Isomorphism of structures in S-toposes

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.

On the strength of the Sikorski extension theorem for Boolean algebras

1983 ◽  
Vol 48 (3) ◽  
pp. 841-846 ◽  
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.

scholarly journals Hunting for Gravitational Quantum Spikes

Universe ◽  
2021 ◽  
Vol 7 (3) ◽  
pp. 49
Author(s):  
Andrzej Góźdź ◽  
Włodzimierz Piechocki ◽  
Grzegorz Plewa ◽  
Tomasz Trześniewski
Keyword(s):  
Initial Data ◽  
Fundamental Importance ◽  
Quantum Structures ◽  
Subtle Effect ◽  
Gravitational Systems ◽  
Numerical Tools ◽  
The Universe

We present the result of our examination of quantum structures called quantum spikes. The classical spikes that are known in gravitational systems, occur in the evolution of the inhomogeneous spacetimes. A different kind of spikes, which we name strange spikes, can be seen in the dynamics of the homogeneous sector of the Belinski–Khalatnikov–Lifshitz scenario. They can be made visible if the so-called inhomogeneous initial data are used. The question to be explored is whether the strange spikes may survive quantization. The answer is in the affirmative. However, this is rather a subtle effect that needs further examination using sophisticated analytical and numerical tools. The spikes seem to be of fundamental importance, both at classical and quantum levels, as they may serve as seeds of real structures in the universe.

Category theory, introduction to

2018 ◽  
Author(s):  
Colin McLarty
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Category Theory ◽  
Standard Set ◽  
Universe Of Sets ◽  
The Universe

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?

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.

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

2002 ◽  
Vol 02 (01) ◽  
pp. 91-112 ◽  
Author(s):  
JEREMY AVIGAD
Keyword(s):  
Set Theory ◽  
Proof Theory ◽  
Admissible Set ◽  
Ordinal Analysis ◽  
Rate Of Growth ◽  
Ordinal Notations ◽  
Universe Of Sets ◽  
The Universe

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.

A report on a game on the universe of sets

2007 ◽  
Vol 81 (5-6) ◽  
pp. 716-719 ◽  
Author(s):  
D. I. Savel’ev
Keyword(s):  
Universe Of Sets ◽  
The Universe

scholarly journals Why is the universe of sets not a set?

Synthese ◽  
2017 ◽  
Vol 197 (2) ◽  
pp. 575-597
Author(s):  
Zeynep Soysal
Keyword(s):  
Universe Of Sets ◽  
The Universe

Some remarks on changing cofinalities

1974 ◽  
Vol 39 (1) ◽  
pp. 27-30 ◽  
Author(s):  
Keith J. Devlin
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Inaccessible Cardinal ◽  
Necessary And Sufficient ◽  
The Universe ◽  
Boolean Extension

AbstractIn [2], Prikry showed that if κ is a weakly inaccessible cardinal which carries a Rowbottom filter, then there is a Boolean extension of V (the universe), having the same cardinals as V, in which cf(κ) = ω. In this note, we obtain necessary and sufficient conditions which a filter D on κ must possess in order that this may be done.

A quasi-intumonistic set theory

1971 ◽  
Vol 36 (3) ◽  
pp. 456-460 ◽  
Author(s):  
Leslie H. Tharp
Keyword(s):  
Set Theory ◽  
Obvious Candidate ◽  
Universe Of Sets ◽  
Basic Philosophy ◽  
The Universe

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.

Sign in / Sign up

Export Citation Format

Share Document