SetTheory is the mathematical science of the infinite.
The language of settheory, in its simplicity, is sufficiently universal to formalize all mathematical concepts and thus settheory, along with Predicate Calculus, constitutes the true Foundations of Mathematics.
The aim of this entry is to present a list of the key objects and concepts used in settheory.
The reader should take care that if the objects under discussion are not just sets (say, groups or schemes) the operations may not be simple set operations, but rather their analogue in the relevant category.
This is version 47 of concepts in settheory, born on 2004-02-29, modified 2006-09-09.
The language of settheory, in its simplicity, is sufficiently universal to formalize all mathematical concepts and thus settheory, along with Predicate Calculus, constitutes the true Foundations of Mathematics.
There are four main directions of current research in settheory, all intertwined and all aiming at the ultimate goal of the theory: to describe the structure of the mathematical universe.
For instance, it is desirable to have the “set of all integers that are divisible by number 3,” the “set of all straight lines in the Euclidean plane that are parallel to a given line”, the “set of all continuous real functions of two real variables” etc.
Readers unfamiliar with model theory may find the glossary in appendix B helpful; throughout the text, uses of terms in a technical sense are linked to their glossary definitions.
If the graphs in the set have no blank nodes in common, then the union of the graphs is a merge; if they do share blank nodes, then it is the union of a set of graphs that is obtained by replacing the graphs in the set by equivalent graphs that share no blank nodes.
Any of the graphs may have some triples added to it; the set of graphs may be extended by extra graphs; or the vocabulary of the graph may be interpreted relative to a stronger notion of vocabulary entailment, i.e.
The Singleton should not be seen as way to implement global variables in the Java programming language; rather, along the lines of the factory design patterns, the Singleton lets you encapsulate and control the creation process by making sure that certain prerequisites are fulfilled or by creating the object lazily on demand.
Singletons that do not hold state but simply control access to resources are also not appropriate for EJBs, since resource management is the role of the EJB container.
The lock is grabbed only if the Singleton instance does not exist, and then the existence of the instance is checked again in case another thread passed the first check an instant before the current thread.
When a singularity is predicted by or elicited from any physical theory, the logical inference to draw is that the theory fails in the locus of the singularity: singularities are not real and physical things, but rather, mathematical things.
Quantum theory has an essential linear feature, superposition, the idea that the state of the system is generally representable by a complex linear combination of mutually exclusive alternatives.
In dealing with a conjectured "quantum settheory", the fundamental concepts leading to classical settheory may have to be eliminated replaced or augmented.
Then [in ZF settheory] you can apply diagonalization to create a new set that can't be in the power set: for every A in the SetOfAllSets, put A in the new set iff A is not in the subset it is matched with.
The problem is not with the set of all sets; rather, it is with the set of all sets that do not contain themselves, and the question as to whether this set contains itself.
In axiomatic formulations of (now so-called "naive") settheory, there used to be a set-construction axiom to the effect that, for any proposition there was a set of all the objects satisfying that proposition.
The intersection of two sets is the "overlap", i.e., the part of the two sets which is common to both.
Often we are interested in relationships between two sets of objects in which the choice of an object in one set completely and unambiguously determines the related object in the second set.
Specifically, he considered the set of all sets that are not elements of themselves, i.e., the set of all collections that are not self-contained.
I'll now write []p to be the sentence of settheory (which is in fact expressible standardly) that says that p is true in every generic extension of the actual universe, and <>p the sentence that says that p is true in some generic extension.
All the settheory books seem to either just do basic stuff with ordinals and cardinals (excluding forcing and large cardinals and determinacy and the like) or put forcing at least 100 or 200 pages in and rely on a lot of the material that's discussed in earlier chapters.
I suppose that here by "settheory" he means to refer to the collection of all true first-order statements in the language of settheory (working from a platonistic perspective).
We have to be able to implement the interface in such a way that this sort of misuse won't result in tests silently doing nothing.
I was playing around with some unit tests to explore Ruby singleton classes on a recent flight when I decided they could be fleshed out into a pretty nice introduction to some basics of Ruby metaprogramming for those who haven't done much with Ruby (which probably includes most people who've only written Rails applications).
singleton class instances of instances of that class.
Singleton, singleton : Encarta® World English Dictionary, North American Edition [home, info]
Phrases that include singleton: john singleton mosby, hoffman singleton graph, lyndhurst john singleton copley baron, singletonset, alexander singleton, more...
Words similar to singleton: one, oner, unit, more...
Some modern theories see the world as being in some sense a continuum and have abandoned any idea that the world is a set of atoms moving through the void.
The primitive language of settheory, a complete description of which has just been given, is adequate for the expression of an enormous number of notions about sets and classes of sets.
The class term 'the class of sets x such that the universe is included in x' I abbreviate as 'the empty class' (or 'f').
Thus the set that the Axiom of Infinity declares to exist is the set:
The idea is that if P(x,y) corresponds to a function from members of a given set to objects, then we can form a new set by replacing every member of the given set with the object the function maps it to.
A somewhat unsettling set.) But it violates the Axiom of Regularity, since its intersection with its only member is not empty.
The Singleton pattern is a solution to the desire to have only one object of a class instantiated during the lifetime of your application.
Though design patterns were invented because solutions were often not reusable on the code level, the singleton pattern actually does have a reusable code version.
We are going to look at a way to generalize the singleton implementation and how we can use the generalization to deal with construction/destruction sequence problems.
