On the Syntax of Logic and Set Theory

Our treatment of propositional calculus in this section will be brief. We study the domain of objects, or (synonymously) expressions. Anything that ``exists", we say, is an object—for example, propositions, desks, and numbers are objects. A handful of objects are designated as signs, these are:

The symbol Ø is the system's basic constant. The next three symbols of the list generate new objects. An object of the form $\alpha{}^{\prime}\mspace{1.0mu}$ for some object $\alpha$ is a variable, an object of the form $\alpha^{\dot{}}$ is an index, and a constant is, in addition to Ø, any object of the form $\alpha\mspace{2.0mu}\hat{}$. We call the first handful of variables $\text{{\O }}{}^{\prime}\mspace{1.0mu},\text{{\O }}{}^{\prime}\mspace{1.0mu}{}^{\prime}\mspace{1.0mu},\text{{\O }}{}^{\prime}\mspace{1.0mu}{}^{\prime}\mspace{1.0mu}{}^{\prime}\mspace{1.0mu},$ etc. with the standard abbreviations $a,b,c,x,y,z,A,B,C,$ etc.; this assignment is taken to be precise, but need not be specified here. In our presentation, when we have a need for metamathematical variables, we use Greek letters $\alpha,\beta,\gamma,$ etc. We call the three unary connectives ${}^{\prime}\mspace{1.0mu},^{\dot{}},\mspace{2.0mu}\hat{}$ stops or logical stops. This name is chosen since they are meant to ``stop" the progress of a logical substitution (defined below). The index stop can be put aside for now; we will not see it again until §3.

The symbol $\rightarrow$ is the fundamental relation, referred to as containment or universal containment. The symbols $\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}$ and $\cap$ are (binary) operations. Table 1 displays the system's basic formation rules. These rules allow, for any objects $\alpha$, $\beta$, $\gamma$, the formation of a new object, having $\alpha$, $\beta$, $\gamma$ as subobjects or subexpressions; this language is heritable in the obvious sense. The object $(\alpha\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}\beta)$ is the conjunction, union, or federation of $\alpha$ and $\beta$.²²2We have searched for an apt name and symbol for the product ($\alpha\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}\beta$): a universal expression of objects-in-multiplicity, akin to ordinary addition but idempotent and, in practice, more basic. The symbol employed is a formal comma modeled after the commas in the notation $\{a,b\}$ for the pair and the Gentzen-Kleene-Rosser notation $\Gamma,A\vdash B$ for the carriage of assumptions. The use of an ordinary comma is tempting but troublesome, so one considers using one or the other of $\cup$ and $\land$. However, because the product should carry neither logical nor set-theoretical connotation in every instance, stultifying and plainly odd-looking expressions arise as a result. The symbol $(\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}\,)$ smooths over these distinctions, and thus helps us achieve the desired conceptual view. Our approach is to use the terminology federation, union, and conjunction (and three distinct symbols) to refer to instances of the same operation appearing at three distinct positions in the order of operations (see footnote, p. 26). It is hoped that a new symbol and a new term will aid the reader and avoid confusion to the greatest possible extent. The object $(\alpha\cap\beta)$ is the intersection of $\alpha$ and $\beta$. We shall suppress these parentheses when they are unnecessary.

We call any object generated from Ø using only the given formation rules (and those that may be added later) formal objects, or well-formed objects. Objects which are not well-formed are informal objects, and include chairs, desks, letters of the alphabet (when not intended as abbreviations of formal objects), and strings of signs which are not well-formed. In this paper, we shall loosely allow consideration of informal or ``ordinary" objects as objects of the system, while giving precedence to the view in which all but the formal objects are laid aside or ignored. Thus the terms object and expression will hereafter refer to formal objects.

We make the following definitions. For $\text{{\O }}\mspace{2.0mu}\hat{}$ we write $\bot$. For $a\rightarrow\bot$ we write $\lnot a$, the negation of $a$. For $\text{{\O }}\rightarrow a$ we write $\underline{a}$, the trivialization of $a$.³³3This notation mirrors the notation $\bar{a}$ for the negation of $a$ in, e.g., Hilbert & Ackermann (1928). We write $a\lor b$ for $\underline{\underline{a}\cap\underline{b}}$, the disjunction of $a$ and $b$. This expression is classically equivalent to the expression $\lnot(\lnot a\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}\lnot b)$; however, the former expression remains serviceable in the intuitionistic case.⁴⁴4Thus $\cap$ can replace Heyting’s connective $\lor$. For $b\rightarrow a$ we may write $a\leftarrow b$, and for $(a\rightarrow b)\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}(a\leftarrow b)$ we write $a=b$.

We call expressions which may be put in the form $a\rightarrow b$⁵⁵5Given an object $\xi$, that is, we may find objects $\alpha$ and $\beta$ such that $\xi=(\alpha\rightarrow\beta)$. formulas; an object that is not a formula is said to be concrete. Expressions of the form $a=b$ may also be called equations or equalities in addition to being called formulas, as usual. In order to improve readability and respect normal usage, we will usually use the synonymous symbol $\leftrightarrow$ instead of $=$ when we denote an equality between formulas. We may also similarly write $a\land b$ in place of $a\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}b$ in the case when $a$ and $b$ are both formulas. In its role, this technique is comparable to the practice (common, for example, in physics) of using modified parentheses (brackets, braces) in lengthy expressions to highlight syntactic units. It is also comparable in its role to the system of dots devised by Peano and featured in the Principia Mathematica. Here, because our work is not too involved, and in order to gain familiarity with the machine language (so to speak), we shall use $\land$ sparingly.

We must pause to clarify the aforementioned notion of logical substitution. First, an instance of an object $\beta$ in the syntax of an object $\alpha$ may be one of two kinds: if $\beta$ is, in this instance, a subobject of $\alpha$, it is said to appear in $\alpha$, and otherwise to merely occur in $\alpha$. We denote the object $\alpha$ modified so that every appearance (note) of the variable $x$ is replaced by the object $\beta$ by $\alpha[\beta/x]$. We refer to $x$ as the target variable, and to $\beta$ as the substituend of the substitution. The reader can verify that this is a well-defined object for its inputs $\alpha$, $\beta$, and $x$, and that no expression which occurs within (``is guarded by") an index, variable, or constant stop is modified by any substitution. If there is no risk of confusion, in place of $\alpha[\beta/x]$, we may sometimes write simply $\alpha(\beta)$.

Formal proofs are greatly simplified through the use of assumptions. An arbitrary object (normally a formula) $\alpha$ is taken or assumed at a given step, and subsequent rules may combine relying on the assumption as if it were proven, until the assumption is discharged. At the discharge step the conclusion of the immediately preceding step is rewritten under the hypothesis of the assumption. A proof with assumptions is not complete until all assumptions have been discharged. That the use of assumptions does not allow any new formulas to be proven is a result known as the deduction theorem, which we will now prove. We first require a lemma.

Because of A11, and because A1 implies that $(\text{{\O }}\rightarrow a)\rightarrow a$, the equation

is characteristic of formulas. Thus, Lemma 2.1 becomes, for all formulas $a$ and $b$, formula (1) of Frege (1879),

The connection between (2) and assumption mechanisms was noticed early on (see van Heijenoort (1967, p. 465)). However, it does not obtain when $a$ and $b$ range over all the objects of our system, as can easily be checked by letting $a$ in (2) be concrete.⁶⁶6Provided that one has already crossed over to the intended conceptual view, this point is obvious. We are considering logic and set theory as they might be conceived in a single common space or worldview in which a commitment exists to a general notion of objecthood encompassing both language and its referents. Consider, then, the content of (2) set-theoretically, e.g., in the case when $a$ and $b$ are ordinary objects (desks, chairs, etc.) and the relation is one of ordinary containment. In our system concrete objects live in the midrange or midst of the lattice. Viewed logically, the lattice encompasses an infinite array of truth values; a concrete expression—a desk, a number, a geometric figure, etc.—is an object whose truth value is neither true, nor false, but rather a thing unique to the object itself. As the formalism is developed in the sequel, the load on the symbol $\rightarrow$ will be decreased by the introduction of the symbols $\supset,\subset,\in$ etc.—though the distinction between these symbols and the family $\rightarrow,\leftarrow,$ etc. shall solely involve properties of their arguments. The formal unity between the concepts of implication and containment, whose naturality we propose in this section and whose fundamental importance to the system should become clear as we proceed, rests undisturbed throughout (echoing in certain respects an early system of Quine, see footnote, p. 18).

Let $(\alpha_{1},\ldots,\alpha_{n})$ be such a proof with assumptions. Consider a sequence of steps $(a_{i},\ldots,$ $a_{i+j})$, $1\leq i<i+j\leq n$, from an assumption step, $\alpha_{i}$, to the first subsequent discharge step, $\alpha_{i+j}$, inclusive, containing no intervening assumption steps. Replacing the entire string with the new string $(\alpha_{i}\rightarrow\alpha_{i+1},\alpha_{i}\rightarrow\alpha_{i+2},\ldots,\alpha_{i}\rightarrow\alpha_{i+j-1})$, it is clear that every prior use of the rule of Exchange can be carried out under the lingering hypothesis $\alpha_{i}$ by inserting the Axiom A1, instantiating it as desired,⁷⁷7As a minor additional hypothesis, one must let variables appearing in assumptions be distinct from those appearing in the axioms. and applying Combination and Exchange. Every Substitution can be executed under the assumption since, by hypothesis, all variables are held constant. All of the axioms, finally, can themselves be written under the assumption of $\alpha_{i}$ by using Lemma 2.1 and Exchange, and we obtain a new formal proof. If it has assumptions, we iterate the process described. If we continue until we have treated all the assumptions in turn, we obtain a formal proof of $\alpha_{n}$.

The stipulation that assumed variables be held constant is perfunctory at this level; there is no motivation to vary assumptions until quantifiers (and/or classes) are introduced. All naturally arising assumptions may in fact be discharged in any order, but one must exercise a bit of caution, since

and these expressions are not equivalent in general,⁸⁸8If your wallet is empty, I will give you my coat. But I am not inclined, otherwise, to give you my coat for your wallet. but are seen to be equivalent when $a$ is a formula. A few details remain which we omit showing that we may generalize these principles, and therefore make and manipulate assumptions naturally, provided that they are formulas.

The following relations easily obtain: conjunctions and intersections of formulas are formulas. The converses of axioms 3, 6, 11, and 12. Of union/conjunction/ federation $(\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}\,)$: associativity, commutativity, idempotence, identity Ø, zero element $\bot$, as well as the relations

Of intersection $\cap$: associativity, commutativity, idempotence, identity $\bot$, zero Ø, and isotonicity (implying substitutivity⁹⁹9The term isotone (and the dual antitone) is found, e.g., in Birkhoff (1940); substitutivity appears, e.g., in Quine (1951); Bernays (1958). One may show that antecedents are antitone, that double antecedents are again isotone, etc.). The same laws therefore hold also for disjunction $\lor$. Finally, both of the distributive laws hold, yielding a distributive lattice of objects under federation and intersection, and a Boolean ring of formulas (under conjunction and disjunction, or equivalently, federation/union and intersection) isomorphic to $\mathbb{Z}_{2}$.

All students of modern mathematics are well accustomed to the assignment of formulas and equations to some value or thing that is in some sense the storing ground of true formulas, and another which is the home and identifier of false formulas. It is a truth which is all the more remarkable given the advanced technology, both industrial and intellectual, available to scientists and philosophers who investigate these instruments in our own age, that whether they are taken to be abstract forms, letters, or empirical entities, they have up to now resisted all efforts to make them rigorously understood and canonized. We do not contest these efforts, which lie well beyond our ken. We follow formalists and intuitionists in distinguishing mathematical discourse from ordinary linguistic discourse, and place before ourselves only the very limited task of calibrating and arranging adequate grounds for the former, laying aside questions surrounding the latter. We make no claims here concerning the concepts of truth and falsehood as they arise in ordinary language.

Table 3 illustrates the syntactic medium in which we are developing logic and set theory. This arrangement may surprise some readers, many of whom may never have paused before over the evocative relations which naturally arise in proof theory (such as $\text{{\O }}\vdash\Delta$, denoting that the set of formulas $\Delta$ follows under no assumptions). The author readily agrees that the arrangement is conceptually surprising (it stirs the imagination even after long familiarity); however, having weighed the matter carefully, he finds arguments which speak for the system as it has been defined, and to condemn it, neither any falsifier, nor such a drastic departure from the norm in foundations. Logic itself is surprising and elusive—it receives these qualities naturally and instantly from the concepts it traffics in. Assigning those formulas which are formally true to the formal constant Ø identifying empty and trivial structures is no more remarkable than assigning them to a letter $\top$, ensconced in a detached logical space. The formulas, in either case, remain what they are; we encounter them, and they inspire us, in the same way. The author believes that it is rare indeed to find an a priori basis for criticism of variations in formal orientation which is invulnerable to a challenge on a priori grounds. One must be practical and, as Hilbert implored us to do, measure theories by their fruits. Given the inconclusive suggestion of patterns in the structure of formal and informal thought, we should be neglecting the due diligence of science to leave either possibility unexamined, in case one orientation should offer distinct utilitarian advantages over its alternative. The author cautions the reader to carefully entertain a wide-ranging field of evidence before adopting the Einsteinian critique that the system is simple, yes—and too simple. So far, we have seen a number of ways in which the present system recommends itself: a compact set of basic symbols, accessible definitions, the felicitous distinction between the formula and the concrete term, as well as the convenience of a unified intuitive presentation. What remains to be discovered is, of course, unknown to the author. In the next section, the reader shall have the opportunity to consider the effects of the novel approach upon the fundamental devices of set theory and model theory.

Aside from these remarks (and with respectful apologies to those in the philosophical domain), the author does not wish to suggest here a single philosophical vantage from which the formal system might best be viewed. According to strict formalism, the system is the article found on the page, dissociated from all semantic interpretations, and whatever metaphysics it might inspire is irrelevant to its use in practice. The formalist, as Carnap once said, is ``among the most pacifistic of mankind," able to partake in the methods of all others, and the author has, in principle, no disagreements with him. We believe that one could incorporate the system into intuitionism by introducing, to begin with, the symbol Ø as the empty hand—the moment before the unfolding construction of proof—while introducing the symbol $\bot$ to represent an impossible object of reflection, or what is called in Gödel (1944) ``the notion of `something' in an unrestricted sense". One may then proceed to define the system of Heyting.¹⁰¹⁰10The intuitionistic case is interesting in its details, but requires additional work; we intend to discuss it elsewhere. The opposite assignment conflicts with intuitionism, for consider: truth is obviously constructive, while the domain in which construction occurs cannot be entirely so. With the early work of Wittgenstein, there is a shared goal of a comprehensive Weltanschauung for analytic thought, and with sustained effort along these lines there might be developed a framework bearing out some of his views on the limitations of language and the world of coherent ideas. The relationship of these ideas to Platonism and its many later manifestations is also quite striking. The contradiction-as-unity ($\text{{\O }}=\bot$) has some curious and remarkable parallels to concepts explored by several ancient Greek philosophers including Plato, later figures such as, for example, Plotinus, Cusanus, and Pascal, and modern figures such as Cantor, Brouwer, Whitehead, and many others. We believe considerable opportunity remains to discuss the system in light of other leading schools. The relationship with Cantor's thought is further touched upon in §4.2, where we shall see that the novel approach adds a note of prescience to one of his more rarely-cited ideas.

In an early attempt to develop the class rigorously for analytic purposes, Peano (1889) refers to the formation of classes as inversion, apparently on the notion that if a formula were taken as a mapping into the truth values 0 and 1, the class derived from the formula would be its kernel. Prima facie, this seems quite natural, perhaps even obvious, but it runs into difficulties having to do with the fact that the domain of definition of such a function remains unclarified. In the particular case of such a system as the one at hand, there is no reason, at present, to expect that any given multiplicity should have a well-defined content. Since it is far from the case that all properties are heritable, this presents quite a grave problem.

Recall that we have chosen to build out from the notion of containment—the principle that objects are formed by fixing an interior. Let an object $a$ have ``content" if there is $b\neq\text{{\O }},\neq a$ such that $a\rightarrow b$. What do we find ``inside" the set of objects $x$ such that $x$ has content? If the set can be formed, the transitivity of containment becomes destabilizing should there exist objects without content. Is such a predicate formally unstatable? It would appear not. In all nontrivial cases some objects indeed have content; if they all have content, the set is fine. With that allowance, however, we seem pushed into the boundless possibilities of a hopelessly ill-founded world. One way to deal with all of this trouble is to suppose that set-theoretical abstraction doesn't care about objects with content in this sense. If it cares only about objects that lack content, we begin to arrive at the notion of a discrete atomic point as a specialized multiplicity, and a theory of classes based upon it.

This approach, a retreat into the safety of wellfoundedness, bears a certain resonance, even if one were hypothetically prepared to dismiss the general tendency of modern set theoretical systems as an historic trend. It remains today an obvious and unavoidable fact (one that no one has ever been quite sure what to make of) that the world of mathematical models is very different, qualitatively speaking, from the world of ``real" objects. The many-pronged efforts during the early twentieth century to clarify the foundations of mathematics (Hilbert, whose school was perhaps preeminent among these, constantly emphasized the need to divest symbols of their ``meaning" in order to establish perfect rigor) has since been met with the runaway success of modern set theory, logic, and computer science, while the outcome of the movement to clear away the ambiguities of language (led by the logical positivists, significists, and others) remains unclear, having been frustrated by a wave of objections which crested during the postwar period and from which, it seems, it is not likely to recover. This suggests that the venture to clarify ordinary language and the venture to clarify mathematics are (at least in some significant respects) distinguishable. Instead of confronting or attempting to repair this dichotomy—and by more tenuous extension, that between mathematical models and empirically observed systems—we would, by enhancing the theoretical significance of the above-mentioned distinction, accept it into our system. In the endeavor to characterize things which are ``one" (robustly ``single" objects, or individuals) contradictions that have beleaguered the study of classes—the paradoxes named after Russell, Curry, Berry, Mirimanoff, Grelling, and so on—shall serve as so many tests to objectively measure candidate definitions. Should they be resolved, we shall have evidence that a satisfactory abstract definition of the intuitive notion of a point or individual is in reach.

Our approach will be based loosely upon the notion of primality in a unique factorization domain.¹¹¹¹11This structure is discussed in most algebra textbooks, e.g. Hungerford (1974, §3.3).

Let $\operatorname{\mathfrak{s}}(a\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}b)$. $(a\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}b)\rightarrow a$, so $a=\text{{\O }}\lor a=(a\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}b)$. If $a\neq\text{{\O }}$, then $a=(a\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}b)$; use $(a\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}b)\rightarrow b$. Now let $\operatorname{\mathfrak{s}}(a)$ and $b\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}c\rightarrow a$. Suppose $\lnot(b\rightarrow a)$. $a\rightarrow a\cap b$. So $a\cap b=a\lor a\cap b=\text{{\O }}$. Since $\lnot(b\rightarrow a)$, $a\cap b=\text{{\O }}$. So $(b\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}c)\cap a=(b\cap a)\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}(c\cap a)=c\cap a$. Since $a\neq\text{{\O }}$ and $(b\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}c)\rightarrow a$, $(b\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}c)\cap a\neq\text{{\O }}$, so $c\cap a\neq\text{{\O }}$, so $c\cap a=a$. So $c\rightarrow(c\cap a)\rightarrow a$.

Let the reader note, we do not see in $\bot$ the set-theoretical universe—it is a still higher maximum, but it serves only as a kind of universal wastepaper basket. We will later encounter an object, the ``universe" $V$; it, too, will fail to be an individual in any nontrivial system. Here, the system echoes the notion, advanced in systems like NBG and MK, that the universe is a proper class, and therefore not an element of any set.

Because we could ask meaningful questions about classes that have not been constructed—for example, whether (under some formal criteria) they may be constructed or not—we could introduce these objects initially as expressions (say, of the symbolic form $\{x\mid\phi\}$) which dwell in the lattice of objects but have ab initio no containment relationships with any previously defined concrete expression, nor with one another. Then we might ensure, proceeding abstractly, that the classes so understood behave formally as though they were objects already in some sense defined—possibly by construction. Where any kind of construction is possible, all conceivable ways of access must converge, and conclude in perfect agreement. In other words, while the introduction of classes may succeed in generating new descriptions or formulations, it must fail to generate new constructions. If the class represents a kind of bridge across a sea of complexity, we wish to ensure that it begins and ends somewhere. We would not build bridges on a planet with no land, nor if it meant giving up the land for bridges.

These ideas will resurface several times as we proceed, particularly in §4.2. Before turning to the formal development, let us consider one more rough analogy. Consider the sequence $1,1+1,1+(1+1),\ldots$ and the sequence $2,2^{2},2^{2^{2}},\ldots$ The syntax of both expressions is very basic, and one should agree that, whatever these symbols mean relative to one another or to other things, if one of the two sequences could be constructed, then the other could as well. Most people would also understand the latter expressions to be defined in terms of the former expressions—however, they need not be. If they were not, then in order to establish equivalence with the ordinary definitions, one would demand that a proof could be supplied showing that we may understand one to be a subsequence of the other, or that a member of the latter sequence takes on a unique location somewhere in the former sequence. Even if one were not exactly sure where in the former sequence, say, the twentieth element of the latter sequence could be correctly placed, one could easily stand about discussing the twentieth element of the latter sequence, more or less as we are doing right now, and proof in hand, refer to it, think of it, regard it, etc., as an element of the former sequence simpliciter.

Now in place of the natural numbers of this example, consider the signs and the formation rules introduced in §2. We add classes to the system by adding the sign $\mid$ to the list of basic signs, and to the formation rules of Table 1 the formation rule

We shall axiomatize these new objects in such a way that in every respect they act precisely as though they were in fact members of the original group of well-formed objects—precisely those members that we think of when we imagine the expression factored with respect to $(\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}\,)$.

We define the class or abstract of variable $x$ and object $\alpha$ (sometimes called the extension of $\alpha$ with respect to $x$), denoted $(x^{\dot{}}\mid\alpha[x^{\dot{}}/x])$ or usually more simply with the abbreviation $\{{}^{x}\,\alpha\}$, to be a product of $x$, the class index, and $\alpha$, the condition of the class. (Though $\alpha$ may range freely over objects, there is little interest in the class if $\alpha$ is not a formula.) The modified—N.B.—condition $\alpha[x^{\dot{}}/x]$ and the modified class index $x^{\dot{}}$ are subobjects of the class. We refer to an appearance of an index $x^{\dot{}}$ as a bound occurence of $x$, as usual, or say that $x$ appears as an index. Thus when $x$ simply appears, we sometimes say that $x$ appears free. This allows us in general to do all our thinking in terms of variables, some of which are ``guarded" from substitution processes.

Having discussed the individual and the class, we can introduce the remaining rules and axioms of this section, and begin to develop the rudiments of a theory of sets. We begin by introducing the simplest class of all:

This class may be referred to as the single-element or type-$0$ universe, or the universal class. The formula Ø which appears as the condition can be thought of as expressing that there are no conditions upon members of the class.

If we select a multiplicity from the object just defined, then we have a set.

Unless specified otherwise, the letters $\phi$ and $\psi$ shall always denote formulas, and we shall use the variables $A,B,C$, etc. to denote sets.¹²¹²12As was stated in §2, all variables are syntactically generic: $\alpha{}^{\prime}\mspace{1.0mu}$ for some object $\alpha$. We shall not make use of typed variables anywhere in the sequel. We define all of the close relatives of $\subset$ and $\in$ in the expected way.¹³¹³13Because the relation $\lnot(a\in A)$ is rarely used, it might be best to let $a\notin A$ denote that $\operatorname{\mathfrak{s}}(a)$ and $a\not\subset A$. However, we shall preserve the usual convention regarding slashed relations here. Schematically, this gives:

In expressions these relations will always bind more tightly than the ones introduced in §2, with the exception of $=$, unless it is denoted by the character $\leftrightarrow$.

Passing for a moment to the context of the integers under ordinary multiplication (the model of primality that everyone knows best): we do not regard the integer 1 as a prime, since it is the multiplicative identity, and since it is what mathematicians call a unit. In a certain sense, it permeates multiplicative space—it is like the solvent in which prime number combinations are dissolved. Similarly, the basic idea of axiom schema A13 (a version of Cantor's comprehension principle, Frege's rule V, Quine's $\ast 202$, and Zermelo's principle of Aussonderong) is to regard Ø as a different kind of object, distinct from ordinary single objects. It falls outside the domain of things which are targeted for membership in classes—a subset of any set, yet not a member of any set. Granting Ø such special status provides a kind of valve for the release of pressure, which checks the advance of Russell's paradox.

Axiom. (Schema) For every well-formed formula $\phi$ and variable $x$,

Comprehension schema A13 is presented in a convenient deployable form in (8), while the formal axiom schema is displayed in Table 4. Note, as mentioned above, that since variables, when they are bound, are encased within an index stop, there is no way that a variable appearing in $a$ can be inadvertently bound in deriving $\phi(a)$ from $a$'s membership in $(x^{\dot{}}\mid\phi(x^{\dot{}}))$.

There is one rule, R3, governing the classes introduced in this section. It states that under a group of assumptions $\gamma$ which do not depend on a given variable $x$ (i.e., in which the variable $x$ does not appear), if an object $\alpha$ contains an object $\beta$, then the class $\{{}^{x}\,\alpha\}$ is contained in the class $\{{}^{x}\,\beta\}$. We point out the immediate derived rule

The use of $\subset$ here is justified, since for any object $\phi$,

and R3 itself may be written

Since these are the last of our rules, we may now note that the deduction theorem continues to hold, permitting the normal introduction and discharge of assumptions. We refer to a variable which appears in an open assumption as a named variable. This vocabulary is chosen naturally, because such a variable is thought of as the ``name" of something, and this name is retained as the steps of the proof flow along. We say that a proof is mindful of names if assumptions open during the occurrence of a substitution which has a named variable as target variable appear, upon discharge, with the named variable modified appropriately, and if the rule R3 is used only when the variable to be converted to an index does not appear in any open assumption. We will assume, from now on, that assumptions are formulas unless stated otherwise.

We present the verifications to be inserted in Theorem 2.2; the setup is identical. Clearly, the rule R2 can be used to target named variables, provided the proof is mindful of names. We have to show that instances of R3 can be carried out under the lingering hypothesis $\alpha_{i}$. Let an occurence of R3 (using variable $x$) act on step $\alpha_{k}$, $i<k<i+j$; then $\alpha_{k}$ is of the form $\gamma\rightarrow(\delta\rightarrow\epsilon)$, where $\gamma,\delta,\epsilon$ are objects, and $x$ does not appear in $\gamma$. Hence $(\alpha_{i}\rightarrow\alpha_{k})=(\alpha_{i}\rightarrow(\gamma\rightarrow(\delta\rightarrow\epsilon)))$. We use this expression to derive a proof (requiring only the postulates of §2, for which the deduction theorem has been verified) of $(\alpha_{i}\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}\gamma)\rightarrow(\delta\rightarrow\epsilon)$, apply R3 (noting that the proof is mindful of names), then insert a proof of $(\alpha_{i}\rightarrow(\gamma\rightarrow\{{}^{x}\,\delta\}\subset\{{}^{x}\,\epsilon\}))$, allowing the proof to proceed under the assumption $\alpha_{i}$. A simple induction argument shows that an arbitrary number of assumptions can be folded up in the same way, to make way for an occurence of R3.¹⁴¹⁴14In fact, because our system is not a resource logic, the open assumptions could simply be treated as an abiding unordered multiplicity. This completes the proof.

In certain respects, we expect all sets to behave like deliberately constructed sets. Hence, one thinks of the set as having a unique decomposition, or of being able to break every set up into its single elements—to ``spread them out" on a universal table top. Since we have defined sets in terms of a class, $V$, this will be an acceptable notion for us, since all sets will be classifiable, well-founded objects. At the present stage of development, however, there might exist sets which have subsets, but which do not have any elements. Suppose, that is, that there exists an object $\xi$ such that

and now suppose that $\xi$ is a set. With such an object present, $\{{}^{x}\,x\in(1\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}2)\}$ could be put equal to $1\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}2\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}\xi$ without contradiction. We exclude the possibility of such objects being defined with a version of the principle of extensionality: that every set ``is determined by its elements" (Zermelo, 1908). The important property is

which asserts that for any given set $A$, the class $\{{}^{x}\,x\in A\}$ is at worst a proper subset of $A$. For definable sets, the two expressions are equivalent, and any finite union of single objects $a_{1}\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}\ldots\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}a_{n}$ is definable (Theorem 3.14). For undefinable sets, however, we must enforce a rather imponderable discipline, and thus we include axiom A14b of Table 4.

Axiom. (Schema) For variables $x$, $\{{}^{x}\,x\in a\}\subset a,$ and $a\subset V\rightarrow a\subset\{{}^{x}\,x\in a\}$.

We can now begin our development in earnest.

The following theorem is easy to prove in the adopted line of development, and is of central importance to what follows.

A straightforward application of $\rightarrow$-transitivity, relying on A14.

We can now examine some well-known antinomies. The Russell set is $s=\{{}^{x}\,x\notin x\}$. A single object $x$ will belong to $s$ if and only if $\lnot(x\in x)$ by A13, but $x\in x$ for all such $x$. Therefore $s$ contains no individuals. Therefore (by extensionality) $s$ is empty.¹⁵¹⁵15Though we avoid Russell’s paradox, Cantor’s theorem on the size of the power set may still be proven (see §4.1),and the theories of computability and decidability appear unaffected, as one might expect. A discussion of several similar paradoxically-self-containing sets is found in Beth (1965, pp. 483-4). Extensions of formulas

(which Quine calls ``epsilon cycles") are under our definitions all equal to Ø. It is not possible under our definitions for any paradox to arise from the relation $s\in s\leftrightarrow(\operatorname{\mathfrak{s}}(s)\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}\phi(s))$, where $\phi(s)$ implies $s\notin s$. More generally, we have:

Curry's paradox arises from considering the extension (with respect to $x$) of the formula $x\in x\rightarrow\phi$, where $\phi$ may be any formula or object. In naive set theory there is a paradox: let the set be called $c$. By naive comprehension, $c\in c\leftrightarrow(c\in c\rightarrow\phi(c))$. Therefore it is true that $c\in c\rightarrow\phi(c)$. So $c\in c$. So $\phi(c)$, which might be any statement at all, is true—all objects, in other words, are trivial. The A13 comprehension schema also implies that $c\in c\rightarrow\phi(c)$. However, it does not then follow from A13 that $c\in c$, only that $c\in c$ if $c$ is single. It then falls to $\phi$ to determine whether in fact $\operatorname{\mathfrak{s}}(c)$ or not.¹⁶¹⁶16In either case, the class is essentially a description which does not affect the underlying ontology. If $c$ is the sole individual $x$ such that $\phi(x)$—or in other words, if there is but one individual $a$ such that $\phi(a)$—then $c$ is single and $c\in c$, whereas if there are at least two such individuals, or no such individual, it is formally proven that $c$ is not single, and therefore that $c\notin c$.

The recurring theme of these solutions is that paradoxes are avoided by countenancing multiplicities which are not individuals. We will see this theme again when we examine other paradoxes in §4. Next, we verify several basic theorems.

The proofs of these formulas offer no difficulty. One simply applies the general method of Theorem 3.12, as in, e.g., the following proof:

Let $y\in\{{}^{x}\,\phi\lor\psi\}$ (name $y$, and the variables appearing in $\phi$ and $\psi$). Then by comprehension $(\phi\lor\psi)[y/x]$, this becomes $\phi[y/x]\lor\psi[y/x]$. Dividing into cases (axiom A6), one concludes $y\in(\{{}^{x}\,\phi\}\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}\{{}^{x}\,\psi\})$.¹⁷¹⁷17The parentheses required here invite the use of $\cup$ (defined below). Discharge the assumption (there are no longer any assumptions, thus no longer any named variables) and apply extensionality. The other direction is similar. For (16), use A6.

As in Theorem 3.15. We note that this theorem, like many mathematical theorems, requires a formal stipulation on the involved variables that is often omitted; in this instance, the variable $x$ cannot appear free in the expressions substituted for $A$ and $B$. For us, however, this restriction is made automatically due to the index-variable distinction drawn at the syntactic level, and the formal statement of the theorem is here given in its entirety.

Since we still lack the ability to form sets of sets, we are still unable to define many familiar devices of set theory, such as the sum set and power set. However, we may define $A\setminus B=\{{}^{x}\,x\in A\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}x\notin B\}$, the difference of $A$ and $B$, $A\mspace{-2.0mu}\mathbin{{}_{\scriptscriptstyle{\Delta}}}\mspace{-2.0mu}B=\{{}^{x}\,(x\in A\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}x\notin B)\lor(x\in B\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}x\notin A)\}$, the symmetric difference or distinction of $A$ and $B$, and $\mathop{{}^{\scriptscriptstyle{\sim}}\mspace{-3.0mu}}\!A=\{{}^{x}\,x\notin A\}$, the complement of $A$. Under our definitions, the union $A\cup B$ of two sets $A$ and $B$ is once again their federation, the fundamental product $A\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}B$ defined for any two objects $A$ and $B$. Elementary formulas involving these can now be proven in the usual way, for example: $\mathop{{}^{\scriptscriptstyle{\sim}}\mspace{-3.0mu}}\{{}^{x}\,\phi\}=\{{}^{x}\,\lnot\phi\}$, etc.

The principle of Extensionality, first proposed by Frege in 1893, and later by Zermelo in 1908, has been subsequently used to provide a rigorous and intuitively natural basis for theorems throughout mathematics. We will use it to ground quantification.¹⁸¹⁸18 During the period just prior to the appearance of his work on NF, Quine (1937) experimented with a similar approach, and proved its equivalence to a system of Tarski. In the system, to the study of which Quine devoted two papers, the simple theory of types is developed using only the connectives of inclusion (containment) and abstraction. For example, the condional $\phi\rightarrow\psi$ is an abbreviation of the formula $\{x\mid\phi\}\subset\{x\mid\psi\}$, where $x$ does not appear in $\phi$ or in $\psi$. Since comprehension is one of the basic assumptions in topos theory and related areas, quantification by means of comprehension is commonly used in category-theoretical developments of deductive systems; see, e.g., Bell (1988, p. 70ff). The author is indebted to Andreas Blass for pointing out this and other references in topos theory.

Before continuing any further, we need to extend R3 to the case when $x$ may appear in $\gamma$:

Let $\{{}^{x}\,\gamma\}\subset\{{}^{x}\,\alpha\rightarrow\beta\}$, $V\subset\{{}^{x}\,\gamma\}$. Then $V=\{{}^{x}\,\alpha\rightarrow\beta\}$. Let $a\in\{{}^{x}\,\alpha\}$. Then $\alpha(a)\rightarrow\beta(a)$, and since $\alpha(a)$, $\beta(a)$ as well, so $a\in\{{}^{x}\,\beta\}$. So $\{{}^{x}\,\alpha\}\subset\{{}^{x}\,\beta\}$ by extensionality.

Once objects may have classes as subobjects, the management of assumptions becomes something of a puzzle. Theorem 3.18 presents a way around this obstacle: it directly follows from it that, given any objects $\gamma,\alpha,\beta$, if $\gamma\rightarrow(\alpha\rightarrow\beta)$ has been proven, then one immediately has $(\mathop{\forall x\,}\gamma)\rightarrow\{{}^{x}\,\alpha\}\subset\{{}^{x}\,\beta\}$. Thus it becomes allowed to convert a named variable to an index, provided the assumptions in which the variable is named are discharged with the introduction of a universal quantifier. In practice this all falls naturally into place most of the time, but it can incite minor technical fallacies.

For example, given formulas $\phi$ and $\psi$, the statement $\phi\rightarrow\psi$ should imply that $\{{}^{x}\,\phi\}\subset\{{}^{x}\,\psi\}$, but as writing $(\phi\rightarrow\psi)\rightarrow\{{}^{x}\,\phi\}\subset\{{}^{x}\,\psi\}$ produces absurdities by substituting for $x$, this will not be a provable formula unless $x$ does not appear in $\phi$ or in $\psi$. On the other hand, if for every individual $x$, $\phi\rightarrow\psi$ (a weaker assumption in general), the expected class relation obtains, as has just been shown.

Kleene (1952) investigates this phenomenon under the rubric variation (p. 102ff). There, one works with a system in which proofs with open assumptions are considered finished formal proofs. In order to keep his assumptions in order, he stipulates that variables which appear in assumptions are ``held constant" until they are substituted for. One is free to do so; however, they are then added to a list appended to the proof-theoretical symbol $\vdash$ which indicates that they have been varied, and that further substitutions are no longer possible. His approach may been influenced by the Peanian pasigraphy (by way of the Principia, perhaps), in which subscripts of the quantified variables are attached to certain logical connectives in place of quantifier operators. Having obtained a structural analog to Kleene's approach, we are prepared to proceed.

We next show how to pass from the universal to the particular, and vice versa.

Let $\operatorname{\mathfrak{s}}(a)\rightarrow\phi(a)$. Suppose $a\in V$; then $\phi(a)$, so $a\in\{{}^{x}\,\phi\}$ by comprehension, and therefore $a\in V\rightarrow a\in\{{}^{x}\,\phi\}$. So by extensionality $V\subset\{{}^{x}\,\phi\}$.

1. Let $\operatorname{\mathfrak{s}}(a)\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}\mathop{\forall x\,}\phi.$ 2. $\operatorname{\mathfrak{s}}(a).$ 3. $\operatorname{\mathfrak{s}}(a)\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}\text{{\O }}.$ 4. $\operatorname{\mathfrak{s}}(a)\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}\text{{\O }}[a/x].$ 5. $a\in V.$ 6. $a\in\{{}^{x}\,\phi\}.$ 7. $\phi(a).$

In the universal case, we have given a backbone to the quasi-reasoning which echoes in the corridors of mathematics departments, where proofs of universal statements proceed by ``choosing" elements ``arbitrarily", ``freely", sometimes even ``at random". Existential statements prompt arguments of a similar style: the mathematician argues that he may ``pick one" and insert it into his algebra; if he is subsequently able to derive an expression that does not depend upon which element he picked, he concludes that the statement is true, usually without accounting for the fate of the assumption naming his choice. This form of reasoning, too, can be filled in with the tools we have defined. We first pause momentarily to verify the inverse direction:

Argue by contradiction, by concluding from $a=\text{{\O }}$ that $\operatorname{\mathfrak{s}}(\text{{\O }})$.

We might state the sense of this statement by saying that ``if it has already been proven that $\mathop{\exists x\,}\phi\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}\phi(a)\rightarrow\psi$"—here, $a$ is simply an independent variable—``then by using R3 and R1 with the theorem as stated, it is proven that $\mathop{\exists x\,}\phi\rightarrow\psi$."²⁰²⁰20In Kleene’s system, one has $\mathop{\exists x\,}\phi,\phi(a)\vdash^{a}\mathop{\exists x\,}\phi\rightarrow\psi$. Notice that $(\mathop{\exists x\,}\phi\mspace{-1.0mu}\mathbin{{}_{\scriptscriptstyle{\setminus}}}\mspace{-3.0mu}\phi(a)\rightarrow\psi)\rightarrow(\mathop{\exists x\,}\phi\rightarrow\psi)$ is false; let $1\in V$, $a=1$, $\phi=(x=1)$, $\psi=(1\neq 1)$.

We need the following.

To prove the lemma, let $\{{}^{x}\,\phi\}\neq\text{{\O }}$, and let $\lnot\phi$. Suppose (choosing a new variable) that $b\in\{{}^{x}\,\phi\}$. Then $\phi[b/x]$, that is, $\phi$. Therefore $\bot$. Therefore $\bot[b/x]$, and $b\in\{{}^{x}\,\bot\}$. So $\{{}^{x}\,\phi\}\subset\{{}^{x}\,\bot\}$ by extensionality. So $\{{}^{x}\,\phi\}=\text{{\O }}$, contrary to hypothesis. Discharging gives $\lnot\lnot\phi$, or $\phi$ by the law of excluded middle.

Let $\phi$ and $\psi$ be defined as stated above.