Up to now, most constructed calculi had the following common property: Whenever the rules of formation of a calculus laid down that expressions of the designs ‘Pa’, ‘Qa’, ‘Pb’ were sentences—‘P’ and ‘Q’ being first-level one-place predicates, ‘a’ and ‘b’ individual-symbols—, ‘Qb’ was a sentence too, according to the same rules. This self-imposed restriction of the logicians is historically understandable, since calculi of this common feature have a certain simplicity which differently constructed calculi will not have. So far, calculi of this type have proved to be sufficient for the formalization of mathematics and small parts of other sciences. We may ask ourselves, however, whether such a restriction will still be desirable when attempting to construct calculi covering more ground. It is quite possible that insistence on this kind of simplicity will involve a greater complexity in other respects. Inquiry into types of calculi which do not possess this simplicity should therefore be of some interest. To such an inquiry we are led also from another point of view. More and more stress has been laid in recent researches on the construction of calculi which should show close connection with ordinary languages, and it is obvious that ordinary language does not have the mentioned property. To use an example given by Carnap: Whereas ‘This stone is red,’ ‘Aluminium is red,’ ‘This stone weighs five pounds’ are all meaningful sentences of ordinary English, ‘Aluminium weighs five pounds’ is not and it does not matter in this connection whether we formulate this fact by saying that ‘Aluminium weighs five pounds,’ though grammatically an impeccable sentence, is logically meaningless, or whether we prefer the more modern formulation that this word-sequence does not form a sentence at all.

It has been proved by Pepis that any formula of the first-order predicate calculus is equivalent (in respect of being satisfiable) to another with a prefix of the form

containing a single existential quantifier. In this paper, we shall improve this theorem in the like manner as the Ackermann and the Gödel reduction theorems have been improved in the preceding papers of the same main title. More explicitly, we shall prove the

Theorem 1. To any given first-order formula it is possible to construct an equivalent one with a prefix of the form (1) and a matrix containing no other predicate variable than a single binary one.

An analogous theorem, but producing a prefix of the form

has been proved in the meantime by Surányi; some modifications in the proof, suggested by Kalmár, led to the above form.

In [1] we have considered a certain system L and shown that although its axioms are considerably weaker than those of [2], it suffices for purposes of the topics covered in [2]. The purpose of the present paper is to consider the system L more carefully and to show that with suitably chosen definitions for numbers, the ordinary theory of real numbers is also obtainable in it. For this purpose, we shall indicate that we can prove in L a certain set of twenty axioms used by Tarski which are sufficient for the arithmetic of real numbers and are to the effect that real numbers form a complete ordered field. Indeed, we cannot prove in L all Tarski's twenty axioms in their full generality. One of them, stating in effect that every bounded class of real numbers possesses a least upper bound, can only be proved as a metatheorem which states that every bounded nameable class of real numbers possesses a least upper bound. However, all the other nineteen axioms can be proved in L without any modification.

This result may be of some interest because the axioms of L are considerably weaker than those commonly employed for the same purpose. In L variables need to take as values only classes each of whose members has no more than two members. In other words, only classes each with no more than two members are to be elements. On the other hand, it is usual to assume for the purpose of natural arithmetic that all finite classes are elements, and, for the purpose of real arithmetic, that all enumerable classes are elements.

The first order functional calculus was proved complete by Gödel in 1930. Roughly speaking, this proof demonstrates that each formula of the calculus is a formal theorem which becomes a true sentence under every one of a certain intended class of interpretations of the formal system.

For the functional calculus of second order, in which predicate variables may be bound, a very different kind of result is known: no matter what (recursive) set of axioms are chosen, the system will contain a formula which is valid but not a formal theorem. This follows from results of Gödel concerning systems containing a theory of natural numbers, because a finite categorical set of axioms for the positive integers can be formulated within a second order calculus to which a functional constant has been added.

By a valid formula of the second order calculus is meant one which expresses a true proposition whenever the individual variables are interpreted as ranging over an (arbitrary) domain of elements while the functional variables of degree n range over all sets of ordered n-tuples of individuals. Under this definition of validity, we must conclude from Gödel's results that the calculus is essentially incomplete.

It happens, however, that there is a wider class of models which furnish an interpretation for the symbolism of the calculus consistent with the usual axioms and formal rules of inference. Roughly, these models consist of an arbitrary domain of individuals, as before, but now an arbitrary class of sets of ordered n-tuples of individuals as the range for functional variables of degree n. If we redefine the notion of valid formula to mean one which expresses a true proposition with respect to every one of these models, we can then prove that the usual axiom system for the second order calculus is complete: a formula is valid if and only if it is a formal theorem.

Es ist bekannt, daβ man aus den Funktionen 0 und n + 1 ausgehend, die in der elementaren Zahlentheorie benutzten Funktionen durch eine endliche Kette von Substitutionen und “primitiven Rekursionen” der Form

(α(a1, …, ar) und β(n, a1, …, ar, ar+1) bereits eingeführte Funktionen) definieren kann. Aber die “mehrfache Rekursion”, wobei die Rekursion gleichzeitig nach mehreren Variablen läuft, führt aus der so entstehenden Klasse der “primitiv-rekursiven Funktionen” hinaus. Die sogenannten “mehrfachen”, und zwar “k-rekursiven” Funktionen (zu welchen als Spezialfall auch die primitiv-rekursiven Funktionen gehören) lassen sich von gewissen Ausgangsfunktionen ausgehend durch eine endliche Kette von Substitutionen und “k-fachen Rekursionen” (wobei die Rekursion gleichzeitig nach k Variablen läuft) einer gewissen Normalform aufbauen. Die k + 1-fache Rekursion führt für ein jedes k aus der Klasse der k-rekursiven Funktionen hinaus. Das kann mit Hilfe des Diagonalverfahrens bewiesen werden. Das Diagonalverfahren auf sämtliche mehrfach-rekursive Funktionen angewandt (auch wenn man sich auf Funktionen von einer gegebenen Variablenanzahl beschränkt) würde als Beispiel einer nicht mehrfachrekursiven Funktion eine Funktion ergeben, bei der es vom ersten Argument abhängt, wievielstellig die Funktion ist.

Man sagt allgemein, daβ eine Funktion durch irgendeine Rekursion definiert wird, wenn ihr Wert an gewissen Anfangsstellen angegeben, und an einer anderen Stelle aus Werten aufgebaut wird, welche die Funktion an vorherigen Stellen angenommen hat. Es kann aber verschieden vorgeschrieben werden, welche Stellen als Vorgänger einer gegebenen Stelle zu betrachten sind.

In this paper a method will be described which is intended to exhibit some relationships between sets of tautologies determined by truth-tables. This method is an attempt to form an algebra of truth-tables. The results sketched below are restricted to sets of tautologies determined by truth-tables with a finite number of elements and involving a single binary connective Δ. However, most of the results can be easily extended to the case of Tarski's logical matrix and even to a more general case.

We denote by S() the set of all tautologies (-tautologies) according to a given truth-table . Let describe a binary connective Δ. Then Δ()(x, y) stands for the truth-value of ΔPQ, when P has the truth-value x and Q has the truth-value y. If no ambiguity may arise we write Δ(x, y) or Δ() for Δ()(x, y).

This paper continues the exposition of a demonstrably consistent foundation for mathematics begun in An extension of basic logic (hereafter referred to as EBL) and in The Heine-Borel theorem in extended basic logic (hereafter referred to as HB). Still earlier papers related to these are A basic logic (referred to as BL) and Representations of calculi (referred to as RC). The main conventions and results of all the above papers will be presupposed in what follows. The numbering of sections and paragraphs will be a continuation of that of EBL and HB.

In a recent note Bergmann states that “nonextensional languages may contain sentences from which contextually denned predicates are not eliminable.” A restatement of his argument is as follows. Suppose L is a functional calculus of fourth order. Included among the definitions occurring in L is

Let ϕ occur in A (where A represents a well-formed formula of L) without an argument. The elimination of ϕ from A in accordance with (1) requires

The proof of (2) depends on the principle of extensionality

Hence Bergmann's conclusion “no such general elimination rule can be constructed for nonextensional languages.”

Theorem. There is an effective procedure to decide whether the set of tautologies determined by a given truth-table with a finite number of elements is empty or not.

Proof. Let W(P) be a w.f.f. with a single variable P and n a given n-valued truth-table with elements (values)

Substitute 1, 2, 3, …, n in succession for P. By the usual contraction process let W(P) assume the truth-values w1, w2, w3, …, wn respectively. The sequence

will be called the value sequence of W(P).

Value sequences consisting of designated elements of exclusively will be called designated; others will be called undesignated.

All the W(P)'s will be classified in the following way:

(a) to the first class CL1 of W(P)'s there belongs the one element P,

(b) to the (t + 1)th class CLt + 1 belong all the w.f.f. which can be built up by means of one generating connective from constituent w.f.f. of which one is an element of CLt and all the others (if any) are elements of CLn ≤ t.

For example, if N and C are the connectives described by a truth-table

etc.

Let ∣CLn∣ stand for the set of value sequences of the elements of CLn.

For Gentzen's natural deduction, a formalized method of deduction in quantification theory dating from 1934, these important advantages may be claimed: it corresponds more closely than other methods of formalized quantification theory to habitual unformalized modes of reasoning, and it consequently tends to minimize the false moves involved in seeking to construct proofs. The object of this paper is to present and justify a simplification of Gentzen's method, to the end of enhancing the advantages just claimed. No acquaintance with Gentzen's work will be presupposed.

A further advantage of Gentzen's method, also somewhat enhanced in my revision of the method, is relative brevity of proofs. In the more usual systematizations of quantification theory, theorems are derived from axiom schemata by proofs which, if rendered in full, would quickly run to unwieldy lengths. Consequently an abbreviative expedient is usually adopted which consists in preserving and numbering theorems for reference in proofs of subsequent theorems. Further brevity is commonly gained by establishing metatheorems, or derived rules, for reference in proving subsequent theorems. In natural deduction, on the other hand, proofs tend to be so short that the abbreviative expedients just now mentioned may conveniently be dispensed with—at least until theorems of extraordinary complexity are embarked upon. In natural deduction accordingly it is customary to start each argument from scratch, without benefit of accumulated theorems or derived rules.

The main purpose of this paper is to present a formal system P in which we enjoy a smooth-running technique and which countenances a universe of classes which is symmetrical as between large and small. More exactly, P is a system which differs from the inconsistent system of [1] only in the introduction of a rather natural new restrictive condition on the defining formulas of the elements (sets, membership-eligible classes). It will be proved that if the weaker system of [2] is consistent, then P is also consistent.

After the discovery of paradoxes, it may be recalled, Russell and Zermelo in the same year proposed two different ways of safeguarding logic against contradictions (see [3], [4]). Since then various simplifications and refinements of these systems have been made. However, in the resulting systems of Zermelo set theory, generation of classes still tends to be laborious and uncertain; and in the systems of Russell's theory of types, complications in the matter of reduplication of classes and meaningfulness of formulas remain. In [2], Quine introduced a system which seems to be free from all these complications. But later it was found out that in it there appears to be an unavoidable difficulty connected with mathematical induction. Indeed, we encounter the curious situation that although we can prove in it the existence of a class V of all classes, and we can also prove particular existence theorems for each of infinitely many classes, nobody has so far contrived to prove in it that V is an infinite class or that there exists an infinite class at all.

Nach Gödel ist der Prädikatenkalkül der ersten Stufe vollständig in dem Sinne, daβ alle identischen Ausdrücke dieses Kalküls ableitbar sind. Für den einstelligen Prädikatenkalkül gilt darüber hinaus, daβ zwei Ausdrücke deduktionsgleich (HB I, p. 149) sind, wenn sie für dieselben Individuenbereiche identisch bzw. nicht identisch sind. Solche Ausdrücke mögen identitätsverbunden heiβen. Daβ umgekehrt deduktionsgleiche Ausdrücke identitätsverbunden sind, gilt auch für den vollen Prädikatenkalkül. Dagegen gibt es dort Ausdrücke, die identitätsverbunden, aber nicht deduktionsgleich sind. Solche Ausdrücke sind die in HB I, p. 123, 124 angegebenen und , die beide genau im Endlichen identisch, also identitätsverbunden sind. Die Frage der gegenseitigen Ableitbarkeit bleibt dort allerdings noch unentschieden. Ich werde im Folgenden die gegenseitige Nichtableitbarkeit von und im Prädikatenkalkül der ersten Stufe beweisen. Daraus folgt dann, daβ für den vollen Prädikatenkalkül die Bernayssche Deduktionsgleichheit nicht mit der Identitätsverbundenheit zusammenfällt. Es sei erwähnt, daβ die Identitätsverbundenheit sich zwar in der “Protosyntax” (d.h. metasprachliches “Alle” und “Es gibt” beschränkt auf Zeichenreihen) definieren läβt. Trotzdem kann man nicht von einer verschärften Vollständigkeit des Prädikatenkalküls sprechen; denn es läβt sich zeigen, daβ die Identitätsverbundenheit keine Beziehung vom Typ der Beweisbarkeit ist.

A formalized theory is called complete if for each sentence expressible in this theory either the sentence itself or its negation is provable.

A theory is called deciddble if there exists an effective procedure (called decision-procedure) which enables one to decide of each sentence, in a finite number of steps, whether or not it is provable in the theory.

It is known that there exist complete but undecidable theories. There exist, namely, the so called essentially undecidable theories, i.e. theories which are undecidable and remain so after an arbitrary consistent extension of the set of axioms. Using the well-known method of Lindenbaum we can therefore obtain from each such theory a complete and undecidable theory.

The aim of this paper is to prove a theorem which shows that complete theories satisfying certain very general conditions are always decidable. In somewhat loose formulation these conditions are: There exist four effective methods M1, M2, M3, M4, such that

(a) M1 enables us to decide in each case whether or not any given formula is a sentence of the theory;

(b) M2 gives an enumeration of all axioms of the theory;

(c) the rules of inference can be arranged in a sequence R1, R2, … such that if p1, … pk, r are arbitrary sentences of the theory, we can decide by M3 whether or not r results from p1, … pk, by the n-th rule;

(d) M4 enables us to construct effectively the negation of each effectively given sentence.

In order to express these conditions more precisely we shall make use of an arithmetization of the considered theory .

Die vorliegenden Untersuchungen haben sich aus solchen entwickelt, die ich an andrer Stelle früher veröffentlichte. Die Kenntnis dieser früheren Arbeit wird in keiner Weise hier vorausgesetzt und ist für das folgende unwesentlich. Es sei mir jedoch gestattet, für den Kenner der früheren Arbeit einige Bemerkungen voraufzuschicken. In der genannten Arbeit hatte ich, angeregt ursprünglich durch Gedankengänge von Herrn Behmann, ein typenfreies System der Logik aufgebaut, das allein auf dem Gedanken fußte, daß der Definitionsbereich der Prädikate im allgemeinen beschränkt ist, sodaß nur bestimmte Zusammenstellungen von Prädikaten und Dingen sinnvolle, d.h. wahre oder falsche Aussagen ergeben, und das diesen Gesichtspunkt meiner Meinung nach konsequent durchführte. Eine anschließende Arbeit sollte dann die Widerspruchsfreiheit des Systems untersuchen. Diese Untersuchung ist inzwischen geschehen. Es zeigte sich, daß sich die Widerspruchsfreiheit des Systems nur dann beweisen ließ, falls die in ihm enthaltene Vollständigkeit von → die sich durch das Bestehen eines Deduktionstheorems zeigte, fallen gelassen wurde. Die Grundformeln (13) und (14) mußten wegbleiben. Die Widerspruchsfreiheit des restlichen Systems ließ sich dann durch eine dem System eigentümliche Methode der Beweistransformation zeigen.

Diese anschließienden Untersuchungen, die ihrer Zeit nicht mehr veröffentlicht werden konnten, sind inzwischen durch die vorliegenden überholt. Es war das Unbefriedigende bei dem genannten Aufbau, daß hinterher doch wieder eine Einschränkung gemacht werden mußte, die sich nicht aus den zugrunde gelegten Prinzipien ergab, sondern von ihnen aus unvorhergesehen war. Diese Einschränkung bewirkte übrigens, daß keine volle Zahlentheorie in dem System zu gewinnen war.

In mathematics, when we want to introduce classes which fulfill certain conditions, we usually prove beforehand that classes fulfilling such conditions do exist, and that such classes are uniquely determined by the conditions. The statements which state such unicity and existence of classes are in mathematical logic consequences of the principles of extensionality and class existence. In order to illustrate how these principles enable us to introduce classes into systems of mathematical logic, let us consider the manner in which Gödel introduces classes in his book on set theory.

For instance, before introducing the definition of the non-ordered pair of two classes

Gödel puts down as its justification the following two axioms:

By A4, for every two classes y and z there exists at least one non-ordered pair w of them; and by A3, w is uniquely determined in A4.

The purpose of the present paper is to construct a fragment of number theory not subject to Gödel incompletability. Originally the system was designed as a metalanguage for classical mathematics (see section 10); but it now appears to the author worthwhile to present it as a mathematical system in its own right, to serve however rather as an instrument of computation than of proof. Its resources in the latter respect seem very extensive, sufficient apparently for the systematic tabulation of every function used in any but the most recondite physics. The author intends to pursue this topic in a later paper; the present one will simply present the system, along with proofs of consistency and completeness and a few metatheorems which will be used as lemmas for future research.

Completeness is achieved by sacrificing the notions of negation and universal quantification customary in number-theoretic systems; the losses consequent upon this are made good in part by the use of the ancestral as a primitive idea. The general outlines of the system follow closely the pattern of Fitch's “basic logic”; however the latter system uses combinatory operators in place of the variables used in the present paper, and if variables are introduced into Fitch's system by definition their range of values will be found to be much more extensive than that of my variables. The present system K is thus a weaker form of Fitch's system.

It is apparently not known whether or not Fitch's system is complete.

In his doctor's thesis [1], Henkin has shown that if a formal logic is consistent, and sufficiently complex (for instance, if it is adequate for number theory), then it must admit a non-standard model. In particular, he showed that there must be a model in which that portion of the model which is supposed to represent the positive integers of the formal logic is not in fact isomorphic to the positive integers; indeed it is not even well ordered by what is supposed to be the relation of ≦.

For the purposes of the present paper, we do not need a precise definition of what is meant by a standard model of a formal logic. The non-standard models which we shall discuss will be flagrantly non-standard, as for instance a model of the sort whose existence is proved by Henkin. It will suffice if we and our readers are in agreement that a model of a formal logic is not a standard model if either:

(a) The relation in the model which represents the equality relation in the formal logic is not the equality relation for objects of the model.

(b) That portion of the model which is supposed to represent the positive integers of the formal logic is not well ordered by the relation ≦.

(c) That portion of the model which is supposed to represent the ordinal numbers of the formal logic is not well ordered by the relation ≦.

The object of this note is to classify the isomorphism types of all complete atomic proper relation algebras. In particular we find that the number of abstractly distinct complete atomic proper finite relation algebras of 2m elements is equal to the number of distinct partitions of the positive integer m into summands which are perfect squares.

A relation algebra is called complete atomic if its Boolean algebra is complete atomic. A proper relation algebra is a relation algebra whose elements are relations. Lyndon has recently proved, by constructing an appropriate finite relation algebra, that not every complete atomic relation algebra is isomorphic to a proper relation algebra.

Henceforth, let us consider a complete atomic proper relation algebra. The atoms {〈a, b〉}, {〈c, d〉} will be called connected if and only if {a, b} ⋂ {c, d} is not empty. Let the relation of connectedness over the set B of all atoms be denoted by ~. For brevity let B = {b1,b2, …}. The relation ~ is obviously reflexive and symmetric. But ~ is not transitive, since a, b, c, d distinct implies {〈a, 0〉} ~ {〈b, c〉}, {〈b, c〉} ~ {〈c, d〉}, and {〈a, b〉} ≁ {〈c, d〉}.