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When I learning forcing theory, I am surprised by various foring definition,I want to know the original purpose, why define partial order in such way and some background material which can help me gain more intuition.

  1. Cohen forcing.It was made by Cohen when he solve CON(ZFC+nonCH),It add many new reals,it is natural to think this if we consider the patial order form,But I heard Cohen used the boolean value model, if condider the boolean value model,how can I understand the intuition?

  2. Random forcing.I just know it made by solovy.But I don't know the background.

  3. Laver forcing.I just know laver forcing used to prove CON(ZFC+BC) by laver.

How about Sacks forcing,Hechler forcing, Mathias forcing,Miller forcing?

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For this question I would suggest instead for the title something like 'Background for classical forcing notions' –  Justin Palumbo Mar 18 '12 at 0:08

3 Answers 3

up vote 8 down vote accepted

The specific notions of forcing you mention are all part of the (now) basic toolbox for getting independence results in set theory at the level of the reals. The standard reference for set theory of this flavor is Bartoszynski and Judah's text "Set theory: on the structure of the real line". If you want to go further in this area (and based on your MO questions so far, perhaps you do) after Kunen's "Set theory: an introduction to independence proofs" this is the book you want to read. It has a wealth of information on the posets you've mentioned.

Each of these forcings is used to add a real of a certain kind to the universe, and in a 'definable way'. There is a unifying view one can take of these forcings: for each there is some ideal $\mathcal{I}$ on the reals so that the forcing is equivalent to forcing with all borel sets not in $\mathcal{I}$ (and the ideal $\mathcal{I}$ has a basis consisting of Borel sets). This perspective on forcing to add reals has been extensively studied with great success by Zapletal and many elegant results characterization properties of the forcing in terms of the relevant ideal have been discovered. See Zapletal's monographs 'Descriptive set theory and definable forcing' or 'Forcing idealized'.

Let me briefly go over the forcings you've mentioned. I'll give the ideal and the original context for the forcing, although each now has many applications for a wide variety of independence results.

Cohen forcing $\mathbb{C}$ was invented by Cohen to produce a model of $\neg\mathrm{CH}$. Of course, it has found many many other uses since then. It is the unique separative countable forcing. The relevant ideal is the collection of meager subsets of $\mathbb{R}$.

Random forcing $\mathbb{B}$ was invented by Solovay. He originally used it to analyze his model where all sets of reals are Lebesgue measurable. The paper "A model of set-theory in which every set of reals is Lebesgue measurable" is still quite readable, though you can also find the proof in Jech or in BJ. The relevant ideal is the collection of null subsets of $\mathbb{R}$.

Sacks forcing $\mathbb{S}$. This was invented by Gerald Sacks in order to produce a minimal real. That is, if $g$ is a Sacks real over the constructible universe $L$, and $x$ is any real in $L[g]$ then either $x\in L$ or $g\in L[x]$. This was done in his paper "Forcing with perfect closed sets". The relevant ideal is the collection of countable subsets of $\mathbb{R}$.

Hechler forcing $\mathbb{D}$ was invented by Stephen Hechler. It is the most basic way of adding a dominating real to the universe, and often it is referred to instead as 'dominating forcing' (including in BJ). Hechler used it produce (consistently) a wide variety of possible cofinal behaviors of the structure $(\omega^\omega,\leq^*)$; this was done in his paper 'On the existence of certain cofinal subsets of $\omega^\omega$'. The relevant ideal was isolated in the paper 'Hechler reals' by Labedzki and Repicky, and is also in BJ.

Laver forcing $\mathbb{L}$ was invented by Richard Laver to produce a model of the Borel conjecture (in his paper "On the consistency of Borel's conjecture", and also in BJ). The ideal is described in Zapletal's monographs.

Mathias forcing $\mathbb{M}$ is due to Adrian Mathias (see: 'Happy families'). He used it to prove (among other things) that the infinite exponent relation $\omega\rightarrow(\omega)^\omega$ holds in the Solovay model mentioned above. For the relevant ideal see either of Zapetal's monographs.

Miller forcing $\mathbb{Q}$ was originally called 'Rational perfect set forcing' by Arnold Miller, and was introduced in his paper with the same name. The paper is on his webpage http://www.math.wisc.edu/~miller/res/rat.pdf where you can read all about it. The ideal is the ideal of $\sigma$-compact subsets of $\omega^\omega$.

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If you want to learn how Cohen invented/discovered forcing, read his 2002 article "The discovery of forcing" (Rocky Mountain journal). Available at Project euclid.

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There is an article by Timothy Chow that gives background motivation and eases you in to forcing called "A beginner's guide to forcing"; it is of an expository nature, but there is still technical detail.

Another article I would recommend would be "A cheerful introduction to forcing and the Continuum Hypothesis" by Kenny Easwaran; it is more technical than the first paper I linked but still very accesible to anyone with basic experience with Set Theory and Model Theory.

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