An indeterminate universe of sets

A ‘category’, in the mathematical sense, is a universe of structures and transformations. Category theory treats such a universe simply in terms of the network of transformations. For example, categorical set theory deals with the universe of sets and functions without saying what is in any set, or what any function ‘does to’ anything in its domain; it only talks about the patterns of functions that occur between sets. This stress on patterns of functions originally served to clarify certain working techniques in topology. Grothendieck extended those techniques to number theory, in part by defining a kind of category which could itself represent a space. He called such a category a ‘topos’. It turned out that a topos could also be seen as a category rich enough to do all the usual constructions of set-theoretic mathematics, but that may get very different results from standard set theory.

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

10.2307/2275400 ◽

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 ◽

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.

Global quantification in Zermelo-Fraenkel set theory

10.2307/2274215 ◽

1985 ◽

Vol 50 (2) ◽

pp. 289-301

Author(s):

John Mayberry

Keyword(s):

Set Theory ◽

Order Theory ◽

Prima Facie ◽

Local Theory ◽

Axiom Schema ◽

First Order ◽

First Order Theory ◽

Naturally Occurring ◽

Universe Of Sets

My aim here is to investigate the role of global quantifiers—quantifiers ranging over the entire universe of sets—in the formalization of Zermelo-Fraenkel set theory. The use of such quantifiers in the formulas substituted into axiom schemata introduces, at least prima facie, a strong element of impredicativity into the thapry. The axiom schema of replacement provides an example of this. For each instance of that schema enlarges the very domain over which its own global quantifiers vary. The fundamental question at issue is this: How does the employment of these global quantifiers, and the choice of logical principles governing their use, affect the strengths of the axiom schemata in which they occur?I shall attack this question by comparing three quite different formalizations of the intuitive principles which constitute the Zermelo-Fraenkel system. The first of these, local Zermelo-Fraenkel set theory (LZF), is formalized without using global quantifiers. The second, global Zermelo-Fraenkel set theory (GZF), is the extension of the local theory obtained by introducing global quantifiers subject to intuitionistic logical laws, and taking the axiom schema of strong collection (Schema XII, §2) as an additional assumption of the theory. The third system is the conventional formalization of Zermelo-Fraenkel as a classical, first order theory. The local theory, LZF, is already very strong, indeed strong enough to formalize any naturally occurring mathematical argument. I have argued (in [3]) that it is the natural formalization of naive set theory. My intention, therefore, is to use it as a standard against which to measure the strength of each of the other two systems.

A proof-theoretic characterization of the primitive recursive set functions

10.2307/2275441 ◽

1992 ◽

Vol 57 (3) ◽

pp. 954-969 ◽

Cited By ~ 10

Author(s):

Michael Rathjen

Keyword(s):

Set Theory ◽

Elementary Proof ◽

Weak Version ◽

Definable Set ◽

Set Functions ◽

Theoretic Characterization ◽

Universe Of Sets ◽

Set Function ◽

Primitive Recursive

AbstractLet KP− be the theory resulting from Kripke-Platek set theory by restricting Foundation to Set Foundation. Let G: V → V (V ≔ universe of sets) be a Δ0-definable set function, i.e. there is a Δ0-formula φ(x, y) such that φ(x, G(x)) is true for all sets x, and V ⊨ ∀x∃!yφ(x, y). In this paper we shall verify (by elementary proof-theoretic methods) that the collection of set functions primitive recursive in G coincides with the collection of those functions which are Σ1-definable in KP− + Σ1-Foundation + ∀x∃!yφ(x, y). Moreover, we show that this is still true if one adds Π1-Foundation or a weak version of Δ0-Dependent Choices to the latter theory.

FRAGMENTS OF FREGE’SGRUNDGESETZEAND GÖDEL’S CONSTRUCTIBLE UNIVERSE

10.1017/jsl.2015.32 ◽

2016 ◽

Vol 81 (2) ◽

pp. 605-628 ◽

Cited By ~ 1

Author(s):

SEAN WALSH

Keyword(s):

Set Theory ◽

The Other ◽

Structure Theory ◽

Sufficient Condition ◽

Weak Version ◽

Power Set ◽

Consistency Problem ◽

Universe Of Sets ◽

The 19Th Century ◽

Abstraction Principles

AbstractFrege’sGrundgesetzewas one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of theGrundgesetzeformed by restricting the comprehension schema are consistent. One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of theGrundgesetze, and our main theorem (Theorem 2.9) shows that there is a model of a fragment of theGrundgesetzewhich defines a model of all the axioms of Zermelo–Fraenkel set theory with the exception of the power set axiom. The proof of this result appeals to Gödel’s constructible universe of sets and to Kripke and Platek’s idea of the projectum, as well as to a weak version of uniformization (which does not involve knowledge of Jensen’s fine structure theory). The axioms of theGrundgesetzeare examples ofabstraction principles, and the other primary aim of this paper is to articulate a sufficient condition for the consistency of abstraction principles with limited amounts of comprehension (Theorem 3.5). As an application, we resolve an analogue of the joint consistency problem in the predicative setting.

AN ORDINAL ANALYSIS OF ADMISSIBLE SET THEORY USING RECURSION ON ORDINAL NOTATIONS

Journal of Mathematical Logic ◽

10.1142/s0219061302000126 ◽

2002 ◽

Vol 02 (01) ◽

pp. 91-112 ◽

Cited By ~ 1

Author(s):

JEREMY AVIGAD

Keyword(s):

Set Theory ◽

Proof Theory ◽

Admissible Set ◽

Ordinal Analysis ◽

Rate Of Growth ◽

Ordinal Notations ◽

Universe Of Sets ◽

The notion of a function from ℕ to ℕ defined by recursion on ordinal notations is fundamental in proof theory. Here this notion is generalized to functions on the universe of sets, using notations for well orderings longer than the class of ordinals. The generalization is used to bound the rate of growth of any function on the universe of sets that is Σ1-definable in Kripke–Platek admissible set theory with an axiom of infinity. Formalizing the argument provides an ordinal analysis.

On the strength of the Sikorski extension theorem for Boolean algebras

10.2307/2273477 ◽

1983 ◽

Vol 48 (3) ◽

pp. 841-846 ◽

Cited By ~ 6

Author(s):

J.L. Bell

Keyword(s):

Boolean Algebra ◽

Prime Ideal ◽

Extension Theorem ◽

Boolean Algebras ◽

Complete Boolean Algebra ◽

The Axiom Of Choice ◽

Universe Of Sets ◽

The Universe ◽

Boolean Extension ◽

Prime Ideal Theorem

The Sikorski Extension Theorem [6] states that, for any Boolean algebra A and any complete Boolean algebra B, any homomorphism of a subalgebra of A into B can be extended to the whole of A. That is,Inj: Any complete Boolean algebra is injective (in the category of Boolean algebras).The proof of Inj uses the axiom of choice (AC); thus the implication AC → Inj can be proved in Zermelo-Fraenkel set theory (ZF). On the other hand, the Boolean prime ideal theoremBPI: Every Boolean algebra contains a prime ideal (or, equivalently, an ultrafilter)may be equivalently stated as:The two element Boolean algebra 2 is injective,and so the implication Inj → BPI can be proved in ZF.In [3], Luxemburg surmises that this last implication cannot be reversed in ZF. It is the main purpose of this paper to show that this surmise is correct. We shall do this by showing that Inj implies that BPI holds in every Boolean extension of the universe of sets, and then invoking a recent result of Monro [5] to the effect that BPI does not yield this conclusion.

Sets in homotopy type theory

Mathematical Structures in Computer Science ◽

10.1017/s0960129514000553 ◽

2015 ◽

Vol 25 (5) ◽

pp. 1172-1202 ◽

Cited By ~ 4

Author(s):

EGBERT RIJKE ◽

BAS SPITTERS

Keyword(s):

Homotopy Type ◽

Type Theory ◽

Connected Components ◽

Homotopy Type Theory ◽

Universe Of Sets ◽

Univalent Foundations ◽

Subobject Classifier ◽

Theoretical Foundations ◽

Special Case

Homotopy type theory may be seen as an internal language for the ∞-category of weak ∞-groupoids. Moreover, weak ∞-groupoids model the univalence axiom. Voevodsky proposes this (language for) weak ∞-groupoids as a new foundation for Mathematics called the univalent foundations. It includes the sets as weak ∞-groupoids with contractible connected components, and thereby it includes (much of) the traditional set theoretical foundations as a special case. We thus wonder whether those ‘discrete’ groupoids do in fact form a (predicative) topos. More generally, homotopy type theory is conjectured to be the internal language of ‘elementary’ of ∞-toposes. We prove that sets in homotopy type theory form a ΠW-pretopos. This is similar to the fact that the 0-truncation of an ∞-topos is a topos. We show that both a subobject classifier and a 0-object classifier are available for the type theoretical universe of sets. However, both of these are large and moreover the 0-object classifier for sets is a function between 1-types (i.e. groupoids) rather than between sets. Assuming an impredicative propositional resizing rule we may render the subobject classifier small and then we actually obtain a topos of sets.

Categoricity regained

10.1017/s0022481200051203 ◽

1976 ◽

Vol 41 (3) ◽

pp. 639-643 ◽

Cited By ~ 4

Author(s):

Erik Ellentuck

Keyword(s):

Infinite Length ◽

Modern Logic ◽

Mathematical Structures ◽

Similarity Type ◽

First Order ◽

Nonstandard Model ◽

Partial Isomorphism ◽

Order Language ◽

Order Arithmetic ◽

Universe Of Sets

One of the earliest goals of modern logic was to characterize familiar mathematical structures up to isomorphism by means of properties expressed in a first order language. This hope was dashed by Skolem's discovery (cf. [6]) of a nonstandard model of first order arithmetic. A theory T such that any two of its models are isomorphic is called categorical. It is well known that if T has any infinite models then T is not categorical. We shall regain categoricity by(i) enlarging our language so as to allow expressions of infinite length, and(ii) enlarging our class of isomorphisms so as to allow isomorphisms existing in some Boolean valued extension of the universe of sets.Let and be mathematical structures of the same similarity type where say R is binary on A. We write if f is an isomorphism of onto , and if there is an f such that . We say that P is a partial isomorphism of onto and write if P is a nonempty set of functions such that(i) if f ∈ P then dom(f) is a substructure of , rng(/f) is a substructure of , and f is an isomorphism of its domain onto its range, and(ii) if f ∈ P, a ∈ A, and b ∈ B then there exist g,h ∈ P, both extending f such that a ∈ dom(g) and b ∈ rng(h). Write if there is a P such that .

A report on a game on the universe of sets

Mathematical Notes ◽

10.1134/s0001434607050197 ◽

2007 ◽

Vol 81 (5-6) ◽

pp. 716-719 ◽

Cited By ~ 1

Author(s):

D. I. Savel’ev

Keyword(s):

Universe Of Sets ◽