On the relative consistency of set theory

1956 ◽  
Vol 21 (3) ◽  
pp. 280-290 ◽  
Author(s):  
Steven Orey
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Simple Theory ◽  
Relative Consistency ◽  
System L

We present a method for demonstrating the consistency of von Neuman-Gödel set theory Σ (or Zermelo set theory Σ′) relative to various other formal set theories. When a model for a logic L1 is constructed, there are two other logics involved: L2, the system which contains the model, and L3, the metalogic in which the argument that L2 contains a model for L1 is carried out. These three systems need not of course all be distinct. In [2] L1 is Σ strengthened by the axiom ∨ = L, L2 is Σ, and L3 is unformalized. In the present paper L1 will be either Σ or Σ′. The basic idea for proving the relative consistency of L1 with respect to some other system L2 is to construct in L2 a model for L1 similar to the model constructed in [2]. Since L2 may be some kind of type theory, the function corresponding to Gödel's function F must be modified in a suitable manner. In verifying the relativized versions of axioms of L1 in L2, we shall have to overcome certain special problems depending on the system L2 and not met with in [2]. Except for the remarks at the end of Section 2, we shall not consider the question of formalizing L3, i.e. we use intuitive logic as our metalogic.In Section 1 we illustrate the method by demonstrating the consistency of Σ relative to the system ML″ obtained by adding E of [4] as an axiom to ML′ of [4]. We use the notation and results of [4] and [7].Section 2 discusses the use of certain other systems for L2. It will be noted that the systems used for L2 are always slightly strengthened versions of well known systems ([5], [6], and the simple theory of types); it is of course to be hoped that modifications may be discovered which would make the method work in the original, unstrengthened systems, or that the relative consistency of the strengthened systems with respect to the original systems can be established.

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.

scholarly journals Homotopy type-theoretic interpretations of constructive set theories

2019 ◽  
pp. 1-32
Author(s):  
Cesare Gallozzi
Keyword(s):  
Comparative Analysis ◽  
Set Theory ◽  
Homotopy Type ◽  
Type Theory ◽  
Axiom Of Choice ◽  
Homotopy Type Theory ◽  
Constructive Set Theory

Abstract We introduce a family of (k, h)-interpretations for 2 ≤ k ≤ ∞ and 1 ≤ h ≤ ∞ of constructive set theory into type theory, in which sets and formulas are interpreted as types of homotopy level k and h, respectively. Depending on the values of the parameters k and h, we are able to interpret different theories, like Aczel’s CZF and Myhill’s CST. We also define a proposition-as-hproposition interpretation in the context of logic-enriched type theories. The rest of the paper is devoted to characterising and analysing the interpretations considered. The formulas valid in the prop-as-hprop interpretation are characterised in terms of the axiom of unique choice. We also analyse the interpretations of CST into homotopy type theory, providing a comparative analysis with Aczel’s interpretation. This is done by formulating in a logic-enriched type theory the key principles used in the proofs of the two interpretations. Finally, we characterise a class of sentences valid in the (k, ∞)-interpretations in terms of the ΠΣ axiom of choice.

Resolution in type theory

1971 ◽  
Vol 36 (3) ◽  
pp. 414-432 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Higher Order Logic ◽  
First Order ◽  
Different Types ◽  
Axiomatic Set Theory ◽  
Order Predicate Calculus

In [8] J. A. Robinson introduced a complete refutation procedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand's Theorem, and provides a very convenient framework in which to search for a proof of a wff believed to be a theorem. Moreover, it has proved possible to formulate many refinements of resolution which are still complete but are more efficient, at least in many contexts. However, when efficiency is a prime consideration, the restriction to first order logic is unfortunate, since many statements of mathematics (and other disciplines) can be expressed more simply and naturally in higher order logic than in first order logic. Also, the fact that in higher order logic (as in many-sorted first order logic) there is an explicit syntactic distinction between expressions which denote different types of intuitive objects is of great value where matching is involved, since one is automatically prevented from trying to make certain inappropriate matches. (One may contrast this with the situation in which mathematical statements are expressed in the symbolism of axiomatic set theory.).

Comparison of Russell's resolution of the semantical antinomies with that of Tarski

1976 ◽  
Vol 41 (4) ◽  
pp. 747-760 ◽  
Author(s):  
Alonzo Church
Keyword(s):  
Type Theory ◽  
Simple Theory ◽  
Vicious Circle ◽  
New Members ◽  
Domain Type ◽  
Individual Variables ◽  
The Individual ◽  
Essential Restriction

In this paper we treat the ramified type theory of Russell [6], afterwards adopted by Whitehead and Russell in Principia mathematica [12], so that we may compare Russell's resolution of the semantical antinomies by ramified type theory with the now widely accepted resolution of them by the method of Tarski in [7], [8], [9].To avoid impredicativity the essential restriction is that quantification over any domain (type) must not be allowed to add new members to the domain, as it is held that adding new members changes the meaning of quantification over the domain in such a way that a vicious circle results. As Whitehead and Russell point out, there is no one particular form of the doctrine of types that is indispensable to accomplishing this restriction, and they have themselves offered two different versions of the ramified hierarchy in the first edition of Principia (see Preface, p. vii). The version in §§58–59 of the writer's [1], which will be followed in this paper, is still slightly different.To distinguish Russellian types or types in the sense of the ramified hierarchy from types in the sense of the simple theory of types, let us call the former r-types.There is an r-type i to which the individual variables belong. If β1, β2, …, βm are any given r-types, m ≧ 0, there is an r-type (β1, β2, …, βm)/n to which there belong m-ary functional variables of level n, n ≧ 1. The r-type (α1, α2, …, αm)/k is said to be directly lower than the r-type (β1, β2, …, βm)/n if α1 = β1, α2 = β2, …, αm = βm, k < n.

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.

On undecidable statements in enlarged systems of logic and the concept of truth

1939 ◽  
Vol 4 (3) ◽  
pp. 105-112 ◽  
Author(s):  
Alfred Tarski
Keyword(s):  
Set Theory ◽  
Modus Ponens ◽  
Logical System ◽  
Natural Numbers ◽  
System L ◽  
Concept Of Truth ◽  
Made In

It is my intention in this paper to add somewhat to the observations already made in my earlier publications on the existence of undecidable statements in systems of logic possessing rules of inference of a “non-finitary” (“non-constructive”) character (§§1–4).I also wish to emphasize once more the part played by the concept of truth in relation to problems of this nature (§§5–8).At the end of this paper I shall give a result which was not touched upon in my earlier publications. It seems to be of interest for the reason (among others) that it is an example of a result obtained by a fruitful combination of the method of constructing undecidable statements (due to K. Gödel) with the results obtained in the theory of truth.1. Let us consider a formalized logical system L. Without giving a detailed description of the system we shall assume that it possesses the usual “finitary” (“constructive”) rules of inference, such as the rule of substitution and the rule of detachment (modus ponens), and that among the laws of the system are included all the postulates of the calculus of statements, and finally that the laws of the system suffice for the construction of the arithmetic of natural numbers. Moreover, the system L may be based upon the theory of types and so be the result of some formalization of Principia mathematica. It may also be a system which is independent of any theory of types and resembles Zermelo's set theory.

scholarly journals Topology in the Alternative Set Theory and Rough Sets via Fuzzy Type Theory

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 432 ◽  
Author(s):  
Vilém Novák
Keyword(s):  
Fuzzy Logic ◽  
Set Theory ◽  
Rough Set ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Rough Sets ◽  
Type Theory ◽  
Rough Set Theory ◽  
Basic Concepts ◽  
Alternative Set

In this paper, we will visit Rough Set Theory and the Alternative Set Theory (AST) and elaborate a few selected concepts of them using the means of higher-order fuzzy logic (this is usually called Fuzzy Type Theory). We will show that the basic notions of rough set theory have already been included in AST. Using fuzzy type theory, we generalize basic concepts of rough set theory and the topological concepts of AST to become the concepts of the fuzzy set theory. We will give mostly syntactic proofs of the main properties and relations among all the considered concepts, thus showing that they are universally valid.

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