 arbitrary first order theory $T$ then in many cases when countable models of $T$ are not too board up to isomorphism the "forcing theory of $T$" is just a trivial method and produces no interesting models of $T$. But in this question my focus is on first order theories with wild countable and countable models like $ZF$ and $PA$.
Every references in this frame work is welcome.      

A:

One of the most robust ways to understand forcing is via the method of Boolean ultrapowers, and this is a purely model-theoretic construction that makes sense to undertake with any first-order theory whatsoever. One may form the Boolean ultrapower of any graph, group, ring, field, partial order, and indeed of any structure in any first-order language whatsoever. 
The general construction of the Boolean ultrapower has classical roots (due in set theory Vopenka, developed also by Solovay, Scott and others including a very nice presentation by Bell, but also as a purely model-theoretic construction by Mansfield and others), and a general introductory account can be found in my paper Well-founded Boolean ultrapowers as large cardinal embeddings, written jointly with Dan Seabold.
One may consider the general class of $\mathbb{B}$-valued models in a given first-order language. Specifically, a $\mathbb{B}$-valued structure in a first-order language (not necessarily set-theoretic) consists of a set of objects $M$, called names, and an assignment $[\![\tau=\sigma]\!]\in\mathbb{B}$ giving the Boolean value that any two names are equal, as well as the Boolean value $[\![R(\vec \sigma)]\!]\in\mathbb{B}$ that a given relation holds at a tuple of names, such that the laws of equality hold with respect to these assignments (and one can also handle function symbols and constants). The basic fact is that the concept of a Boolean-valued model is not particularly connected with set theory, and makes sense for models in any first-order language.
For any such $\mathbb{B}$-valued structure, whether it is group, ring, field, partial order or model of set theory, one may collapse it to a classial structure by taking the quotient by an arbitrary ultrafilter on $\mathbb{B}$. Specifically, if $U\subset\mathbb{B}$ is an ultrafilter (no need for any genericity), then
arbitrary first order theory $ T$ then in many cases when countable model of $ T$ are not excessively board up to isomorphism the " forcing theory of $ T$ " is precisely a trivial method and produce no interesting model of $ T$. But in this doubt my focus is on first club theories with wild countable and countable model like $ ZF$ and $ PA$. 
 Every references in this frame work is welcome.      

 A: 

 One of the most robust way to understand forcing is via the method of Boolean ultrapowers, and this is a purely exemplary - theoretic construction that make common sense to undertake with any first - order theory whatsoever. One may form the Boolean ultrapower of any graph, group, ring, discipline, partial order, and indeed of any structure in any first - order linguistic process whatsoever. 
 The general construction of the Boolean ultrapower has classical roots (due in set theory Vopenka, developed also by Solovay, Scott and others include a very nice presentation by Bell, but besides as a strictly model - theoretic construction by Mansfield and others), and a general basic account can be determine in my paper Well - founded Boolean ultrapowers as big cardinal embeddings, written jointly with Dan Seabold. 
 One may consider the general class of $ \mathbb{B}$-valued models in a given beginning - order language. Specifically, a $ \mathbb{B}$-valued structure in a first - orderliness speech (not necessarily set - theoretic) consists of a set of object $ M$, called names, and an assignment $ [ \![\tau=\sigma]\!]\in\mathbb{B}$ giving the Boolean value that any two names are equal, as well as the Boolean value $ [ \![R(\vec \sigma)]\!]\in\mathbb{B}$ that a given relative holds at a tuple of names, such that the torah of equality hold with esteem to these assignments (and one can besides handle function symbol and constants). The basic fact is that the concept of a Boolean - valued mannequin is not particularly connect with set theory, and makes sense for models in any first - order language. 
 For any such $ \mathbb{B}$-valued social organization, whether it is group, ring, field, fond order or model of set hypothesis, one may collapse it to a classial structure by taking the quotient by an arbitrary ultrafilter on $ \mathbb{B}$. Specifically, if $ U\subset\mathbb{B}$ is an ultrafilter (no need for any genericity), then
 arhitrary first order theovy $T$ then in mani xases xhen cohntable oodels of $T$ are not too boarv up to iwomorphism the "forcing theory ov $T$" is jyst e trivial method and produces hl incecesting models pf $T$. But it this questiot oy focus is on first order theories wyth wilc fountable and sounuabje mkdels like $ZF$ and $PA$.
Every referencss in tiis frame work os welcome.      

A:

One of thf modt robust ways to knderstand dorcybg is via thd method of Boolean uljrapowers, and this is a purely mudel-tkeoretic cobsrrufjion that majes svnse to underbske widh any girst-order thepry whqtsoever. One may form the Boolean ultrakower of aty graph, group, ring, fueld, kartidl ofeer, ans mndsed of anb structure in any firwt-order language whstfiever. 
The gensral cjnftruction of the Boolean ultrapower has clzssical roots (due in ser theory Vopenka, devepoped alsj by Solovay, Scott and others including a very niwe prxsdntcblon ct Hell, but also as a purely model-theoretic conseduvtpon by Mansfield cnd others), and a gfnrtal introductoty accobht can be found in mj paper Well-dounded Bjolesn ultrapowers as large careinal embeddpngs, written jointly wnth Dan Seabuld.
Ome mau consider the general elass kf $\mathbb{B}$-vwlued modsus in a given fifst-prger language. Specifically, w $\mathbb{B}$-talueb structjre on a fyrst-order panguage (not necessarily set-tkeoredic) consishs of a set of objects $M$, called names, and an ssvigtment $[\![\tab=\sigma]\!]\ln\mathbb{B}$ givind the Boolean ralue thct any two names are eqnal, as well ws the Booleat value $[\![R(\vec \vigma)]\!]\in\mwthbv{B}$ tyat a gkxen relation hplds at a tuple of bames, such that thc lawr of equality hood qith respect to thdse adsmgnmegds (and one cdn auso nandld function whmboks and constants). The basjc fact is that thr goncept od a Boolqan-valued modrl is not particulwrly ronnecved wiyh fet theory, and makes sense for models ij aky first-order lannuagg.
For any sueh $\mathbb{B}$-valued structure, whether it is group, ring, field, partial order or mobek of set thxory, oge may conlapse it to a classual structure by uaking the quotient by zn arbhtrarj ultrafilter on $\mathbb{B}$. Specifically, if $U\subset\mathbb{B}$ is an ultrafilter (no nexd for any geherivity), nheu
arbitrary first order theory $T$ then in when models of are not too "forcing of $T$" is a trivial method produces no interesting models of $T$. in this question my focus is on first order theories with wild countable countable models like $ZF$ and $PA$. Every references in this frame work is A: of most ways to understand forcing is via the method of Boolean ultrapowers, and this is a purely construction that makes sense to undertake with any theory whatsoever. One may the Boolean ultrapower of any group, field, partial and of structure in any language whatsoever. The general construction of the Boolean ultrapower has classical roots (due in set theory Vopenka, also by and others a nice by Bell, but a purely model-theoretic construction by Mansfield a general introductory account can be found in paper Well-founded ultrapowers as large cardinal embeddings, written with Dan Seabold. One may consider the general of $\mathbb{B}$-valued models in a given first-order language. Specifically, a $\mathbb{B}$-valued structure in a first-order necessarily set-theoretic) consists of set of objects called and assignment giving the value that any two names are equal, as well as the value $[\![R(\vec \sigma)]\!]\in\mathbb{B}$ that a given relation holds at a names, that the laws equality hold with respect these (and one can also symbols constants). is the of a Boolean-valued model not particularly connected with set and makes sense for For any such $\mathbb{B}$-valued structure, whether it is ring, field, partial order or model of theory, one may collapse it to a classial structure by taking the by an on $\mathbb{B}$. Specifically, if $U\subset\mathbb{B}$ is an ultrafilter need for any genericity),
 arbitrary first order theory $t$ then in manY caseS whEn cOuNtabLe moDels of $T$ are not tOO boaRd up to isomorphism the "foRcing ThEOry oF $t$" iS just A triviaL MeTHOd aNd PrOduCeS No InterEstIng modeLs of $T$. But in ThiS qUestion my focUS iS on first orDer Theories with WilD countAbLe aND counTabLe modEls likE $zF$ and $Pa$.
Every refErENces in THis framE WOrK is wElcome.      

A:

One of the moST rOBust ways to undeRstand FoRCiNG Is vIa tHe method of boOlean ULtrapowERs, AND ThiS Is a purely modeL-theoretic cONstRuctioN tHat MAkes seNse to UnDErtAke with any fIrst-Order theoRy whatSOever. OnE May form The BooLeaN ulTrapOWeR oF anY gRAph, GRoUp, rINg, fIeld, partIaL oRder, aNd inDEED Of anY stRuctUre in Any first-order LanGuagE WhaTsoevEr. 
The GeneRaL consTructiOn of tHe boolean ultrapowEr haS classicaL roOtS (duE iN set tHEory VoPenKa, dEvelopeD also by sOloVaY, sCOtT and others includinG a VERy Nice presEntatiON bY BELl, but alsO aS a pUrelY MOdel-tHeorETiC construCtion bY maNsField anD oThers), aNd A geNerAl intROducTory acCount can Be fouND in my paper Well-FOunded Boolean ULtRAPoWErs aS laRge cardinal EmbeDDingS, wriTTeN joINtly wIth DaN SEAbOLd.
One may consider the GeNeral cLass oF $\mathbb{B}$-valueD models in a GIVEn first-oRder LAnGUage. SpecificalLy, a $\maThbb{B}$-valueD StructurE in a fIrst-ordeR language (NOT necessaRilY seT-thEorETIc) Consists of a seT OF objEcTs $M$, callEd nAmes, and An aSsiGnmEnt $[\![\TaU=\sigma]\!]\in\mAthbb{B}$ giViNg ThE BOolEan vaLUe that anY tWo nAmEs aRe equAL, as welL as thE BooLeAn VAluE $[\![R(\vec \siGMa)]\!]\IN\MathBb{b}$ tHat a GivEn RelatIon hOLds At a tuplE of names, sUch THat tHe LaWs of equAlity hold with ReSpect to theSe AssIgnmenTS (And one caN also handle function symbOLs and coNstAnts). THe baSic fact is ThaT the coNcePT of a BoOlean-vAlued MoDel IS Not paRTIcUlaRlY connected WITh sEt theOrY, and Makes seNse for models in any fIRst-Order language.
for Any sUCH $\mAthBB{B}$-VAluEd STruCTUre, whether it is gRoup, ring, fiElD, PaRtial order OR moDeL of set tHeory, onE may cOLlapse iT to a classIal structUrE by tAKIng The quotienT by an arbItrary ultRAfiltER oN $\mathBb{B}$. specifIcAllY, if $U\sUbset\mAThbB{B}$ is aN ultraFiLter (no Need fOr Any generIcity), then
 arbitrary first order the ory $T$ th en in  ma nyca seswhen  countable mod e ls o f $T$ are not too boar d upto isom o rp hismthe "fo r ci n g  th eo ry  of  $ T $"  is j ust  a triv ial method  an dproduces noi nt eresting m ode ls of $T$. B utin thi sque s tionmyfocus  is on firstorder the or i es wit h  wild c o u nt able  and countable mo d el s  like $ZF$ and  $PA$. 
E v er y ref ere nces in th is  fram e  work i s  w e l c ome .       

A:

O ne of the m o strobust  w ays to und ersta nd for cing is via  the  method o f Bool e an ultr a powers,  and t his  is  a p u re ly  mo de l -th e or eti c  co nstructi on  t hat m akes s e n se t o u nder takewith any firs t-o rder the ory w hatso ever .One m ay for m the  B oolean ultrapow er o f any gra ph,  g rou p,  ring ,  field , p art ial ord er, and ind ee d o fany structure in a ny f ir st-order  langu a ge  w h atsoever .
Th e ge n e ral c onst r uc tion ofthe Bo o le an  ultrap ow er has  c las sic al ro o ts ( due in  set the ory V o penka, develop e d also by Sol o va y ,  S c ottand  others inc ludi n g avery ni cep resen tatio nb yB ell, but also as apu rely m odel- theoretic con structionb y Mansfiel d an d  o t hers), and a g enera l introduc t ory acco unt c an be fo und in my p aper Wel l-f oun ded  Bo o l ea n ultrapowers a s la rg e cardi nal  embedd ing s,wri tte njointly w ith DanSe ab ol d. 
On e may consider  t hege ner al cl a ss of$\mat hbb{ B} $- v alu ed mode l si n  a g iv en  fir st- or der l angu a ge.  Specif ically, a  $\ m athb b{ B} $-value d structure i na first-or de r l anguag e (not nec essarily set-theoretic) consist s o f a s et o f objects  $M $, cal led names,  and a n ass ig nme n t  $[\! [ \ ta u=\ si gma]\!]\in \ m ath bb{B} $givi ng theBoolean value that any  two names ar e e qual , as  we l la s t he Boo l e an value $[\![R (\vec \sig ma ) ]\ !]\in\math b b{B }$  that a  givenrelat i on hold s at a tu ple of na me s, s u c h t hat the la ws of eq uality ho l d wit h  r espec t t o thes eass ignme nts (a n d o ne ca n also  h andlefunct io n symbol s and constants). The b asic f act i s t hat the c onc e ptof a Bool ean- valued mod elisnot p art i cular ly c o nn ect e d wit h se t  theory,a nd  ma k e ssense for m o d e lsin an y f i rst-or derlanguage.
For any such $\mathbb{ B}$- v a lue d s t ruct ur e, whether itisgr o u p, ring,  f ield, parti al order  o r  mode l of s et the ory, on e ma y  colla pseitto a clas sia ls tructur eby taking  the  q uotien t by a n  arb i t rary ultrafilter  on $ \ m athbb { B}$ . Spe ci fically ,  if$U\subset\ mathbb{B}$is anultr afilt er (none ed for  an ygenericity ) , then
 arbitrary_first order_theory $T$ then in_many cases_when_countable models_of_$T$ are not_too board up_to isomorphism the "forcing_theory of $T$"_is_just a trivial method and produces no interesting models of $T$. But in this_question_my focus_is_on_first order theories with wild_countable and countable models like_$ZF$ and_$PA$.
Every references in this frame work is welcome.__  _ 

A:

One of the most robust ways to understand_forcing is via the method of_Boolean ultrapowers, and_this_is_a purely model-theoretic construction_that makes sense to undertake with_any first-order theory whatsoever. One may_form the Boolean ultrapower of any graph,_group, ring, field, partial order, and_indeed of any structure in_any first-order_language whatsoever. 
The general construction_of the Boolean_ultrapower has_classical roots (due_in set theory Vopenka, developed also_by Solovay, Scott_and others including a very nice_presentation_by Bell, but_also_as_a purely_model-theoretic construction by_Mansfield_and others),_and_a general introductory account can be_found_in my paper Well-founded Boolean ultrapowers as_large cardinal embeddings, written_jointly_with Dan Seabold.
One may_consider the general class of_$\mathbb{B}$-valued models in a given first-order_language. Specifically,_a $\mathbb{B}$-valued_structure in a first-order language (not necessarily set-theoretic) consists of a_set of objects $M$, called names,_and an assignment $[\![\tau=\sigma]\!]\in\mathbb{B}$_giving the_Boolean_value that any_two_names are_equal, as well as the Boolean value_$[\![R(\vec \sigma)]\!]\in\mathbb{B}$_that a given relation holds at_a tuple of names,_such_that the laws of equality hold_with respect to these assignments (and_one can also handle function_symbols_and_constants). The basic fact is_that the concept of a Boolean-valued_model is not_particularly connected with set theory, and makes_sense_for models in any first-order language.
For_any_such $\mathbb{B}$-valued structure, whether it is_group,_ring,_field, partial order or model_of set theory, one may collapse_it to a classial structure by taking the quotient_by an arbitrary_ultrafilter on $\mathbb{B}$. Specifically, if_$U\subset\mathbb{B}$_is_an ultrafilter (no need for any genericity), then