scholarly journals MODAL STRUCTURALISM AND REFLECTION

2018 ◽  
Vol 12 (4) ◽  
pp. 823-860 ◽  
Author(s):  
SAM ROBERTS
Keyword(s):  
Set Theory ◽  
Reflection Principle ◽  
Reflection Principles

AbstractModal structuralism promises an interpretation of set theory that avoids commitment to abstracta. This article investigates its underlying assumptions. In the first part, I start by highlighting some shortcomings of the standard axiomatisation of modal structuralism, and propose a new axiomatisation I call MSST (for Modal Structural Set Theory). The main theorem is that MSST interprets exactly Zermelo set theory plus the claim that every set is in some inaccessible rank of the cumulative hierarchy. In the second part of the article, I look at the prospects for supplementing MSST with a modal structural reflection principle, as suggested in Hellman (2015). I show that Hellman’s principle is inconsistent (Theorem 5.32), and argue that modal structural reflection principles in general are either incompatible with modal structuralism or extremely weak.

scholarly journals A STRONG REFLECTION PRINCIPLE

2017 ◽  
Vol 10 (4) ◽  
pp. 651-662 ◽  
Author(s):  
SAM ROBERTS
Keyword(s):  
Set Theory ◽  
Reflection Principle ◽  
Higher Order ◽  
Second Order ◽  
Proper Class ◽  
Strong Reflection ◽  
Reflection Principles ◽  
Extendible Cardinals ◽  
Order Set

AbstractThis article introduces a new reflection principle. It is based on the idea that whatever is true in all entities of some kind is also true in a set-sized collection of them. Unlike standard reflection principles, it does not re-interpret parameters or predicates. This allows it to be both consistent in all higher-order languages and remarkably strong. For example, I show that in the language of second-order set theory with predicates for a satisfaction relation, it is consistent relative to the existence of a 2-extendible cardinal (Theorem 7.12) and implies the existence of a proper class of 1-extendible cardinals (Theorem 7.9).

On a set theory of bernays

1967 ◽  
Vol 32 (3) ◽  
pp. 319-321 ◽  
Author(s):  
Leslie H. Tharp
Keyword(s):  
Set Theory ◽  
Free Variable ◽  
Reflection Principle ◽  
Formal Definition ◽  
Von Neumann ◽  
Starting Point ◽  
Definition Of ◽  
Image Position

We are concerned here with the set theory given in [1], which we call BL (Bernays-Levy). This theory can be given an elegant syntactical presentation which allows most of the usual axioms to be deduced from the reflection principle. However, it is more convenient here to take the usual Von Neumann-Bernays set theory [3] as a starting point, and to regard BL as arising from the addition of the schema where S is the formal definition of satisfaction (with respect to models which are sets) and ┌φ┐ is the Gödel number of φ which has a single free variable X.

Forcing for the impredicative theory of classes

1972 ◽  
Vol 37 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Rolando Chuaqui
Keyword(s):  
General Theory ◽  
Set Theory ◽  
Continuum Hypothesis ◽  
Reflection Principle ◽  
Inaccessible Cardinal ◽  
Consistency Proof ◽  
Suitable Form ◽  
Definition Of ◽  
Class Construction ◽  
The Continuum

The purpose of this work is to formulate a general theory of forcing with classes and to solve some of the consistency and independence problems for the impredicative theory of classes, that is, the set theory that uses the full schema of class construction, including formulas with quantification over proper classes. This theory is in principle due to A. Morse [9]. The version I am using is based on axioms by A. Tarski and is essentially the same as that presented in [6, pp. 250–281] and [10, pp. 2–11]. For a detailed exposition the reader is referred there. This theory will be referred to as .The reflection principle (see [8]), valid for other forms of set theory, is not provable in . Some form of the reflection principle is essential for the proofs in the original version of forcing introduced by Cohen [2] and the version introduced by Mostowski [10]. The same seems to be true for the Boolean valued models methods due to Scott and Solovay [12]. The only suitable form of forcing for found in the literature is the version that appears in Shoenfield [14]. I believe Vopěnka's methods [15] would also be applicable. The definition of forcing given in the present paper is basically derived from Shoenfield's definition. Shoenfield, however, worked in Zermelo-Fraenkel set theory.I do not know of any proof of the consistency of the continuum hypothesis with assuming only that is consistent. However, if one assumes the existence of an inaccessible cardinal, it is easy to extend Gödel's consistency proof [4] of the axiom of constructibility to .

An intuitionistic theory of types

1998 ◽  
Author(s):  
Per Martin-Löf
Keyword(s):  
Set Theory ◽  
Predicate Logic ◽  
Mathematical Object ◽  
Present Theory ◽  
Reflection Principle ◽  
Simple Type ◽  
Elimination Rule ◽  
Mathematical Objects ◽  
Special Cases ◽  
Intuitionistic Mathematics

The theory of types with which we shall be concerned is intended to be a full scale system for formalizing intuitionistic mathematics as developed, for example, in the book by Bishop 1967. The language of the theory is richer than the language of first order predicate logic. This makes it possible to strengthen the axioms for existence and disjunction. In the case of existence, the possibility of strengthening the usual elimination rule seems first to have been indicated by Howard 1969, whose proposed axioms are special cases of the existential elimination rule of the present theory. Furthermore, there is a reflection principle which links the generation of objects and types and plays somewhat the same role for the present theory as does the replacement axiom for Zermelo-Fraenkel set theory. An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongly impredicative axiom that there is a type of all types whatsoever, which is at the same time a type and an object of that type. This axiom had to be abandoned, however, after it was shown to lead to a contradiction by Jean Yves Girard. I am very grateful to him for showing me his paradox. The change that it necessitated is so drastic that my theory no longer contains intuitionistic simple type theory as it originally did. Instead, its proof theoretic strength should be close to that of predicative analysis. Mathematical objects and their types. We shall think of mathematical objects or constructions. Every mathematical object is of a certain kind or type. Better, a mathematical object is always given together with its type, that is, it is not just an object, it is an object of a certain type.

Iterated reflection principles and the ω-rule

1982 ◽  
Vol 47 (4) ◽  
pp. 721-733 ◽  
Author(s):  
Ulf R. Schmerl
Keyword(s):  
Reflection Principle ◽  
Peano Arithmetic ◽  
Equivalent System ◽  
Formal Similarity ◽  
Sound System ◽  
Formal Systems ◽  
Ordinal Notations ◽  
Axiomatizable Theory ◽  
Reflection Principles ◽  
Image Position

The ω-rule,with the meaning “if the formula A(n) is provable for all n, then the formula ∀xA(x) is provable”, has a certain formal similarity with a uniform reflection principle saying “if A(n) is provable for all n, then ∀xA(x) is true”. There are indeed some hints in the literature that uniform reflection has sometimes been understood as a “formalized ω-rule” (cf. for example S. Feferman [1], G. Kreisel [3], G. H. Müller [7]). This similarity has even another aspect: replacing the induction rule or scheme in Peano arithmetic PA by the ω-rule leads to a complete and sound system PA∞, where each true arithmetical statement is provable. In [2] Feferman showed that an equivalent system can be obtained by erecting on PA a transfinite progression of formal systems PAα based on iterations of the uniform reflection principle according to the following scheme:Then T = (∪dЄ, PAd, being Kleene's system of ordinal notations, is equivalent to PA∞. Of course, T cannot be an axiomatizable theory.

The inconsistency of a certain axiom system for set theory

1972 ◽  
Vol 37 (3) ◽  
pp. 538-542
Author(s):  
James D. Davis
Keyword(s):  
Set Theory ◽  
Axiom System ◽  
Reflection Principle ◽  
Axiom Of Choice ◽  
Standard Notation ◽  
The Axiom Of Choice

We prove in this paper that the -system, an axiom system for set theory suggested for investigation by Takeuti in [2], is inconsistent. We also show that this system without the ω-rule is consistent if Zermelo-Fraenkel set theory with the axiom of choice and an axiom due to Reinhardt and Silver is consistent. The -system is an effort to strengthen Bernays-Gödel set theory by adding a reflection principle.In addition to the standard notation of set theory, we write X″{x) to mean {y∣〈x, y〉 Є X}.

One Absolutely Infinite Universe to Rule Them All

2020 ◽  
pp. 52-86
Author(s):  
Joseph Almog
Keyword(s):  
Set Theory ◽  
Reflection Principle ◽  
Large Cardinals ◽  
Top Down ◽  
Bottom Up

We contrast two Universe-outlooks and universality-sources. The first—localism—runs bottom-up and is in the vein of modern iterative set theory, generating ever more sets but all limited unities and barring an ur-Universe taken as a primary—the prime-object/unity. This contrasts with an absolutely infinite Universe-first outlook, globalism, inspired by some remarks on Cantor but later exiled by Zermelo. The metaphysics is now all top-down, and all sets (e.g. large cardinals) are regarded as generated fragments. The role of the (Godel-central) reflection principle is dissected.?

Reflecting on incompleteness

1991 ◽  
Vol 56 (1) ◽  
pp. 1-49 ◽  
Author(s):  
Solomon Feferman
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Elementary Number Theory ◽  
Finite Generation ◽  
Formal Systems ◽  
Mathematical Thought ◽  
Reflection Principles ◽  
Simple Notion ◽  
Generation Procedure

To what extent can mathematical thought be analyzed in formal terms? Gödel's theorems show the inadequacy of single formal systems for this purpose, except in relatively restricted parts of mathematics. However at the same time they point to the possibility of systematically generating larger and larger systems whose acceptability is implicit in acceptance of the starting theory. The engines for that purpose are what have come to be called reflection principles. These may be iterated into the constructive transfinite, leading to what are called recursive progressions of theories. A number of informative technical results have been obtained about such progressions (cf. Feferman [1962], [1964], [1968] and Kreisel [1958], [1970]). However, for some years I had hoped to give a more realistic and perspicuous finite generation procedure. This was first done in a rather special way in Feferman [1979] for the characterization of predicativity, which may be regarded as that part of mathematical thought implicit in our acceptance of elementary number theory. What is presented here is a new and simple notion of the reflective closure of a schematic theory which can be applied quite generally.Two examples of schematic theories in the sense used here are versions of Peano arithmetic and Zermelo set theory.

Indescribable cardinals and elementary embeddings

1991 ◽  
Vol 56 (2) ◽  
pp. 439-457 ◽  
Author(s):  
Kai Hauser
Keyword(s):  
Set Theory ◽  
Predicate Symbol ◽  
Large Cardinal ◽  
Unary Predicate ◽  
Finite Sequences ◽  
Elementary Embeddings ◽  
Reflection Principles ◽  
The One ◽  
Image Position ◽  
The Universe

Indescribability is closely related to the reflection principles of Zermelo-Fränkel set theory. In this axiomatic setting the universe of all sets stratifies into a natural cumulative hierarchy (Vα: α ϵ On) such that any formula of the language for set theory that holds in the universe already holds in the restricted universe of all sets obtained by some stage.The axioms of ZF prove the existence of many ordinals α such that this reflection scheme holds in the world Vα. Hanf and Scott noticed that one arrives at a large cardinal notion if the reflecting formulas are allowed to contain second order free variables to which one assigns subsets of Vα. For a given collection Ω of formulas in the ϵ language of set theory with higher type variables and a unary predicate symbol they define an ordinal α to be Ω indescribable if for all sentences Φ in Ω and A ⊆ VαSince a sufficient coding apparatus is available, this definition is (for the classes of formulas that we are going to consider) equivalent to the one that one obtains by allowing finite sequences of relations over Vα, some of which are possibly k-ary. We will be interested mainly in certain standardized classes of formulas: Let (, respectively) denote the class of all formulas in the language introduced above whose prenex normal form has n alternating blocks of quantifiers of type m (i.e. (m + 1)th order) starting with ∃ (∀, respectively) and no quantifiers of type greater than m. In Hanf and Scott [1961] it is shown that in ZFC, indescribability is equivalent to inaccessibility and indescribability coincides with weak compactness.

Sign in / Sign up

Export Citation Format

Share Document