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| In mathematics, [[forcing (mathematics)|forcing]] is a method of constructing new models ''M''[''G''] of [[set theory]] by adding a generic subset ''G'' of a [[poset]] ''P'' to a model ''M''. The poset ''P'' used will determine what statements hold in the new universe (the 'extension'); to force a statement of interest thus requires construction of a suitable ''P''. This article lists some of the posets ''P'' that have been used in this construction.
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| ==Notation==
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| *''P'' is a poset with order <.
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| *''V'' is the universe of all sets
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| *''M'' is a countable transitive model of set theory
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| *''G'' is a generic subset of ''P'' over ''M''.
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| ==Definitions==
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| *''P'' satisfies the '''[[countable chain condition]]''' if every antichain in ''P'' is at most countable. This implies that ''V'' and ''V''[''G''] have the same cardinals (and the same cofinalities).
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| *A subset ''D'' of ''P'' is called '''dense''' if for every ''p'' <math>\in</math> ''P'' there is some ''q'' <math>\in</math> ''D'' with ''q'' ≤ ''p''.
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| *A '''filter''' on ''P'' is a nonempty subset ''F'' of ''P'' such that if ''p'' < ''q'' and ''p'' <math>\in</math> ''F'' then ''q'' <math>\in</math> ''F'', and if ''p'' <math>\in</math> ''F'' and ''q'' <math>\in</math> ''F'' then there is some ''r'' <math>\in</math> ''F'' with ''r'' ≤ ''p'' and ''r'' ≤ ''q''.
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| *A subset ''G'' of ''P'' is called '''generic''' over ''M'' if it is a filter that meets every dense subset of ''P'' in ''M''.
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| ==Cohen forcing==
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| In Cohen forcing (named after [[Paul Cohen (mathematician)|Paul Cohen]]) ''P'' is the set of functions from a finite subset of ω<sub>2</sub> × ω to {0,1}
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| and ''p'' < ''q'' if ''p'' <math>\supseteq</math> ''q''.
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| This poset satisfies the countable chain condition. Forcing with this poset adds ω<sub>2</sub> distinct reals to the model; this was the poset used by Cohen in his original proof of the independence of the continuum hypothesis.
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| More generally, one can replace ω<sub>2</sub> by any cardinal κ so construct a model where the continuum has size at least κ. Here, the only restriction is that κ does not have cofinality ω.
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| ==Grigorieff forcing==
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| Grigorieff forcing (after Serge Grigorieff) destroys a free ultrafilter on ω
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| ==Hechler forcing==
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| Hechler forcing (after Stephen Herman Hechler) is used to show that Martin's axiom implies that every family of less than ''c'' functions from ω to ω is eventually dominated by some such function.
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| ''P'' is the set of pairs (''s'',''E'') where ''s'' is a finite sequence of natural numbers (considered as functions from a finite ordinal to ω) and ''E'' is an element of some fixed set ''G'' of functions from ω to ω. The element (''s'', ''E'') is stronger than (''t'',''F'') if ''t'' is contained in ''s'', ''F'' is contained in ''E'', and if ''k'' is in the domain of ''s'' but not of ''t'' then ''s''(''k'')>''h''(''k'') for all ''h'' in ''F''.
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| ==Jockusch–Soare forcing<!--Linked from 'Robert I. Soare' and 'Carl Jockusch'-->==
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| Forcing with <math>\Pi^0_1</math> classes was invented by [[Robert Soare]] and [[Carl Jockusch]] to prove, among other results, the [[low basis theorem]]. Here ''P'' is the set of nonempty <math>\Pi^0_1</math> subsets of [[Cantor space|<math>2^{\omega}</math>]] (meaning the sets of paths through infinite, [[computable function|computable]] [[Tree (descriptive set theory)|subtrees]] of <math>2^{<\omega}</math>), ordered by inclusion.
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| ==Iterated forcing==
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| {{Empty section|date=July 2010}}
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| ==Laver forcing==
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| Laver forcing was used by [[Richard Laver|Laver]] to show that Borel's conjecture that all strong measure zero sets are countable is consistent with ZFC. (Borel's conjecture is not consistent with the continuum hypothesis.)
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| *''P'' is the set of Laver trees, ordered by inclusion.
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| A '''Laver tree''' ''p'' is a subset of the finite sequences of natural numbers such that
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| * ''p'' is a tree: ''p'' contains any initial sequence of any element of ''p''
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| * ''p'' has a stem: a maximal node ''s''(''p'') = ''s'' <math>\in</math> ''p'' such that ''s'' <math>\leq</math> ''t'' or ''t'' <math>\leq</math> ''s'' for all ''t'' in ''p'',
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| *If ''t'' <math>\in</math> ''p'' and ''s'' <math>\leq</math> ''t'' then ''t'' has an infinite number of immediate successors ''tn'' in ''p'' for ''n'' <math>\in</math> ω.
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| If ''G'' is generic for (''P'',≤), then the real {''s''(''p'') : p <math>\in</math> ''G''}, called a ''Laver-real'', uniquely determines ''G''.
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| ==Levy collapsing==
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| These posets will collapse various cardinals, in other words force them to be equal in size to smaller cardinals.
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| *'''Collapsing a cardinal to ω:''' ''P'' is the set of all finite sequences of ordinals less than a given cardinal λ. If λ is uncountable then forcing with this poset collapses λ to ω.
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| *'''Collapsing a cardinal to another:''' ''P'' is the set of all functions from a subset of κ of cardinality less than κ to λ (for fixed cardinals κ and λ). Forcing with this poset collapses λ down to κ.
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| *'''Levy collapsing:''' If κ is regular and λ is inaccessible, then ''P'' is the set of functions ''p'' on subsets of λ× κ with domain of size less than κ and ''p''(α,ξ)<α for every (α,ξ) in the domain of ''p''. This poset collapses all cardinals less than λ onto κ, but keeps λ as the successor to κ.
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| Levy collapsing is named for [[Azriel Levy]].
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| ==Magidor forcing==
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| Amongst many forcing notions developed by [[Menachem Magidor|Magidor]], one of the best known is a generalization of Prikry forcing used to change the cofinality of a cardinal to a given smaller regular cardinal.
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| ==Mathias forcing==
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| *An element of ''P'' is a pair consisting of a finite set ''s'' of natural numbers and an infinite set ''A'' of natural numbers such that every element of ''s'' is less than every element of ''A''. The order is defined by (''s'', ''A'') < (''t'',''B'') if ''t'' is an initial segment of ''s'', ''A'' is a subset of ''B'', and ''s'' is contained in ''t'' <math>\cup</math> ''B''.
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| Mathias forcing is named for Adrian Richard David Mathias.
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| ==Namba forcing==
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| Namba forcing (after Kanji Namba) is used to change the cofinality of ω<sub>2</sub> to ω without collapsing ω<sub>1</sub>.
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| *''P'' is the set of perfect trees in the set of finite sequences of ordinals less than ω<sub>2</sub>. ''P'' is ordered by inclusion.
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| ==Prikry forcing==
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| In Prikry forcing (after Karel Prikry) ''P'' is the set of pairs (''s'',''A'') where ''s'' is a finite subset of a fixed measurable cardinal κ, and ''A'' is an element of a fixed normal measure ''D'' on κ. A condition (''s'',''A'') is stronger than (''t'', ''B'') if ''t'' is an initial segment of ''s'', ''A'' is contained in ''B'', and ''s'' is contained in ''t'' <math>\cup</math> ''B''. This forcing notion can be used to change to cofinality of κ while preserving all cardinals.
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| ==Product forcing==
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| Taking a product of forcing conditions is a way of simultaneously forcing all the conditions.
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| *'''Finite products''': If ''P'' and ''Q'' are posets, the product poset ''P''× ''Q'' has the partial order defined by (''p''<sub>1</sub>, ''q''<sub>1</sub>) ≤ (''p''<sub>2</sub>, ''q''<sub>2</sub>) if ''p''<sub>1</sub> ≤ ''p''<sub>2</sub> and ''q''<sub>1</sub> ≤ ''q''<sub>2</sub>.
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| *'''Infinite products''': The product of a set of posets ''P''<sub>''i''</sub>, ''i'' <math>\in</math> ''I'', each with a largest element 1 is the set of functions ''p'' on ''I'' with ''p''(''i'') <math>\in</math> ''P''(''i'') and such that ''p''(''i'') = 1 for all but a finite number of ''i''. The order is given by ''p'' ≤ ''q'' if ''p''(''i'') ≤ ''q''(''i'') for all ''i''.
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| *The '''Easton product''' (after William Bigelow Easton) of a set of posets ''P''<sub>''i''</sub>, ''i'' <math>\in</math> ''I'', where ''I'' is a set of cardinals is the set of functions ''p'' on ''I'' with ''p''(''i'') <math>\in</math> ''P''(''i'') and such that for every regular cardinal γ the number of elements α of γ with ''p''(α) ≠ 1 is less than γ.
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| ==Radin forcing==
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| Radin forcing (after Lon Berk Radin), a technically involved generalization of Magidor forcing, adds a closed, unbounded subset to some regular cardinal λ.
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| If λ is a sufficiently large cardinal, then the forcing keeps λ regular, [[measurable cardinal|measurable]], [[supercompact cardinal|supercompact]], etc.
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| ==Random forcing==
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| *''P'' is the set of Borel subsets of [0,1] of positive measure, where ''p'' is called stronger than ''q'' if it is contained in ''q''. The generic set ''G'' then encodes a "random real": the unique real ''x''<sub>''G''</sub> in all rational intervals [''r'',''s'']<sup>''V''[''G'']</sup> such that [''r'',''s'']<sup>''V''</sup> is in ''G''. This real is "random" in the sense that if ''X'' is any subset of [0,1]<sup>''V''</sup> of measure 1, lying in ''V'', then ''x''<sub>''G''</sub> ∈ ''X''.
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| ==Sacks forcing==
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| *''P'' is the set of all perfect trees contained in the set of finite {0,1} sequences. (A tree ''T'' is a set of finite sequences containing all initial segments of its members, and is called perfect if for any element ''t'' of ''T'' there is a tree ''s'' containing it so that both ''s''0 and ''s''1 are in ''T''.) A tree ''p'' is stronger than ''q'' if ''p'' is contained in ''q''. Forcing with perfect trees was used by [[Gerald Sacks|Gerald Enoch Sacks]] to produce a real ''a'' with minimal degree of constructibility.
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| ==Shooting a fast club==
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| For ''S'' a stationary subset of <math>\omega_1</math> we set <math>P=\{\langle \sigma,
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| C\rangle\,\colon\sigma</math> is
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| a closed sequence from ''S'' and ''C'' is a closed unbounded subset of
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| <math>\omega_1\}</math>, ordered by <math>\langle \sigma',C'\rangle\leq\langle\sigma,
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| C\rangle</math> iff <math>\sigma'</math> end-extends <math>\sigma</math> and <math>C'\subseteq
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| C</math> and <math>\sigma'\subseteq\sigma\cup C</math>. In <math>V[G]</math>, we have that <math>\bigcup\{\sigma\,
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| \colon(\exists C)(\langle\sigma,C\rangle\in
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| G)\}</math> is a closed unbounded subset of ''S'' almost contained in each club set in ''V''. <math>\aleph_1</math> is preserved.
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| ==Shooting a club with countable conditions==
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| For ''S'' a stationary subset of <math>\omega_1</math> we set ''P'' equal to the set of closed countable sequences from ''S''. In <math>V[G]</math>, we have that <math>\bigcup G</math> is a closed unbounded subset of ''S'' and <math>
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| \aleph_1</math> is preserved, and if CH holds then all cardinals are preserved.
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| ==Shooting a club with finite conditions==
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| For ''S'' a stationary subset of <math>\omega_1</math> we set ''P'' equal to the set of finite sets of pairs of countable ordinals, such that if <math>p\in P</math> and <math>\langle\alpha,\beta\rangle\in p</math> then <math>\alpha\leq\beta</math>
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| and <math>\alpha\in S</math>, and whenever <math>\langle\alpha,
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| \beta\rangle</math> and <math>\langle\gamma,\delta\rangle</math> are distinct elements of ''p'' then either
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| <math>\beta<\gamma</math> or <math>\delta<\alpha</math>. ''P'' is ordered by reverse inclusion.
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| In <math>V[G]</math>, we have that <math>\{\alpha\,\colon(\exists\beta)(\langle
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| \alpha,\beta\rangle\in\bigcup G)\}</math> is a closed unbounded subset of ''S'' and all cardinals are preserved.
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| ==Silver forcing==
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| Silver forcing (after [[Jack Silver|Jack Howard Silver]]) satisfies Fusion, the Sacks property, and
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| is minimal with respect to reals (but not minimal).
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| ==References==
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| *{{Citation | last1=Jech | first1=Thomas | author1-link=Thomas Jech | title=Set Theory: Millennium Edition | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-44085-7 | year=2003}}
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| *{{Citation | last1=Kunen | first1=Kenneth | author1-link=Kenneth Kunen | title=[[Set Theory: An Introduction to Independence Proofs]] | publisher=Elsevier | isbn=978-0-444-86839-8 | year=1980}}
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| ==External links==
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| *A.Miller (2009), [http://www.math.wisc.edu/~miller/old/m873-09/forcing.pdf ''Forcing Tidbits.'']
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| [[Category:Forcing (mathematics)]]
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