Inner models for set theory – Part III

1953 ◽  
Vol 18 (2) ◽  
pp. 145-167 ◽  
Author(s):  
J. C. Shepherdson
Keyword(s):  
Set Theory ◽  
Original System ◽  
Axiom System ◽  
Complete Model ◽  
Universal Class ◽  
Inner Models ◽  
Consistency Results ◽  
Image Position ◽  
The Universe ◽  
New Hypothesis

In this third and last paper on inner models we consider some of the inherent limitations of the method of using inner models of the type defined in 1.2 for the proof of consistency results for the particular system of set theory under consideration. Roughly speaking this limitation may be described by saying that practically no further consistency results can be obtained by the construction of models satisfying the conditions of theorem 1.5, i.e., conditions 1.31, 1.32, 1.33, 1.51, viz.:This applies in particular to the ‘complete models’ defined in 1.4. Before going on to a precise statement of these limitations we shall consider now the theorem on which they depend. This is concerned with a particular type of complete model examples of which we call “proper complete models”; they are those complete models which are essentially interior to the universe, those whose classes are sets of the universe constituting a class thereof, i.e., those for which the following proposition is true:The main theorem of this paper is that the statement that there are no models of this kind can be expressed formally in the same way as the axioms A, B, C and furthermore it can be proved that if the axiom system A, B, C is consistent then so is the system consisting of axioms A, B, C, plus this new hypothesis that there exist no proper complete models. When combined with the axiom ‘V = L’ introduced by Gödel in (1) this new hypothesis yields a system in which any normal complete model which exists has for its universal class V, the universal class of the original system.

Inner models for set theory—Part II

1952 ◽  
Vol 17 (4) ◽  
pp. 225-237 ◽  
Author(s):  
J. C. Shepherdson
Keyword(s):  
Set Theory ◽  
Complete Model ◽  
Universal Class ◽  
Propositional Function ◽  
Class V ◽  
Completeness Condition ◽  
Inner Models ◽  
Condition C ◽  
Definition Of ◽  
Image Position

In this paper we continue the study of inner models of the type studied inInner models for set theory—Part I.The present paper is concerned exclusively with a particular kind of model, the ‘super-complete models’ defined in section 2.4 of I (page 186). The condition (c) of 2.4 and the completeness condition 1.42 imply that such a model is uniquely determined when its universal class Vmis given. Writing condition (c) and the completeness conditions 1.41, 1.42 in terms of Vm, we may state the definition in the form:3.1. Dfn.A classVmis said to determine a super-complete model if the model whose basic notions are defined by,satisfies axiomsA, B, C.N. B. This definition is not necessarily metamathematical in nature. If desired, it could be written out quite formally as the definition of a notion ‘SCM(U)’ (‘Udetermines a super-complete model’) thus:whereψ(U) is the propositional function expressing in terms ofUthe fact that the model determined byUaccording to 3.1 satisfies the relativization of axioms A, B, C. E.g. corresponding to axiom A1m, i.e.,,ψ(U) contains the equivalent term. All the relativized axioms can be similarly expressed in this way by first writing out the relativized form (after having replaced all defined symbols which occur by the corresponding formulae in primitive notation) and then replacing ‘(Am)ϕ(Am) bywhich is in turn replaced by, and similarly replacing ‘(xm)ϕ(xm)’ by ‘(xm)ϕ(xm)’ by ‘(X)(X ϵ U ▪ ⊃ ▪ ϕ(X)), andThusψ(U) is obtained in primitive notation.

INNER MODEL THEORETIC GEOLOGY

2016 ◽  
Vol 81 (3) ◽  
pp. 972-996 ◽  
Author(s):  
GUNTER FUCHS ◽  
RALF SCHINDLER
Keyword(s):  
Set Theory ◽  
Main Theme ◽  
Solid Core ◽  
Strong Cardinal ◽  
The Third ◽  
Inner Models ◽  
Inner Model ◽  
Basic Concepts ◽  
Woodin Cardinal ◽  
The Universe

AbstractOne of the basic concepts of set theoretic geology is the mantle of a model of set theory V: it is the intersection of all grounds of V, that is, of all inner models M of V such that V is a set-forcing extension of M. The main theme of the present paper is to identify situations in which the mantle turns out to be a fine structural extender model. The first main result is that this is the case when the universe is constructible from a set and there is an inner model with a Woodin cardinal. The second situation like that arises if L[E] is an extender model that is iterable in V but not internally iterable, as guided by P-constructions, L[E] has no strong cardinal, and the extender sequence E is ordinal definable in L[E] and its forcing extensions by collapsing a cutpoint to ω (in an appropriate sense). The third main result concerns the Solid Core of a model of set theory. This is the union of all sets that are constructible from a set of ordinals that cannot be added by set-forcing to an inner model. The main result here is that if there is an inner model with a Woodin cardinal, then the solid core is a fine-structural extender model.

Natural internal forcing schemata extending ZFC: Truth in the universe?

1994 ◽  
Vol 59 (2) ◽  
pp. 461-472
Author(s):  
Garvin Melles
Keyword(s):  
Set Theory ◽  
Mathematical Truth ◽  
Unified Theory ◽  
Rule Of Thumb ◽  
Guiding Principle ◽  
Survey Article ◽  
Inner Models ◽  
Independence Results ◽  
Universe Of Sets ◽  
The Universe

Mathematicians have one over on the physicists in that they already have a unified theory of mathematics, namely, set theory. Unfortunately, the plethora of independence results since the invention of forcing has taken away some of the luster of set theory in the eyes of many mathematicians. Will man's knowledge of mathematical truth be forever limited to those theorems derivable from the standard axioms of set theory, ZFC? This author does not think so, he feels that set theorists' intuition about the universe of sets is stronger than ZFC. Here in this paper, using part of this intuition, we introduce some axiom schemata which we feel are very natural candidates for being considered as part of the axioms of set theory. These schemata assert the existence of many generics over simple inner models. The main purpose of this article is to present arguments for why the assertion of the existence of such generics belongs to the axioms of set theory.Our central guiding principle in justifying the axioms is what Maddy called the rule of thumb maximize in her survey article on the axioms of set theory, [8] and [9]. More specifically, our intuition conforms with that expressed by Mathias in his article What is Maclane Missing? challenging Mac Lane's view of set theory.

Reduction to a dyadic predicate

1954 ◽  
Vol 19 (3) ◽  
pp. 180-182 ◽  
Author(s):  
W. V. Quine
Keyword(s):  
Set Theory ◽  
Predicate Calculus ◽  
Quantification Theory ◽  
Logical Type ◽  
Image Position ◽  
The Universe

Consider any interpreted theory Θ, formulated in the notation of quantification theory (or lower predicate calculus) with interpreted predicate letters. It will be proved that Θ is translatable into a theory, likewise formulated in the notation of quantification theory, in which there is only one predicate letter, and it a dyadic one.Let us assume a fragment of set theory, adequate to assure the existence, for all x and y without regard to logical type, of the set {x, y) whose members are x and y, and to assure the distinctness of x from {x, y} and {{x}}. ({x} is explained as {x, x}.) Let us construe the ordered pair x; y in Kuratowski's fashion, viz. as {{x}, {x, y}}, and then construe x;y;z as x;(y;z), and x;y;z;w as x;(y;z;w), and so on. Let us refer to w, w;w, w;w;w, etc. as 1w, 2w, 3w, etc.Suppose the predicates of Θ are ‘F1’, ‘F2’, …, finite or infinite in number, and respectively d1-adic, d2-adic, …. Now let Θ′ be a theory whose notation consists of that of quantification theory with just the single dyadic predicate ‘F’, interpreted thus:The universe of Θ′ is to comprise all objects of the universe of Θ and, in addition, {x, y) for every x and y in the universe of Θ′. (Of course the universe of Θ may happen already to comprise all this.)Now I shall show how the familiar notations ‘x = y’, ‘x = {y, z}’, etc., and ultimately the desired ‘’, ‘’, etc. themselves can all be defined within Θ′.

Inner models for set theory—Part I

1951 ◽  
Vol 16 (3) ◽  
pp. 161-190 ◽  
Author(s):  
J. C. Shepherdson
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Large Family ◽  
Axiom System ◽  
Axiom Of Choice ◽  
Axiomatic System ◽  
Additional Hypothesis ◽  
Inner Models ◽  
Inner Model ◽  
The Axiom Of Choice

One of the standard ways of proving the consistency of additional hypotheses with the basic axioms of an axiom system is by the construction of what may be described as ‘inner models.’ By starting with a domain of individuals assumed to satisfy the basic axioms an inner model is constructed whose domain of individuals is a certain subset of the original individual domain. If such an inner model can be constructed which satisfies not only the basic axioms but also the particular additional hypothesis under consideration, then this affords a proof that if the basic axiom system is consistent then so is the system obtained by adding to this system the new hypothesis. This method has been applied to axiom systems for set theory by many authors, including v. Neumann (4), Mostowski (5), and more recently Gödel (1), who has shown by this method that if the basic axioms of a certain axiomatic system of set theory are consistent then so is the system obtained by adding to these axioms a strong form of the axiom of choice and the generalised continuum hypothesis. Having been shown in this striking way the power of this method it is natural to inquire whether it has any limitations or whether by the construction of a sufficiently ingenious inner model one might hope to decide other outstanding consistency questions, such as the consistency of the negations of the axiom of choice and continuum hypothesis. In this and two following papers we prove some general theorems concerning inner models for a certain axiomatic system of set theory which lead to the result that as far as a fairly large family of inner models are concerned this method of proving consistency has been exhausted, that no essentially new consistency results can be obtained by the use of this kind of model.

scholarly journals LOGIC IN THE TRACTATUS

2017 ◽  
Vol 10 (1) ◽  
pp. 1-50 ◽  
Author(s):  
MAX WEISS
Keyword(s):  
Set Theory ◽  
Classical Logic ◽  
Logical System ◽  
Image Position ◽  
Countably Infinite ◽  
The Universe

AbstractI present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named.There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably infinite, then the property of being a tautology is $\Pi _1^1$-complete. But third, it is only granted the assumption of countability that the class of tautologies is ${\Sigma _1}$-definable in set theory.Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects.

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.

Term models for weak set theories with a universal set

1987 ◽  
Vol 52 (2) ◽  
pp. 374-387 ◽  
Author(s):  
T. E. Forster
Keyword(s):  
Boolean Algebra ◽  
Set Theory ◽  
Axiom Of Choice ◽  
The Other ◽  
Free Variables ◽  
Image Position ◽  
The Axiom Of Choice ◽  
Axiomatic Systems ◽  
The Universe

We shall be concerned here with weak axiomatic systems of set theory with a universal set. The language in which they are expressed is that of set theory—two primitive predicates, = and ϵ, and no function symbols (though some function symbols will be introduced by definitional abbreviation). All the theories will have stratified axioms only, and they will all have Ext (extensionality: (∀x)(∀y)(x = y· ↔ ·(∀z)(z ϵ x ↔ z ϵ y))). In fact, in addition to extensionality, they have only axioms saying that the universe is closed under certain set-theoretic operations, viz. all of the formand these will always include singleton, i.e., ι′x exists if x does (the iota notation for singleton, due to Russell and Whitehead, is used here to avoid confusion with {x: Φ}, set abstraction), and also x ∪ y, x ∩ y and − x (the complement of x). The system with these axioms is called NF2 in the literature (see [F]). The other axioms we consider will be those giving ⋃x, ⋂x, {y: y ⊆x} and {y: x ⊆ y}. We will frequently have occasion to bear in mind that 〈 V, ⊆ 〉 is a Boolean algebra in any theory extending NF2. There is no use of the axiom of choice at any point in this paper. Since the systems with which we will be concerned exhibit this feature of having, in addition to extensionality, only axioms stating that V is closed under certain operations, we will be very interested in terms of the theories in question. A T-term, for T such a theory, is a thing (with no free variables) built up from V or ∧ by means of the T-operations, which are of course the operations that the axioms of T say the universe is closed under.

scholarly journals Internal Consistency and the Inner Model Hypothesis

2006 ◽  
Vol 12 (4) ◽  
pp. 591-600 ◽  
Author(s):  
Sy-David Friedman
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Weak Sense ◽  
Large Cardinals ◽  
Model Method ◽  
Inner Models ◽  
Inner Model ◽  
The Relationship ◽  
The Universe ◽  
The Way

There are two standard ways to establish consistency in set theory. One is to prove consistency using inner models, in the way that Gödel proved the consistency of GCH using the inner model L. The other is to prove consistency using outer models, in the way that Cohen proved the consistency of the negation of CH by enlarging L to a forcing extension L[G].But we can demand more from the outer model method, and we illustrate this by examining Easton's strengthening of Cohen's result:Theorem 1 (Easton's Theorem). There is a forcing extensionL[G] of L in which GCH fails at every regular cardinal.Assume that the universe V of all sets is rich in the sense that it contains inner models with large cardinals. Then what is the relationship between Easton's model L[G] and V? In particular, are these models compatible, in the sense that they are inner models of a common third model? If not, then the failure of GCH at every regular cardinal is consistent only in a weak sense, as it can only hold in universes which are incompatible with the universe of all sets. Ideally, we would like L[G] to not only be compatible with V, but to be an inner model of V.We say that a statement is internally consistent iff it holds in some inner model, under the assumption that there are innermodels with large cardinals.

The Problem of Conceptual Understanding

2015 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Real Number ◽  
Set Theory ◽  
Conceptual Understanding ◽  
Philosophy Of Mathematics ◽  
Axiom System ◽  
The Real ◽  
Iterative Conception ◽  
Universe Of Sets ◽  
The Universe ◽  
The Web

This chapter considers one of the most intriguing questions that philosophy of mathematics in practice must, sooner or later, confront: how understanding of mathematics is obtained. In particular, it examines how issues of meaning and understanding in relation to practice and use relate to the question of the acceptability of “classical” or postulational mathematics, a question usually formulated in terms of consistency. The chapter begins with a discussion of the iterative conception of the universe of sets and its presuppositions, analyzing it from the standpoint of the web of practices. It then addresses the issue of conceptual understanding in mathematics, as exemplifid by the theory Zermelo–Fraenkel axiom system (ZFC). Finally, it looks at arguments based on the idea of the real-number continuum as a source of justification for the axioms of set theory.

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