Forcing and the Universe of Sets: Must We Lose Insight?

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.

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.

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 ◽

Why is the universe of sets not a set?

Synthese ◽

10.1007/s11229-017-1513-x ◽

2017 ◽

Vol 197 (2) ◽

pp. 575-597

Author(s):

Zeynep Soysal

Keyword(s):

Universe Of Sets ◽

A quasi-intumonistic set theory

10.2307/2269954 ◽

1971 ◽

Vol 36 (3) ◽

pp. 456-460 ◽

Cited By ~ 8

Author(s):

Leslie H. Tharp

Keyword(s):

Set Theory ◽

Obvious Candidate ◽

Universe Of Sets ◽

Basic Philosophy ◽

It is natural, given the usual iterative description of the universe of sets, to investigate set theories which in some way take account of the unfinished character of the universe. We do not here consider any arguments aimed at justifying one system over another, or at clarifying the basic philosophy. Rather, we look at an obvious candidate which is similar to a system discussed by L. Pozsgay in [1]. Pozsgay sketched the development of the ordinary theorems in such a system and attempted to show it equiconsistent with ZF. In this paper we show that the consistency of the system we call IZF can be proved in the usual ZF set theory.

Forcing isomorphism

10.2307/2275144 ◽

1993 ◽

Vol 58 (4) ◽

pp. 1291-1301 ◽

Cited By ~ 1

Author(s):

J. T. Baldwin ◽

M. C. Laskowski ◽

S. Shelah

Keyword(s):

Partially Ordered Set ◽

Ordered Set ◽

Order Theory ◽

Dependence Structure ◽

Order Property ◽

First Order ◽

First Order Theory ◽

Omitting Types ◽

Universe Of Sets ◽

If two models of a first-order theory are isomorphic, then they remain isomorphic in any forcing extension of the universe of sets. In general however, such a forcing extension may create new isomorphisms. For example, any forcing that collapses cardinals may easily make formerly nonisomorphic models isomorphic. However, if we place restrictions on the partially-ordered set to ensure that the forcing extension preserves certain invariants, then the ability to force nonisomorphic models of some theory T to be isomorphic implies that the invariants are not sufficient to characterize the models of T.A countable first-order theory is said to be classifiable if it is superstable and does not have either the dimensional order property (DOP) or the omitting types order property (OTOP). If T is not classifiable, Shelah has shown in [5] that sentences in L∞,λ do not characterize models of T of power λ. By contrast, in [8] Shelah showed that if a theory T is classifiable, then each model of cardinality λ is described by a sentence of L∞,λ. In fact, this sentence can be chosen in the . ( is the result of enriching the language by adding for each μ < λ a quantifier saying the dimension of a dependence structure is greater than μ) Further work ([3], [2]) shows that ⊐+ can be replaced by ℵ1.

Isomorphism of structures in S-toposes

10.2307/2273748 ◽

1981 ◽

Vol 46 (3) ◽

pp. 449-459

Author(s):

J. L. Bell

Keyword(s):

Fundamental Importance ◽

General Context ◽

Universe Of Sets ◽

The Universe ◽

Boolean Extension

It is a well-known fact that two structures are ∞ω-equivalent if and only if they are isomorphic in some Boolean extension of the universe of sets (cf. [4]; an early allusion to this result appears in [8]). My principal object here is to show that arbitrary toposes defined over the category of sets may be used instead. Thus ∞ω-equivalence means isomorphism in the extremely general context of some universe of "variable" sets in which not only is much of the usual set-theoretic machinery unavailable but the underlying logic is not even classical. This provides further support for the view that ∞ω-equivalence is a relation between structures of fundamental importance.

Omega-inconsistency and the Universe of Sets

10.1063/1.3037086 ◽

2008 ◽

Author(s):

Willard Mittelman ◽

Theodore E. Simos ◽

George Psihoyios

Keyword(s):

Universe Of Sets ◽