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In 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.

Notation

P is a poset with order <.
V is the universe of all sets
M is a countable transitive model of set theory
G is a generic subset of P over M.

Definitions

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).
A subset D of P is called dense if for every p $\in$ P there is some q $\in$ D with q ≤ p.
A filter on P is a nonempty subset F of P such that if p < q and p $\in$ F then q $\in$ F, and if p $\in$ F and q $\in$ F then there is some r $\in$ F with r ≤ p and r ≤ q.
A subset G of P is called generic over M if it is a filter that meets every dense subset of P in M.

Cohen forcing

In Cohen forcing (named after Paul Cohen) P is the set of functions from a finite subset of ω₂ × ω to {0,1} and p < q if p $\supseteq$ q.

This poset satisfies the countable chain condition. Forcing with this poset adds ω₂ distinct reals to the model; this was the poset used by Cohen in his original proof of the independence of the continuum hypothesis.

More generally, one can replace ω₂ 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.

Grigorieff forcing

Grigorieff forcing (after Serge Grigorieff) destroys a free ultrafilter on ω

Hechler forcing

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.

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.

Jockusch–Soare forcing

Forcing with $Π_{1}^{0}$ 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 $Π_{1}^{0}$ subsets of $2^{ω}$ (meaning the sets of paths through infinite, computable subtrees of $2^{< ω}$ ), ordered by inclusion.

Iterated forcing

Template:Empty section

Laver forcing

Laver forcing was used by 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.)

P is the set of Laver trees, ordered by inclusion.

A Laver tree p is a subset of the finite sequences of natural numbers such that

p is a tree: p contains any initial sequence of any element of p
p has a stem: a maximal node s(p) = s $\in$ p such that s $\leq$ t or t $\leq$ s for all t in p,
If t $\in$ p and s $\leq$ t then t has an infinite number of immediate successors tn in p for n $\in$ ω.

If G is generic for (P,≤), then the real {s(p) : p $\in$ G}, called a Laver-real, uniquely determines G.

Levy collapsing

These posets will collapse various cardinals, in other words force them to be equal in size to smaller cardinals.

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.
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.
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.

Levy collapsing is named for Azriel Levy.

Magidor forcing

Amongst many forcing notions developed by 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.

Mathias forcing

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 $\cup$ B.

Mathias forcing is named for Adrian Richard David Mathias.

Namba forcing

Namba forcing (after Kanji Namba) is used to change the cofinality of ω₂ to ω without collapsing ω₁.

P is the set of perfect trees in the set of finite sequences of ordinals less than ω₂. P is ordered by inclusion.

Prikry forcing

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 $\cup$ B. This forcing notion can be used to change to cofinality of κ while preserving all cardinals.

Product forcing

Taking a product of forcing conditions is a way of simultaneously forcing all the conditions.

Finite products: If P and Q are posets, the product poset P× Q has the partial order defined by (p₁, q₁) ≤ (p₂, q₂) if p₁ ≤ p₂ and q₁ ≤ q₂.
Infinite products: The product of a set of posets P_i, i $\in$ I, each with a largest element 1 is the set of functions p on I with p(i) $\in$ 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.
The Easton product (after William Bigelow Easton) of a set of posets P_i, i $\in$ I, where I is a set of cardinals is the set of functions p on I with p(i) $\in$ 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.

Radin forcing

Radin forcing (after Lon Berk Radin), a technically involved generalization of Magidor forcing, adds a closed, unbounded subset to some regular cardinal λ.

If λ is a sufficiently large cardinal, then the forcing keeps λ regular, measurable, supercompact, etc.

Random forcing

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_G in all rational intervals [r,s]^V[G] such that [r,s]^V is in G. This real is "random" in the sense that if X is any subset of [0,1]^V of measure 1, lying in V, then x_G ∈ X.

Sacks forcing

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 s0 and s1 are in T.) A tree p is stronger than q if p is contained in q. Forcing with perfect trees was used by Gerald Enoch Sacks to produce a real a with minimal degree of constructibility.

Shooting a fast club

For S a stationary subset of $ω_{1}$ we set $P = {⟨ σ, C ⟩ : σ$ is a closed sequence from S and C is a closed unbounded subset of $ω_{1}}$ , ordered by $⟨ σ^{'}, C^{'} ⟩ \leq ⟨ σ, C ⟩$ iff $σ^{'}$ end-extends $σ$ and $C^{'} \subseteq C$ and $σ^{'} \subseteq σ \cup C$ . In $V [G]$ , we have that $⋃ {σ : (\exists C) (⟨ σ, C ⟩ \in G)}$ is a closed unbounded subset of S almost contained in each club set in V. $ℵ_{1}$ is preserved.

Shooting a club with countable conditions

For S a stationary subset of $ω_{1}$ we set P equal to the set of closed countable sequences from S. In $V [G]$ , we have that $⋃ G$ is a closed unbounded subset of S and $ℵ_{1}$ is preserved, and if CH holds then all cardinals are preserved.

Shooting a club with finite conditions

For S a stationary subset of $ω_{1}$ we set P equal to the set of finite sets of pairs of countable ordinals, such that if $p \in P$ and $⟨ α, β ⟩ \in p$ then $α \leq β$ and $α \in S$ , and whenever $⟨ α, β ⟩$ and $⟨ γ, δ ⟩$ are distinct elements of p then either $β < γ$ or $δ < α$ . P is ordered by reverse inclusion. In $V [G]$ , we have that ${α : (\exists β) (⟨ α, β ⟩ \in ⋃ G)}$ is a closed unbounded subset of S and all cardinals are preserved.

Silver forcing

Silver forcing (after Jack Howard Silver) satisfies Fusion, the Sacks property, and is minimal with respect to reals (but not minimal).

References

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To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

External links

A.Miller (2009), Forcing Tidbits.

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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.
+==Notation==
+*''P'' is a poset with order <.
+*''V'' is the universe of all sets
+*''M'' is a countable transitive model of set theory
+*''G'' is a generic subset of ''P'' over ''M''.
+==Definitions==
+*''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).
+*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''.
+*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''.
+*A subset ''G'' of ''P'' is called '''generic''' over ''M'' if it is a filter that meets every dense subset of ''P'' in ''M''.
+==Cohen forcing==
+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> &times; ω to {0,1}
+and ''p'' < ''q'' if ''p'' <math>\supseteq</math> ''q''.
+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.
+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.
+==Grigorieff forcing==
+Grigorieff forcing (after Serge Grigorieff) destroys a free ultrafilter on &omega;
+==Hechler forcing==
+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.
+''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''.
+==Jockusch–Soare forcing<!--Linked from 'Robert I. Soare' and 'Carl Jockusch'-->==
+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.
+==Iterated forcing==
+{{Empty section|date=July 2010}}
+==Laver forcing==
+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.)
+*''P'' is the set of Laver trees, ordered by inclusion.
+A '''Laver tree''' ''p'' is a subset of the finite sequences of natural numbers such that
+* ''p'' is a tree: ''p'' contains any initial sequence of any element of ''p''
+* ''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'',
+*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.
+If ''G'' is generic for (''P'',≤), then the real {''s''(''p'') : p <math>\in</math> ''G''}, called a ''Laver-real'', uniquely determines ''G''.
+==Levy collapsing==
+These posets will collapse various cardinals, in other words force them to be equal in size to smaller cardinals.
+*'''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.
+*'''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.
+*'''Levy collapsing:''' If κ is regular and λ is inaccessible, then ''P'' is the set of functions ''p'' on subsets of λ&times; 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.
+Levy collapsing is named for [[Azriel Levy]].
+==Magidor forcing==
+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.
+==Mathias forcing==
+*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''.
+Mathias forcing is named for Adrian Richard David Mathias.
+==Namba forcing==
+Namba forcing (after Kanji Namba) is used to change the cofinality of ω<sub>2</sub> to ω without collapsing ω<sub>1</sub>.
+*''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.
+==Prikry forcing==
+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.
+==Product forcing==
+Taking a product of forcing conditions is a way of simultaneously forcing all the conditions.
+*'''Finite products''': If ''P'' and ''Q'' are posets, the product poset ''P''&times; ''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>.
+*'''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''.
+*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.
+==Radin forcing==
+Radin forcing (after Lon Berk Radin), a technically involved generalization of Magidor forcing, adds a closed, unbounded subset to some regular cardinal  λ.
+If λ is a sufficiently large cardinal, then the forcing keeps  λ regular, [[measurable cardinal|measurable]], [[supercompact cardinal|supercompact]],  etc.
+==Random forcing==
+*''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''.
+==Sacks forcing==
+*''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.
+==Shooting a fast club==
+For ''S'' a stationary subset of <math>\omega_1</math> we set <math>P=\{\langle \sigma,
+ C\rangle\,\colon\sigma</math> is
+a closed sequence from ''S'' and ''C'' is a closed unbounded subset of
+<math>\omega_1\}</math>, ordered by <math>\langle \sigma',C'\rangle\leq\langle\sigma,
+C\rangle</math> iff <math>\sigma'</math> end-extends <math>\sigma</math> and <math>C'\subseteq
+C</math> and <math>\sigma'\subseteq\sigma\cup C</math>.  In <math>V[G]</math>, we have that <math>\bigcup\{\sigma\,
+\colon(\exists C)(\langle\sigma,C\rangle\in
+ G)\}</math> is a closed unbounded subset of ''S'' almost contained in each club set in ''V''. <math>\aleph_1</math> is preserved.
+==Shooting a club with countable conditions==
+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>
+\aleph_1</math> is preserved, and if CH holds then all cardinals are preserved.
+==Shooting a club with finite conditions==
+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>
+and <math>\alpha\in S</math>, and whenever <math>\langle\alpha,
+\beta\rangle</math> and <math>\langle\gamma,\delta\rangle</math> are distinct elements of ''p'' then either
+<math>\beta<\gamma</math> or <math>\delta<\alpha</math>.  ''P'' is ordered by reverse inclusion.
+In <math>V[G]</math>, we have that <math>\{\alpha\,\colon(\exists\beta)(\langle
+\alpha,\beta\rangle\in\bigcup G)\}</math> is a closed unbounded subset of ''S'' and all cardinals are preserved.
+==Silver forcing==
+Silver forcing (after [[Jack Silver|Jack Howard Silver]]) satisfies Fusion, the Sacks property, and
+is minimal with respect to reals (but not minimal).
+==References==
+*{{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}}
+*{{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}}
+==External links==
+*A.Miller (2009), [http://www.math.wisc.edu/~miller/old/m873-09/forcing.pdf ''Forcing Tidbits.'']
+[[Category:Forcing (mathematics)]]

Post's lattice: Difference between revisions

en>EmilJ

Revision as of 18:25, 14 October 2013

Contents

Notation

Definitions

Cohen forcing

Grigorieff forcing

Hechler forcing

Jockusch–Soare forcing

Iterated forcing

Laver forcing

Levy collapsing

Magidor forcing

Mathias forcing

Namba forcing

Prikry forcing

Product forcing

Radin forcing

Random forcing

Sacks forcing

Shooting a fast club

Shooting a club with countable conditions

Shooting a club with finite conditions

Silver forcing

References

External links

Navigation menu

Post's lattice: Difference between revisions

en>EmilJ

Revision as of 18:25, 14 October 2013

Notation

Definitions

Cohen forcing

Grigorieff forcing

Hechler forcing

Jockusch–Soare forcing

Iterated forcing

Laver forcing

Levy collapsing

Magidor forcing

Mathias forcing

Namba forcing

Prikry forcing

Product forcing

Radin forcing

Random forcing

Sacks forcing

Shooting a fast club

Shooting a club with countable conditions

Shooting a club with finite conditions

Silver forcing

References

External links

Navigation menu

Search