Forcing

Judith Roitman

Invited Contribution

Forcing was invented by Paul Cohen in the early 1960's in order to prove that the negation of the Axiom of Choice (AC) and that the negation of the Continuum Hypothesis (CH) are consistent with the axioms of set theory. (AC and CH were already known to be consisent). For example, to prove the negation of CH consistent, Cohen started with the transitive collapse N of a countable elementary submodel of the constructible universe L (in which CH holds) and constructed X = { x_i : i < omega₂^N }, a subset of 2^omega, where no element of X is in N. The smallest transitive model N' containing the set X as an element N as a subset is the desired model. The only internal property of X we care about is that x_i = x_j iff i = j. The major thing to check is that for all ordinals i, if N thinks i is a cardinal, so does N'; in particular, omega₂N = omega₂^N', ensuring this is what makes the construction ofX difficult.

The literature on or using forcing in topology is vast, and in this brief note weonly mention a few examples, none recent.

The machinery that makes all this work is profound and complex. The standard reference for the basic machinery is [Kunen]. For more advanced material, the standard reference is [Shelah], although the reader is better off searching out other people's explications.

There are three ways to use forcing.

The first is the brute force approach, in which you construct an object with certain properties, or a model in which there are no objects with certain properties, by forcing what you want directly - either constructing the object, or iteratively destroying any such object. The solution to the Suslin conjecture provides examples of both: Jech and Tennenbaum independently forced the existence of a Suslin tree; Solovay and Tennenbaum iteratively destroyed them. [Jech] is a good reference for this.

The second is to look at a model that is not obviously directly related to the problem - perhaps someone else has already constructed it in another context - and notice that it has the desired attribute. Much of [Miller] uses this approach.

The third is to work from combinatorial principles whose consistency is established by forcing; these combinatorial principles may themselves codify forcing arguments, e.g. Martin's axiom, or the Proper Forcing Axiom. Much of [Todorcevic] uses this approach.

References

[Kunen]

K. Kunen, Set theory: an introduction to independence proofs, North-Holland, New York, 1980

[Jech]

T. Jech, Set theory, Academic Press, New York, 1978

[Miller]

A. Miller, Some properties of measure and category, Trans. Amer. Math. Soc., 266, 93-114

Shelah]

S. Shelah, Proper forcing, Lecture notes in mathematics, 940, Spring-Verlag, New York, 1982

[Todorcevic]

S. Todorcevic, Partition problems in topology, Contemporary Mathematics, 84, AMS, Providence, 1989

University of Kansas, Lawrence, KS, USA 66045
roitman@math.ukans.edu

Date received: May 8, 1996
Date published: November 4, 1999.