Concept
Forcing (mathematics)
Overview
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
V
{\displaystyle V}
to a larger universe
V
[
G
]
{\displaystyle V[G]}
by introducing a new "generic" object
G
{\displaystyle G}
.
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 computability theory. Descriptive set theory uses the notions of forcing from both computability 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.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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Such a resultant force, as it is called, may be found for any pair of forces throughout the group. Similarly, for any pair of such resultants a single resultant may be found. And by repeating this course, all of them may be reduced to two.
Herbert Spencer, First Principles
States what Forcing is.
There are nevertheless, in mathematics, some examples of so-called Induction, in which the conclusion does bear the appearance of a generalization grounded on some of the particular cases included in it. A mathematician, when he has calculated a sufficient number of the terms of an algebraical or arithmetical series to have ascertained what is called the _law_ of the series, does not hesitate to fill up any number…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Forcing is.
Our Sages have already stated—for him who has understanding—that all forces that reside in a body are angels, much more the forces that are active in the Universe. The theory that each force acts only in one particular way, is expressed in Bereshit Rabba (chap. l.) as follows: “One angel does not perform two things, and two angels do not perform one thing”; this is exactly the property of all forces. We may find a…
Moses Maimonides, The guide for the perplexed
States what Forcing is.
Philosophers who discussed Forcing (mathematics)
Related ideas
- First-order logic
- Skolem's paradox
- Infinite set
- If and only if
- Compactness theorem
- Zermelo–Fraenkel set theory
- Computability theory
- Truth value
- Dana Scott
- Mathematical logic
- Model theory
- Partially ordered set
- Power set
- Set theory
- Subset
- Cantor's theorem
- Löwenheim–Skolem theorem
- Bijection
- Boolean algebra (structure)
- Structure (mathematical logic)
- Dedekind cut
- Continuum hypothesis
- Gödel's incompleteness theorems
- Cardinal number
Place in the tradition
Part of Outline of logic.