A formal system of logic

1950 ◽  
Vol 15 (1) ◽  
pp. 25-32 ◽  
Author(s):  
Hao Wang
Keyword(s):  
Set Theory ◽  
Existence Theorems ◽  
Formal System ◽  
Mathematical Induction ◽  
Restrictive Condition ◽  
Infinite Class ◽  
Class V ◽  
Inconsistent System ◽  
Theory Generation

The main purpose of this paper is to present a formal systemPin which we enjoy a smooth-running technique and which countenances a universe of classes which is symmetrical as between large and small. More exactly,Pis 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, thenPis 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.

scholarly journals Discrete Mathematics into K-11 and K-12 Grade Education

2021 ◽  
Vol 23 (05) ◽  
pp. 319-324
Author(s):  
Mr. Balaji. N ◽  
◽  
Dr. Karthik Pai B H ◽  
Keyword(s):  
Set Theory ◽  
Mathematical Reasoning ◽  
Mathematical Induction ◽  
Discrete Mathematics ◽  
College Classrooms ◽  
Underlying Principle ◽  
Education Study ◽  
Strongly Connected ◽  
Proof Techniques ◽  
K 12

Discrete mathematics is one of the significant part of K-11 and K-12 grade college classrooms. In this contribution, we discuss the usefulness of basic elementary, some of the intermediate discrete mathematics for K-11 and K-12 grade colleges. Then we formulate the targets and objectives of this education study. We introduced the discrete mathematics topics such as set theory and their representation, relations, functions, mathematical induction and proof techniques, counting and its underlying principle, probability and its theory and mathematical reasoning. Core of this contribution is proof techniques, counting and mathematical reasoning. Since all these three concepts of discrete mathematics is strongly connected and creates greater impact on students. Moreover, it is potentially useful in their life also out of the college study. We explain the importance, applications in computer science and the comments regarding introduction of such topics in discrete mathematics. Last part of this article provides the theoretical knowledge and practical usability will strengthen the made them understand easily.

Certain predicates defined by induction schemata

1953 ◽  
Vol 18 (1) ◽  
pp. 49-59 ◽  
Author(s):  
Hao Wang
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Formal System ◽  
General Notion ◽  
Constant Number ◽  
Natural Numbers ◽  
System L ◽  
Truth Definition ◽  
Consistency Proofs ◽  
Special Case

It is known that we can introduce in number theory (for example, the system Z of Hilbert-Bernays) by induction schemata certain predicates of natural numbers which cannot be expressed explicitly within the framework of number theory. The question arises how we can define these predicates in some richer system, without employing induction schemata. In this paper a general notion of definability by induction (relative to number theory), which seems to apply to all the known predicates of this kind, is introduced; and it is proved that in a system L1 which forms an extension of number theory all predicates which are definable by induction (hereafter to be abbreviated d.i.) according to the definition are explicitly expressible.In order to define such predicates and prove theorems answering to their induction schemata, we have to allow certain impredicative classes in L1. However, if we want merely to prove that for each constant number the special case of the induction schema for a predicate d.i. is provable, we do not have to assume the existence of impredicative classes. A certain weaker system L2, in which only predicative classes of natural numbers are allowed, is sufficient for the purpose. It is noted that a truth definition for number theory can be obtained in L2. Consistency proofs for number theory do not seem to be formalizable in L2, although they can, it is observed, be formalized in L1.In general, given any ordinary formal system (say Zermelo set theory), it is possible to define by induction schemata, in the same manner as in number theory, certain predicates which are not explicitly definable in the system. Here again, by extending the system in an analogous fashion, these predicates become expressible in the resulting system. The crucial predicate instrumental to obtaining a truth definition for a given system is taken as an example.

A formalization of the theory of ordinal numbers

1965 ◽  
Vol 30 (3) ◽  
pp. 295-317 ◽  
Author(s):  
Gaisi Takeuti
Keyword(s):  
Set Theory ◽  
Natural Extension ◽  
Mathematical Induction ◽  
Predicate Calculus ◽  
Transfinite Induction ◽  
First Order ◽  
Recursive Functions ◽  
Ordinal Numbers ◽  
Primitive Recursive ◽  
Order Predicate Calculus

Although Peano's arithmetic can be developed in set theories, it can also be developed independently. This is also true for the theory of ordinal numbers. The author formalized the theory of ordinal numbers in logical systems GLC (in [2]) and FLC (in [3]). These logical systems which contain the concept of ‘arbitrary predicates’ or ‘arbitrary functions’ are of higher order than the first order predicate calculus with equality. In this paper we shall develop the theory of ordinal numbers in the first order predicate calculus with equality as an extension of Peano's arithmetic. This theory will prove to be easy to manage and fairly powerful in the following sense: If A is a sentence of the theory of ordinal numbers, then A is a theorem of our system if and only if the natural translation of A in set theory is a theorem of Zermelo-Fraenkel set theory. It will be treated as a natural extension of Peano's arithmetic. The latter consists of axiom schemata of primitive recursive functions and mathematical induction, while the theory of ordinal numbers consists of axiom schemata of primitive recursive functions of ordinal numbers (cf. [5]), of transfinite induction, of replacement and of cardinals. The latter three axiom schemata can be considered as extensions of mathematical induction.In the theory of ordinal numbers thus developed, we shall construct a model of Zermelo-Fraenkel's set theory by following Gödel's construction in [1]. Our intention is as follows: We shall define a relation α ∈ β as a primitive recursive predicate, which corresponds to F′ α ε F′ β in [1]; Gödel defined the constructible model Δ using the primitive notion ε in the universe or, in other words, using the whole set theory.

scholarly journals On the existence of Stone-Čech compactification

2010 ◽  
Vol 75 (4) ◽  
pp. 1137-1146 ◽  
Author(s):  
Giovanni Curi
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Natural Extension ◽  
Formal System ◽  
Topological Spaces ◽  
Completely Regular ◽  
Constructive Type Theory ◽  
Constructive Type ◽  
Constructive Set Theory ◽  
In Virtue Of

Introduction. In 1937 E. Čech and M.H. Stone, independently, introduced the maximal compactification of a completely regular topological space, thereafter called Stone-Čech compactification [8, 23]. In the introduction of [8] the non-constructive character of this result is so described: “It must be emphasized that β(S) [the Stone-Čech compactification of S] may be defined only formally (not constructively) since it exists only in virtue of Zermelo's theorem”.By replacing topological spaces with locales, Banaschewski and Mulvey [4, 5, 6], and Johnstone [14] obtained choice-free intuitionistic proofs of Stone-Čech compactification. Although valid in any topos, these localic constructions rely—essentially, as is to be demonstrated—on highly impredicative principles, and thus cannot be considered as constructive in the sense of the main systems for constructive mathematics, such as Martin-Löf's constructive type theory and Aczel's constructive set theory.In [10] I characterized the locales of which the Stone-Čech compactification can be defined in constructive type theory CTT, and in the formal system CZF+uREA+DC, a natural extension of Aczel's system for constructive set theory CZF by a strengthening of the Regular Extension Axiom REA and the principle of Dependent Choice.

The independence of Ramsey's theorem

1969 ◽  
Vol 34 (2) ◽  
pp. 205-206 ◽  
Author(s):  
E. M. Kleinberg
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Infinite Class ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
The Axiom Of Choice

In [3] F. P. Ramsey proved as a theorem of Zermelo-Fraenkel set theory (ZF) with the Axiom of Choice (AC) the following result:(1) Theorem. Let A be an infinite class. For each integer n and partition {X, Y} of the size n subsets of A, there exists an infinite subclass of A all of whose size n subsets are contained in only one of X or Y.

scholarly journals Towards a dual ontology: duality, a case study

Sofia ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 62-79
Author(s):  
Francesco Maria Ferrari
Keyword(s):  
Set Theory ◽  
Category Theory ◽  
Formal System ◽  
Basic Mechanism ◽  
Spontaneous Breaking ◽  
Theoretic Approach ◽  
Non Linear ◽  
Linear Thermodynamics ◽  
Far From Equilibrium

The main aim of this work is to depict the interconnection of the most relvantformal concepts of modal logic and category theory, i.e., bisimulation andduality, arising from the mathematical analysis of physical processes and toshow their relevance with respect to some foundational issues related to the actual ontological debates. Current foundamental physics concerns the non-linear thermodynamics of the quantum eld, whose range is made of far from equilibrium systems and whose basic mechanism of symmetries (patterns) formation supposes the spontaneous breaking of symmetries (SBS). SBS implies that such systems reach unpredictable states. Thus, evolutive and/or far from equilibrium systems are to be conceived primarily as processes and just in a secondary way as objects, for the information they display is always incomplete with respect to their evolution. Formally, this is due to their non-linear mathematical behaviour.This make a question about the ontology of such systems, given thatthe actual most widespread ontologies conceive existent entities just as objects(actualist ontologies). It is claimed that the fundamental dierence and advantage of category theoretic approach to foundation is that, instead of considering objects and operations for what they 'are', as it is in set theory, in and through category theory we are considering them for what they 'do'. This, of course, would constitute a signicative shifting in mathematical philosophy and in foundationof mathematical physics: from a Platonic to an Aristotelian ontology ofmathematics (and, then, of physics). Actually, providing a contribution to thisvery shift is what this paper want to be focused on. In fact, the implicit pointthe present investigation is concerned with is how to treat the potential innite:the modalization of the existence of each object of the domain of quanticationmeans a potentially innite variation of the domain of quantication. The Aristotelian notion of potentiality diers with the usual one (employed by Platonism and/or formalism and/or conceptualism) inasmuch it does not presupposes any actuality. For instance, it is well known that the Platonic presupposition of set theory consists in the fact "that each potential innite, if it is rigorously applicable mathematically, presupposes an actual innite" [Hallett (1984, p. 25)]. In turn, the formalist notion of (absolute) completeness derives directly from that, if only for the actuality of the information a formal system was intended to dispaly.

Logical reflection and formalism

1958 ◽  
Vol 23 (3) ◽  
pp. 241-249 ◽  
Author(s):  
P. Lorenzen
Keyword(s):  
Nineteenth Century ◽  
Seventeenth Century ◽  
Set Theory ◽  
Basic Concept ◽  
Special Class ◽  
Formal System ◽  
One Step ◽  
Greek Mathematics ◽  
Axiomatic Set Theory ◽  
Modern Mathematics

A “foundational crisis” occurred already in Greek mathematics, brought about by the Pythagorean discovery of incommensurable quantities. It was Eudoxos who provided new foundations, and since then Greek mathematics has been unshakeable. If one reads modern mathematical textbooks, one is normally told that something very similar occurred in modern mathematics. The calculus invented in the seventeenth century had to go through a crisis caused by the use of divergent series. One is told that by the achievements of the nineteenth century from Cauchy to Cantor this crisis has definitely been overcome. It is well known, but it is nevertheless very often not taken seriously into account, that this is an illusion. The so-called ε-δ-definitions of the limit concepts are an admirable achievement, but they are only one step towards the goal of a final foundation of analysis. The nineteenth century solution of the problem of foundations consists of recognizing, in addition to the concept of natural number as the basis of arithmetic, another basic concept for analysis, namely the concept of set. By the inventors of set theory it was strongly held that these sets are self-evident to our intuition; but very soon the belief in their self-evidence was destroyed by the set-theoretic paradoxes. After that, about 1908, the period of axiomatic set theory began. In analogy to geometry there was put forward an uninterpreted system of axioms, a formal system. This, of course, is quite possible. A formal system contains strings of marks; and a special class of these strings, the class of the so-called “theorems”, is inductively defined.

On the axiom of extensionality – Part I

1956 ◽  
Vol 21 (1) ◽  
pp. 36-48 ◽  
Author(s):  
R. O. Gandy
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Formal System ◽  
Free Variable ◽  
Simple Theory ◽  
Predicate Calculus ◽  
Simple System ◽  
System A ◽  
Image Position

In part I of this paper it is shown that if the simple theory of types (with an axiom of infinity) is consistent, then so is the system obtained by adjoining axioms of extensionality; in part II a similar metatheorem for Gödel-Bernays set theory will be proved. The first of these results is of particular interest because type theory without the axioms of extensionality is fundamentally rather a simple system, and it should, I believe, be possible to prove that it is consistent.Let us consider — in some unspecified formal system — a typical expression of the axiom of extensionality; for example:where A(h) is a formula, and A(f), A(g) are the results of substituting in it the predicate variagles f, g for the free variable h. Evidently, if the system considered contains the predicate calculus, and if h occurs in A(h) only in parts of the form h(t) where t is a term which lies within the range of the quantifier (x), then 1.1 will be provable. But this will not be so in general; indeed, by introducing into the system an intensional predicate of predicates we can make 1.1 false. For example, Myhill introduces a constant S, where ‘Sϕψχω’ means that (the expression) ϕ is the result of substituting ψ for χ in ω.

Frege's Theorem and the Peano Postulates

1995 ◽  
Vol 1 (3) ◽  
pp. 317-326 ◽  
Author(s):  
George Boolos
Keyword(s):  
Set Theory ◽  
Cardinal Number ◽  
Formal System ◽  
Second Order ◽  
Order Logic ◽  
Russell’S Paradox ◽  
Russell's Paradox ◽  
Second Order Logic ◽  
Number Of Authors

Two thoughts about the concept of number are incompatible: that any zero or more things have a (cardinal) number, and that any zero or more things have a number (if and) only if they are the members of some one set. It is Russell's paradox that shows the thoughts incompatible: the sets that are not members of themselves cannot be the members of any one set. The thought that any (zero or more) things have a number is Frege's; the thought that things have a number only if they are the members of a set may be Cantor's and is in any case a commonplace of the usual contemporary presentations of the set theory that originated with Cantor and has become ZFC.In recent years a number of authors have examined Frege's accounts of arithmetic with a view to extracting an interesting subtheory from Frege's formal system, whose inconsistency, as is well known, was demonstrated by Russell. These accounts are contained in Frege's formal treatise Grundgesetze der Arithmetik and his earlier exoteric book Die Grundlagen der Arithmetik. We may describe the two central results of the recent re-evaluation of his work in the following way: Let Frege arithmetic be the result of adjoining to full axiomatic second-order logic a suitable formalization of the statement that the Fs and the Gs have the same number if and only if the F sand the Gs are equinumerous.

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