from language , is a function such that: (i) ; (ii) for any (if there is any); (iii) ; (iv) ; and (v) for the remaining values of .

Proof:

Item (i) is a direct consequence from axiom P2. If , then , for , , and . If , then . So, item (ii) is satisfied. That means is different of , which entails , according to P5. On the other hand, if, e.g., (which is different of , of and of ), then , which entails . That means is not either. Therefore, item (iii) is satisfied. Besides, for any , which satisfies item (iv). For the remaining terms , all we have to do is to remember only for those such that , where either or . But those cases were already analysed. Therefore, . That concludes item (v).

The next diagram represents function . If , then (first arrow from left to right), , and so on; until , and . The remaining values have images . That is why do no not represent them in the diagram.

\begin{picture}(60.0,60.0)\par\put(3.0,13.0){\framebox(130.0,40.0){}} \par\put(10.0,43.0){$\phi_{n}\circ\sigma$} \par\put(14.0,25.0){$\phi_{0}$} \par\put(-5.0,27.0){\vector(1,0){15.0}} \par\put(44.0,25.0){$\phi_{1}$} \par\put(25.0,27.0){\vector(1,0){15.0}} \par\put(55.0,27.0){\vector(1,0){15.0}} \par\put(73.0,24.0){$\cdots$} \par\put(85.0,27.0){\vector(1,0){15.0}} \par\put(104.0,25.0){$\phi_{n-1}$} \par\put(125.0,27.0){\vector(1,0){15.0}} \par\put(144.0,25.0){$\phi_{n}$} \par\end{picture}

Theorem 20

.

Proof:

Suppose . According to definition 2, . That happens only if . That means the successor of does not share all images of . That happens because the successor of is not a non- term.

Theorem 21

.

Proof:

That is a corollary from the fact that for any , we have , if is neither nor .

Definition 5

iff