forcing

Also known as unramified forcing

In set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand the set theoretical universe to a larger universe by introducing a new "generic" object .

Overview

In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand the set theoretical universe to a larger universe by introducing a new "generic" object . Forcing was first used by Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory. It has been considerably reworked and simplified in the following years, and has since served as a powerful technique, both in set theory and in areas of mathematical logic such as recursion theory. Descriptive set theory uses the notions of forcing from both recursion theory and set theory. Forcing has also been used in model theory, but it is common in model theory to define genericity directly without mention of forcing. Intuition Forcing is usually used to construct an expanded universe that satisfies some desired property. For example, the expanded universe might contain many new real numbers (at least of them), identified with subsets of the set of natural numbers, that were not there in the old universe, and thereby violate the continuum hypothesis. In order to intuitively justify such an expansion, it is best to think of the "old universe" as a model of the set theory, which is itself a set in the "real universe" . By the Löwenheim–Skolem theorem, can be chosen to be a "bare bones" model that is externally countable, which guarantees that there will be many subsets (in ) of that are not in . Specifically, there is an ordinal that "plays the role of the cardinal " in , but is actually countable in . Working in , it should be easy to find one distinct subset of per each element of . (For simplicity, this family of subsets can be characterized with a single subset .)

From Wikipedia (CC BY-SA 4.0).

Also known as unramified forcing

Notable quotes

“Violence … is the last refuge of the incompetent.”
— Isaac Asimov, in Foundation (1951)
“The danger is becoming greater. As the arsenals of the superpowers grow in size and sophistication and as other governments—perhaps even, in the future, dozens of governments—acquire these weapons, it may be only a matter of time before madness, desperation, greed or miscalculation lets loose the terrible force.”
— Jimmy Carter, as quoted in The Watchtower magazine, (15 August 1981)
“I am strongly drawn to the simple life and am often oppressed by the feeling that I am engrossing an unnecessary amount of the labour of my fellow-men. I regard class differences as contrary to justice and, in the last resort, based on force. I also consider that plain living is good for everybody, physically and mentally.”
— Albert Einstein, in Mein Weltbild (1931)
“It seems safe to look forward to the time when the conception of attractive and repulsive forces, having served its purpose as a useful piece of scientific scaffolding, will be replaced by the deduction of the phenomena known as attraction and repulsion, from the general laws of motion.”
— Thomas Henry Huxley, The Advance of Science in the Last Half-Century (1889)
“Who overcomes by force, hath overcome but half his foe.”
— John Milton in Paradise Lost
“Occult historians generally agree that V.V.V.V.V. signified Vi Veri Vniversum Vivus Vici ("By the force of truth I have conquered the universe"), one of the eleven magic mottoes of Aleister Crowley.”
— Robert Anton Wilson, Everything Is Under Control : Conspiracies, Cults, and Cover-Ups (1998), p. 22

Quotes via Wikiquote (CC BY-SA), each with its original source.