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Thirty Years of Foundational Studies Lectures on the Development of Mathematical Logic and the Study of the Foundations of Mathematics in 1930–1964
Abstract
This chapter presents sixteen lectures on the development of mathematical logic and the study of the foundations of mathematics in the years 1930–1964, delivered by the author in the Summer School in Vaasa, Finland in the summer of 1964. The chapter distinguishes three major movements in the philosophy of mathematics: the intuitionism of Brouwer, the logicism of Frege and Russell, and the formalism of Hilbert. The logicism of Frege and Russell tries to reduce mathematics to logic. This seemed to be an excellent program, but when it was put into effect, it turned out that there is simply no logic strong enough to encompass the whole of mathematics. Thus what remained from this program is a reduction of mathematics to set theory. The formalism of Hilbert sets up a program which requires that the whole of mathematics be axiomatized and, that these axiomatic theories be then proved consistent by using very simple combinatorial arguments. The chapter discusses the changes that the three schools underwent during the years 1930–960.
... Mostowski [101] gives an example of a Σ 0 1 provability predicate for which G2 fails. Let Pr M T (x) be the Σ 0 1 formula "∃y(Prf T (x, y) ∧ ¬Prf T ( 0 = 0 , y))" where Prf T (x, y) is a ∆ 0 1 formula saying that "y is a proof of x". ...
... Since the formula Pr M T (x) satisfies D1 and D3, it does not satisfy D2. Mostowski's example [101] shows that G2 may fail for Σ 0 1 provability predicates satisfying D1 and D3. One important non-standard provability predicate is Rosser provability predicate Pr R T (x) introduced by Rosser [116] to improve Gödel's first incompleteness theorem. ...
We give a survey of current research on G\"{o}del's incompleteness theorems from the following three aspects: classifications of different proofs of G\"{o}del's incompleteness theorems, the limit of the applicability of G\"{o}del's first incompleteness theorem, and the limit of the applicability of G\"{o}del's second incompleteness theorem.
... Mostowski [101] gives an example of a Σ 0 1 provability predicate for which G2 fails. Let Pr M T (x) be the Σ 0 1 formula "∃y(Prf T (x, y) ∧ ¬Prf T ( 0 = 0 , y))" where Prf T (x, y) is a ∆ 0 1 formula saying that "y is a proof of x". ...
... Since the formula Pr M T (x) satisfies D1 and D3, it does not satisfy D2. Mostowski's example [101] shows that G2 may fail for Σ 0 1 provability predicates satisfying D1 and D3. One important non-standard provability predicate is Rosser provability predicate Pr R T (x) introduced by Rosser [116] to improve Gödel's first incompleteness theorem. ...
We give a survey of current research on Gödel's incompleteness theorems from the following three aspects: classifications of different proofs of Gödel's incompleteness theorems, the limit of the applicability of Gödel's first incompleteness theorem, and the limit of the applicability of Gödel's second incompleteness theorem.
... Mostowski [98] gave an example of a Σ 0 1 provability predicate for which G2 fails. Let Pr M T (x) be the Σ 0 1 formula "∃y(Prf T (x, y) ∧ ¬Prf T ( 0 = 0 , y))" where Prf T (x, y) is a ∆ 0 1 formula saying that "y is a proof of x". ...
... We know that G2 holds for provability predicates satisfying D1-D3. However, Mostowski's example [98] shows that G2 may fail for Σ 0 1 provability predicates satisfying D1 and D3. ...
In this paper, we give a survey of current research on Gödel's incompleteness theorems from the following three aspects: (1) classifications of different proofs of Gödel's incompleteness theorems; (2) the limit of applicability of Gödel's first incompleteness theorem; (3) the limit of applicability of Gödel's second incompleteness theorem.
... Indeed, as early as the first nonclassical logics appeared, the possibility of building mathematics upon them was conceived. As mentioned by Mostowski [31], J. Lukasiewicz hoped that there would be some nonclassical logics which can be properly used in mathematics as non-Euclidean geometry does. In 1952, Rosser and Turquette [36, p. 109] proposed a similar and even more explicit idea: ...
... Unfortunately, the above idea has not attracted much attention in logical community. For such a situation, Mostowski [31] pointed out that most of nonclassical logics invented so far have not been really used in mathematics, and intuitionistic logic seems the unique one of nonclassical logics which still has an opportunity to carry out the Lukasiewicz's project. A similar opinion was also expressed by Dieudonne [12], and he said that mathematical logicians have been developing a variety of nonclassical logics such as second-order logic, modal logic and many-valued logic, but these logics are completely useless for mathematicians working in other research areas. ...
We present a basic framework of automata theory based on quantum logic. Inparticular, we introduce the orthomodular lattice-valued (quantum) predicate ofrecognizability and establish some of its fundamental properties.
... Not all provability predicates satisfy the second incompleteness theorem. For example, Mostowski [22] showed that there exists a Σ 1 provability predicate Pr T (x) of T satisfying Σ 1 C such that PA ¬Pr T ( 0 = 1 ). This fact together with clause 2 of Theorem 2.2 implies that the two consistency statements ¬Pr T ( 0 = 1 ) and ¬ Pr T ( ϕ ) ∧ Pr T ( ¬ϕ ) are different in general, and the conclusion of clause 2 of Theorem 2.2 cannot be strengthened to T ¬Pr T ( 0 = 1 ). ...
... Not all provability predicates satisfy the second incompleteness theorem. For example, Mostowski [22] showed that there exists a Σ 1 provability predicate Pr T (x) of T satisfying Σ 1 C such that PA ¬Pr T ( 0 = 1 ). This fact together with clause 2 of Theorem 2.2 implies that the two consistency statements ¬Pr T ( 0 = 1 ) and ¬ Pr T ( ϕ ) ∧ Pr T ( ¬ϕ ) are different in general, and the conclusion of clause 2 of Theorem 2.2 cannot be strengthened to T ¬Pr T ( 0 = 1 ). ...
We investigate modal logical aspects of provability predicates satisfying the following condition: : If , then . We prove the arithmetical completeness theorems for monotonic modal logics , , , , and with respect to provability predicates satisfying the condition . That is, we prove that for each logic L of them, there exists a provability predicate satisfying such that the provability logic of is exactly L. In particular, the modal formulas : and : are not equivalent over non-normal modal logic and correspond to two different formalizations and of consistency statements, respectively. Our results separate these formalizations in terms of modal logic.
... An example of a Σ 1 provability predicate for which the second incompleteness theorem does not hold was given by Mostowski [22]. Let Pr M T (x) be the Σ 1 formula ∃y(Prf T (x, y) ∧ ¬Prf T ( 0 = 0 , y)) where Prf T (x, y) is a ∆ 1 (PA) formula saying that "y is a T -proof of x". ...
This paper is a continuation of Arai's paper on derivability conditions for Rosser provability predicates.
We investigate the limitations of the second incompleteness theorem by constructing three different Rosser provability predicates satisfying several derivability conditions.
... Mostowski (p. 24 in [20]) introduced the formula Pr M T (x) :≡ ∃y(Prf T (x, y) ∧ ¬Prf T ( 0 = 0 , y)) as an example of a Σ 1 provability predicate for which the second incompleteness theorem does not hold. Notice that ...
We investigate relationships between versions of derivability conditions for provability predicates. We show several implications and non-implications between the conditions, and we discuss unprovability of consistency statements induced by derivability conditions. First, we classify already known versions of the second incompleteness theorem, and exhibit some new sets of conditions which are sufficient for unprovability of Hilbert–Bernays’ consistency statement. Secondly, we improve Buchholz’s schematic proof of provable -completeness. Then among other things, we show that Hilbert–Bernays’ conditions and Löb’s conditions are mutually incomparable. We also show that neither Hilbert–Bernays’ conditions nor Löb’s conditions accomplish Gödel’s original statement of the second incompleteness theorem.
... An example of a Σ 1 provability predicate for which the second incompleteness theorem does not hold was given by Mostowski [20]. Let Pr M T (x) be the Σ 1 formula ∃y(Prf T (x, y) ∧ ¬Prf T ( 0 = 0 , y)) where Prf T (x, y) is a ∆ 1 (PA) formula saying that "y is a proof of x". ...
This paper is a continuation of Arai's paper on derivability conditions for Rosser provability predicates. We investigate the limitations of the second incompleteness theorem by constructing three different Rosser provability predicates satisfying several derivability conditions.
... Mostowski (p. 24 in [20]) introduced the formula Pr M T (x) :≡ ∃y(Prf T (x, y) ∧ ¬Prf T ( 0 = 0 , y)) as an example of a Σ 1 provability predicate for which the second incompleteness theorem does not hold. Notice that ...
We investigate relationships between versions of derivability conditions for provability predicates.
We show several implications and non-implications between the conditions, and we discuss unprovability of consistency statements induced by derivability conditions.
First, we classify already known versions of the second incompleteness theorem, and exhibit some new sets of conditions which are sufficient for unprovability of Hilbert-Bernays' consistency statement.
Secondly, we improve Buchholz's schematic proof of provable -completeness.
Then among other things, we show that Hilbert-Bernays' conditions and L\"ob's conditions are mutually incomparable.
We also show that neither Hilbert-Bernays' conditions nor L\"ob's conditions accomplish G\"odel's original statement of the second incompleteness theorem.
... vii). In the time point in which that book ends, there starts the story told in [Mostowski 1965], covering the period from Gödel's results to the sixties. Thus each of these books confirms the other in the claim that a new period of modern logic starts in the thirties. ...
The foundations of mathematics cover mathematical as well as philosophical problems. At the turn of the 20th century logicism, formalism and intuitionism, main foundational schools were developed. A natural problem arose, namely of how much the foundations of mathematics influence the real practice of mathematicians. Although mathematics was and still is declared to be independent of philosophy, various foundational controversies concerned some mathematical axioms, e.g. the axiom of choice, or methods of proof (particularly, non-constructive ones) and sometimes qualified them as admissible (or not) in mathematical practice, relatively to their philosophical (and foundational) content. Polish Mathematical School was established in the years 1915–1920. Its research program was outlined by Zygmunt Janiszewski (the Janiszewski program) and suggested that Polish mathematicians should concentrate on special branches of studies, including set theory, topology and mathematical logic. In this way, the foundations of mathematics became a legitimate part of mathematics. In particular, the foundational investigations should be conducted independently of philosophical assumptions and apply all mathematically accepted methods, finitary or not, and the same concerns other branches of mathematics. This scientific ideology contributed essentially to the remarkable development of logic, set theory and topology in Poland.
This Element takes a deep dive into Gödel's 1931 paper giving the first presentation of the Incompleteness Theorems, opening up completely passages in it that might possibly puzzle the student, such as the mysterious footnote 48a. It considers the main ingredients of Gödel's proof: arithmetization, strong representability, and the Fixed Point Theorem in a layered fashion, returning to their various aspects: semantic, syntactic, computational, philosophical and mathematical, as the topic arises. It samples some of the most important proofs of the Incompleteness Theorems, e.g. due to Kuratowski, Smullyan and Robinson, as well as newer proofs, also of other independent statements, due to H. Friedman, Weiermann and Paris-Harrington. It examines the question whether the incompleteness of e.g. Peano Arithmetic gives immediately the undecidability of the Entscheidungsproblem, as Kripke has recently argued. It considers set-theoretical incompleteness, and finally considers some of the philosophical consequences considered in the literature.
Needles to say, there is a vast literature on model theory of first-order logic and its applications. Some references have already been given throughout the text. I will repeat some of them and will add other recommendations.
This is a short biography of the distinguished representative of Warsaw School of Logic and his activity in post-war Poland.
Die beiden vorangehenden Teile dieses Buches befaßten sich mit Grundlagenthemen der Informatik, die aus dem Bereich der sog. „mathematischen Logik“ stammen. Die mathematische Logik ist, wie es ihr Name schon sagt, ein Teilgebiet der Mathematik - und natürlich konnten die beiden Logikteile dieses Buches nicht all das darstellen, was in diesem Bereich der Mathematik behandelt wird. Eine solche Darstellung hätte selbst dann den Rahmen dieses Buches gesprengt, wenn man sich auf die Teile der mathematischen Logik konzentriert hätte, die aus der Sicht der Informatik besonders relevant und interessant sind.
This paper discusses the problem of availability of constructive metamathematics for constructive theories. Since intuitionism is usually considered as the most important kind of constructivism, we have the following question: Is it possible to give intuitionistically acceptable proofs of metamathematical theorems? This issue is illustrated by the completeness theorem for intuitionistic predicate logic and the consistency of arithmetic. The conclusion is that hitherto collected evidence justifies the claim that the universal intuitionistic metamathematics is very problematic.
This essay is offered in the spirit of interdisciplinary goodwill that prevailed during the Jerusalem meetings. It is not intended for specialists in the formal approach, though it will certainly stand in need of their corrections and comments; it is addressed rather to others who study language and philosophical problems concerning language — linguists, psycholinguists, ‘ordinary-language’ philosophers, etc. I am prompted to express the thoughts that follow, first of all, because the non-specialist already has available to him some general accounts of the nature of the formal approach that claim to show its ultimate fruitlessness for the study of natural language.1 I think these accounts are fundamentally wrong, and I hope to show this. I ain motivated, secondly, by the belief that some recent developments in the formal approach could be extremely suggestive to those who adopt other approaches. Much of what has been written on these developments is difficult to follow, and I propose to say something about some of them later in reasonably non-technical language.
The assumptions of symbolic logic must have their beginnings somewhere. But why should we look in Frege’s logic, why not in some other? This question must be answered before it will be possible to examine the Ideography as part of an inquiry into fundamentals.
The formation of the current concept of computability (recursivity) is one of the major achievements of twentieth-century logical theory (see e.g. Davis 1965, Rogers 1967, ch. 1). What is especially impressive about this notion is its robustness. Originally, different logicians proposed different and prima facie completely unrelated concepts of computability, among them computability by a Turing machine, definability à la Kleene through a finite number of recursion equations and expressibility in Church’s lambda-calculus. As if by miracle, these different characterizations of computability turned out to coincide. This consilience of apparently unrelated definitions is a telling indication that the notion so captured is indeed theoretically important. It lends prima facie support to the thesis of Church (1936) that mechanically determined (finitely computable) functions (in a pretheoretical sense) are precisely the recursive ones.
The purpose of this entry is to outline the field of logic by showing its historical development. Items which recur in this process constantly can be plausibly supposed to constitute the core of logic. It will suffice to start the story from the beginnings of modern logic, while referring to its historical antecedents at relevant points.
We can find in several places an assertion that Gödel's second incompleteness theorem defeated Hilbert's program. But, (as M. Detlefsen argued in his book) in order to establish this assertion, we need to address additional issues. First we formulate Hilbert's program. Second we reconstruct a standard argument for the claim that Gödel's second incompleteness theorem defeated Hilbert's program. In doing so, we formulate a critical, and problematic assumption which we call "DCT" (Derivability Conditions Thesis). Finally we examine three arguments whose aims are to justify DCT. We show that the first and the second argument are not valid, and discuss the third argument, which is based on Kreisel's idea. We identify a difficulty in this argument as well. After examining the difficulty, we conclude that we cannot claim that Gödel's second incompleteness theorem defeats Hilbert's program. Moreover we clarify what is essentially needed for such an argument to succeed.
In this essay1 I briefly discuss some developments in logic in Poland after the Second World War at the period of the communist domination. By no means the essay is meant to be a technical report on problems and results. Rather, I shall try to present some of the main trends and ideas, as well as to give some flavor of those times which happily enough are over.
During the 10th Congress of Logic, Methodology and Philosophy of Science (Florence, August 19–25, 1995), I took part in a panel discussion on the situation of logic in Eastern Europe during the time of Soviet domination. This essay, originally written to celebrate the 50th anniversary of the Polish Academy of Sciences1, is primarily based on the paper I presented on that occasion2. The list of people with whom I consulted while preparing first the Florence paper and then the present one is rather long.3 I appreciate the assistance of all of them. My special thanks are due to Wojciech Buszkowski, Andrzej Grzegorczyk, Witold Marciszewski, Wiktor Marek, Roman Murawski, Jerzy Tiuryn and Jan Zygmunt, who in addition to offering various suggestions, remarks and criticism provided me with brief overviews of selected areas of logical investigations carried out by Polish logicians. The postwar Polish logic is too rich and too diversified for one person to be able to present it in an adequate manner, and I would not have been able to complete this paper without these people’s kind assistance. Yet, it goes without saying that the final responsibility for this paper is mine. I tried as far as I was able to evaluate critically all pieces of information I was offered and occasionally revise them. In this manner, the Polish version of this survey
was prepared in cooperation with Jan Zygrnunt, who was also the author of the initial draft of its English version. The latter was accessible on the Studia Logica home page for quite some time. Eventually, I rewrote its various parts taking into account suggestions offered to me by its readers. Of those, I particularly appreciate the remarks and comments offered by Z. Adamowicz, J.M. Dunn, W. Marek, R. Murawski, Z. Pawlak, J. van Benthem, and H. Wansing. I also express my gratitude to Tom Brunty who took the effort to polish this translation.
A most natural view is that systems sci-ence is a science dealing with systems and their systems properties. What, then, is sci-ence that it can deal with systems proper-ties which by nature are deeply penetrating also into the systemic nature of our concep-tual processes, of language, and of science itself. We start out from an independent view of the concept of science. Namely, that sciences are deductive in the sense that the results of scientific activity are deductively presentable as in descriptive theories. This is what allows wide communication of scien-tific propositions, necessary for their acces-sibility, examination and tests by other sci-entists, eventually to be intersubjectively ac-cepted. On the other hand, the scientific ac-tivity is in general a process which is beyond full deductive description. We examplify a tendency. The more introspectively oriented the domain of inquiry for a science is, the more difficult is it to isolate the deductive-result part of a science from the scientific ac-tivity. A systemic view of science arizes with science referring not only to the deductive-result part of science but also to the scien-tific activity. We make it explicit that "sci-ence" in "systems science" be systemically conceived, and develop the concepts of sys-tems science, general systems, and system, accordingly.
According to the mainstream in the 20th century, the foundations of mathematics were identified with logic and set theory.
Indeed, results concerning philosophically most interesting questions are often negative: the first order axiomatic set-theoretical
universe is deductively incomplete, inevitably non-standard, and we have no clear idea of what the intended models of set
theory are (part I). So, the foundational view of mathematics itself might be suspect. But in the spirit of Poincaré, one
should look for an other solution. He remarks that the varieties of classical first order theories is unable to deal with
the most common modes of mathematical reasoning such as complete induction and model building. For such a purpose, Hintikka's
IF-Logic seems to be an adequate way-out.
The success of set theory as a foundation for mathematics inspires its use in artificial intelligence, particularly in commonsense reasoning. In this survey, we briefly review classical set theory from an AI perspective, and then consider alternative set theories. Desirable properties of a possible commonsense set theory are investigated, treating different aspects like cumulative hierarchy, self-reference, cardinality, etc. Assorted examples from the ground-breaking research on the subject are also given.
Simple formula should contain only few quantifiers. In the paper the methods to estimate quantity and quality of quantifiers needed to express a sentence equivalent to given one.
Scientific knowledge develops in an increasingly fragmentary way.A multitude of scientific disciplines branch out. Curiosity for thisdevelopment leads into quests for a unifying understanding. To a certain extent, foundational studies provide such unification. There is a tendency, however, also of a fragmentary growth of foundational studies, like in a multitude of disciplinaryfoundations. We suggest to look at the foundational problem, not primarily as a search for foundations for one discipline in another, as in some reductionist approach, but as a steady revelation ofpresuppositions for individual scientific theories – which are boundto meet, sooner or later, in a common language. A decisive point hereis our holistic conception of language, as a whole of description-interpretation processes which are entangled(complementary) in the language itself. For every language there is alinguistic complementarity. We suggest this unique form ofentanglement as a unifying presupposition, ultimately foundational forall communicable knowledge. Involved is a linguistic realism, in terms ofwhich we critically examine ``language-world'' problems, as exposed byWittgenstein, and Russell, about a foundational interdependence of language andreality (world). Throughout, we attach to the developmentof foundational studies of mathematics, logics, and the naturalsciences. In particular, we study the interpretation problem foraxioms of infinity in some detail. We emphasize that the holistic concept of language contradicts Carnap's semiotic fragmentation thesis (thus, no clean cutbetween syntax, semantics, pragmatics).
The (meta)logic underlying classical theory of computation is Boolean (two-valued) logic. Quantum logic was proposed by Birkhoff and von Neumann as a logic of quantum mechanics more than sixty years ago. The major difference between Boolean logic and quantum logic is that the latter does not enjoy distributivity in general. The rapid development of quantum computation in recent years stimulates us to establish a theory of computation based on quantum logic. The present paper is the first step toward such a new theory and it focuses on the simplest models of computation, namely finite automata. It is found that the universal validity of many properties of automata depend heavily upon the distributivity of the underlying logic. This indicates that these properties does not universally hold in the realm of quantum logic. On the other hand, we show that a local validity of them can be recovered by imposing a certain commutativity to the (atomic) statements about the automata under consideration. This reveals an essential difference between the classical theory of computation and the computation theory based on quantum logic.
A back and forth condition on interpretations for those second-order languages without functional variables whose non-logical vocabulary is finite and excludes functional constants is presented. It is shown that this condition is necessary and sufficient for the interpretations to be equivalent in the language. When applied to second-order languages with an infinite non-logical vocabulary, excluding functional constants, the back and forth condition is sufficient but not necessary. It is shown that there is a class of infinitary second-order languages whose non-logical vocabulary is infinite for which the back and forth condition is both necessary and sufficient. It is also shown that some applications of the back and forth construction for second-order languages can be extended to the infinitary second-order languages. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
At first sight it might seem strange to devote in a handbook of philosophical logic a chapter to algorithms. For, algorithms are traditionally the concern of mathematicians and computer scientists. There is a good reason, however, to treat the material here, because the study of logic presupposes the study of languages, and languages are by nature discrete inductively defined structures of words over an alphabet. Moreover, the derivability relation has strong algorithmic features. In almost any (finitary) logical system, the consequences of a statement can be produced by an algorithm. Hence questions about derivability, and therefore also underivability, ask for an analysis of possible algorithms. In particular, questions about decidability (is there an algorithm that automatically decides if ψ is derivable from φ?) boil down to questions about all algorithms. This explains the interest of the study of algorithms for logicians.
This chapter discusses ordered structures and related concepts. The methods discussed in this chapter arose from an attempt to evolve a general metamathematical principle on the basis of which the completeness of certain concepts, including that of a real-closed ordered field, can be deduced without recourse to a detailed procedure of elimination. For the (non-empty and consistent) set of statements K be model-incomplete it is necessary and sufficient that there exist models M, M' of K, and a statement X, which contains only relations and constants of M, such that M ' is an extension of M, and such that X is satisfied by M' although it is not satisfied by M.
Im Folgenden wird eine Vereinfachung der Beweisanordnung für Henkin's Theorem [2] p. 162 angegeben (Abschnitt I). – Die Vereinfachung zeigt sich u.a. darin, daß für den abgeänderten Beweis eine Abschätzung der “projektiven Klasse” (im Sinne der Kleene-Mostowski'schen Theorie der projektiven Mengen von natürlichen Zahlen) für das zu konstruierende Modell gelingt (Abschnitte II–IV). Die Kenntnis der Henkin'schen Arbeit wird vorausgesetzt.
Um eine maximale widerspruchsfreie Menge Γ ω von geschlossenen Formeln, in der zu jeder Existenzaussage (∃ x ) A ( x ) eine Beispielaussage A ( u ij ) vorhanden ist, zu erhalten, führt Henkin einen Turm von Kalkülen S 0 , S 1 , …, S i , … und dessen Vereinigung S ω ein. Abwechselnd wird dann die gegebene Menge von Formeln in S i zu einer maximalen widerspruchsfreien Menge Γ i erweitert und in S i +1 durch Hinzunahme von Beispielaussagen für alle Existenzaussagen aus Γ i ergänzt, für die das nicht schon in einem früheren Schritt geschehen ist. Γ ω ist dann die Vereinigung aller Γ i .
Die Vereinfachung besteht nun darin, daß – nach Erweiterung von S 0 zu einem Kalkül S * durch Hinzunahme einer einfachen Folge u 1 , u 2 , … von “neuen” Konstanten – in S * zu jeder Existenzaussage eine bedingte Beispielaussage gebildet wird. K sei die Klasse dieser Aussagen, Λ die gegebene widerspruchsfreie Menge; dann wird Λ * = K ◡ Λ in S * einmal zu einer maximalen widerspruchsfreien Menge ergänzt.
Unter Verwendung der Standard-Anordnung aller geschlossenen Formeln von S * erhalten wir für diejenigen von der Form (∃ x ) A ( x ) eine Aufzählung (∃ x ) A i ( x ), dazu eine Indexfolge j i ( i =1, 2, …), definiert durch:
This chapter discusses various notions of models that appeared in the recent investigations of formal systems. The discussion is applied to the study of the following problem: given a formal system S based on an infinite number of axioms A1, A2, A3,… it is possible to prove in S the consistency of the system based on a finite number A1, A2 …, An of these axioms. The chapter considers two systems S and s based on the functional calculus of the first order. The question arises whether the assumptions concerning the form of axioms and rules of proof are general enough to cover the cases of standard formal systems based on a finite number of axioms. The answer is affirmative. To see this, the chapter remarks that the ɛ-rule enables to get rid of quantifiers in the axioms provided that introduce a sufficient number of ɛ-term.