 prime theories. This is not very useful in itself, since prime theories are rather elusive entities. What we are interested in, however, is measuring the distances between general theories in such a way that their order relations are respected. To cope with this task, first note that IN is not just a metric space but even a pospace with respect to the order relation ≤ defined by x ≤ y : xi ≤ yi for all i ∈ N. Secondly, the metric of Definition 4.3 is radially convex with respect to this order relation. Moreover, it satisfies the condition d(x, z) ≥ d(x∧y, y∧z)+d(x∨y, y∨z). In the next section it will be shown that the space of theories T (L) is still a second countable compact Hausdorff space. Hence it can be embedded into the Hilbert cube. Moreover, due to the UrysohnCarruth metrization theorem this can be done in such a way that the space of theories inherits the radially convex metric of IN. 5 The logical space of theories as a Vietoris space The final step for constructing a metric suitable for truth approximation is to endow the set of theories, that is, the set C(ST (L)) of closed sets of ST (L) with an appropriate metric, that is, with a metric that respects the existing topology and the order structure on C(ST (L)). Recall that C(ST (L)) can be conceived as an order structure 〈C(ST (L)),≤〉 where ≤ is the opposite order to set-theoretical inclusion ⊆.9 In order to apply the Urysohn-Carruth metrization Theorem 2.4 one has to show that 〈C(ST (L)),≤〉 is a pospace. That is to say, it has to be endowed with a topology, and this topology has to be compatible with the order relation ≤ in the sense that the graph of ≤ is closed with respect to the product topology of C(ST (L)) × C(ST (L)) (see Definition 2.5 (ii)). The recipe for endowing C(ST (L)) with a topological structure may be traced back to Vietoris and Hausdorff in the twenties of the last century. It goes like this. Theorem 5.1 Assume 〈X,O(X)〉 to be a second countable compact Haus
prime theories. This is not very useful in itself, since prime theory are preferably elusive entities. What we are interested in, however, is quantify the distances between general theory in such a way that their order relation are respected. To cope with this task, beginning note that IN is not just a measured space but even a pospace with respect to the order relation ≤ defined by x ≤ y: xi ≤ yi for all i ∈ N. Secondly, the metric of Definition 4.3 is radially convex with esteem to this order relation. Moreover, it meet the condition d(x, z) ≥ d(x∧y, y∧z)+d(x∨y, y∨z). In the next section it will be show that the space of theories T (L) is still a second countable compact Hausdorff space. therefore it can be embedded into the Hilbert cube. Moreover, due to the UrysohnCarruth metrization theorem this can be done in such a way that the space of theory inherits the radially convex metric of IN. 5 The logical space of theories as a Vietoris space The final step for constructing a metric desirable for accuracy approximation is to endow the set of theories, that is, the located C(ST (L) ) of closed sets of ST (L) with an appropriate metric, that is, with a system of measurement that respects the existing topology and the order social organization on C(ST (L) ). Recall that C(ST (L) ) can be conceived as an order structure 〈 C(ST (L)),≤ 〉 where ≤ is the opposite order to set - theoretical inclusion ⊆.9 In order to apply the Urysohn - Carruth metrization Theorem 2.4 one has to show that 〈 C(ST (L)),≤ 〉 is a pospace. That is to allege, it induce to be endowed with a topology, and this topology has to be compatible with the order relation back ≤ in the sense that the graph of ≤ is closed with respect to the merchandise topology of C(ST (L) ) × C(ST (L) ) (see Definition 2.5 (ii) ). The recipe for endowing C(ST (L) ) with a topological structure may be hound back to Vietoris and Hausdorff in the twenties of the final century. It goes like this. Theorem 5.1 Assume 〈 X, O(X) 〉 to be a second countable compendious Haus
 prlme theories. This is not very useful in itself, since prime tfeories are rather elusive eititues. Wyat we are interested kn, howeveg, is measyrinj the distances ustween nzneram themcies in such a eay that tveir order reldtkous are respected. To cope with this twsk, firxt note that IN ys npe juan c metric space but even a pospade with respect to tne order relation ≤ defined by d ≤ y : xi ≤ yi for apl i ∈ N. Secindlr, the metric uf Definitpmn 4.3 is radjally convex with respect to thks orber relatiob. Norfmver, it satmsfies the condition d(x, z) ≥ d(x∧y, y∧z)+c(x∨y, y∨z). In the kext vecrion it will be shown that the space of theories T (L) is still a secine couttabne cunpazt Gansdkrff soacx. Hence it dan be embeeded into the Hilbett bibe. Moreover, due tj ehe UrysohnCarruth metrization theorem uhis dan be done in such a wqy that the space of jheories igherits the radially convex metric of IN. 5 The loghcal aoact if thdirles as a Vietoris space The final step for cogattubting a metric sultable for truth alpgocymation is to endow ths set of theories, hhat is, the wet C(ST (L)) of vlosed sets of ST (L) with an appropriate netric, that is, witk a metric tkat rexpectx the existing topology and the order dtructure un C(ST (L)). Recall ghan C(SD (L)) can be conceived as an order stcuctuxe 〈C(ST (L)),≤〉 whete ≤ is ehe opposihe order to set-theoreticwl inelusimn ⊆.9 In ordfr to apply the Urysohn-Carruth metrization Thgoram 2.4 one has to smow that 〈C(ST (L)),≤〉 ys a pospace. Tkat is tj say, it has to be endmwed with a topology, and jhis topology has to fe cimparible wkgh the order rrlation ≤ pn the sense that the greph ur ≤ is closed wick twspect to the lroauce noppljcy of C(ST (L)) × C(ST (L)) (rre Dewinition 2.5 (il)). Tfe rrcipe for endowing C(VT (L)) with a topologicak ftructurg may be eraced back tp Vietoris and Hauddorfh in tie twemtigs of the last century. It goes mike this. Thcorem 5.1 Assume 〈X,O(X)〉 to be a secpnd countable compact Haus
prime theories. This is not very useful since theories are elusive entities. What is the distances between theories in such way that their order relations are To cope with this task, first note that IN is not just a space but even a pospace with respect to the order relation ≤ defined x y xi yi for all i ∈ N. Secondly, the metric of Definition 4.3 is radially convex with to this order relation. Moreover, it satisfies the d(x, z) ≥ d(x∧y, y∨z). In the next section will shown that space theories (L) is still second countable compact Hausdorff space. Hence it can be embedded into the Hilbert cube. Moreover, due to UrysohnCarruth metrization can be in a that the space inherits the radially convex metric of logical space of theories as a Vietoris space final step constructing a metric suitable for truth is to endow the set of theories, that the set C(ST (L)) of closed sets of ST (L) with an appropriate metric, that a metric that respects existing topology and order on (L)). that C(ST can be conceived as an order structure 〈C(ST (L)),≤〉 where ≤ the opposite order to set-theoretical inclusion ⊆.9 In order to Urysohn-Carruth Theorem 2.4 one to show that 〈C(ST is pospace. That is to has be topology, this has to be compatible the order relation ≤ in sense that the graph respect to the product topology of C(ST (L)) C(ST (L)) (see Definition 2.5 (ii)). The for endowing C(ST (L)) with a topological structure may be traced back Vietoris and the twenties of the last century. It goes this. Theorem 5.1 Assume to be a second countable compact Haus
 prime theories. This is not verY useful in iTself, SinCe pRiMe thEoriEs are rather eluSIve eNtities. What we are intereSted iN, hOWeveR, Is MeasuRing the DIsTANceS bEtWeeN gENeRal thEorIes in suCh a way that TheIr Order relatioNS aRe respecteD. To Cope with this TasK, first NoTe tHAt IN iS noT just A metriC Space bUt even a poSpACe with REspect tO THe OrdeR relation ≤ defined bY X ≤ y : XI ≤ yi for all i ∈ N. SecOndly, tHe MEtRIC of defInition 4.3 is rAdIally COnvex wiTH rESPEct TO this order relAtion. MoreovER, it SatisfIeS thE ConditIon d(x, Z) ≥ d(X∧Y, y∧z)+D(x∨y, y∨z). In the nExt sEction it wIll be sHOwn that THe space Of theoRieS T (L) Is stILl A sEcoNd COunTAbLe cOMpaCt HausdoRfF sPace. HEnce IT CAN be eMbeDded Into tHe Hilbert cube. morEoveR, Due To the urysoHnCaRrUth meTrizatIon thEoRem this can be donE in sUch a way thAt tHe SpaCe Of theORies inHerIts The radiAlly conVEx mEtRIC Of iN. 5 The logical space oF tHEOrIes as a ViEtoris SPaCe tHe final sTeP foR conSTRuctiNg a mETrIc suitabLe for tRUtH aPproximAtIon is tO eNdoW thE set oF TheoRies, thAt is, the sEt C(ST (l)) Of closed sets of st (L) with an approPRiATE mETric, ThaT is, with a metRic tHAt reSpecTS tHe eXIstinG topoLoGY aND the order structure oN C(sT (L)). RecAll thAt C(ST (L)) can be coNceived as aN ORDer strucTure 〈c(sT (l)),≤〉 Where ≤ is the oppoSite oRder to set-tHEoreticaL inclUsion ⊆.9 In oRder to appLY The UrysoHn-CArrUth MetRIZaTion Theorem 2.4 onE HAs to ShOw that 〈C(sT (L)),≤〉 Is a pospAce. thaT is To sAy, It has to be Endowed wItH a ToPoLogY, and tHIs topoloGy Has To Be cOmpatIBle witH the oRder ReLaTIon ≤ In the seNSe THAt thE gRaPh of ≤ Is cLoSed wiTh reSPecT to the pRoduct topOloGY of C(sT (l)) × C(sT (L)) (see DEfinition 2.5 (ii)). ThE rEcipe for enDoWinG C(ST (L)) wITH a topoloGical structure may be tracED back to vieToris And HAusdorff iN thE twentIes OF the laSt centUry. It GoEs lIKE this. tHEoRem 5.1 asSume 〈X,O(X)〉 to bE A SecOnd coUnTablE compacT Haus
 prime theories. This is n ot very us efulinits el f, s ince  prime theorie s  are  rather elusive entiti es. W ha t  wea re  inte restedi n, h owe ve r,  is  m e as uring  th e dista nces betwe enge neral theori e sin such away  that theirord er rel at ion s  areres pecte d. Toc ope wi th this t as k , firs t  note t h a tIN i s not just a metr i cs pace but evena posp ac e  w i t h r esp ect to the  o rderr elation ≤d e f ine d  by x ≤ y : x i ≤ yi fora lli ∈ N.  S eco n dly, t he me tr i c o f Definitio n 4. 3 is radi ally c o nvex wi t h respe ct tothi s o rder re la tio n. Mor e ov er, itsatisfie sth e con diti o n d (x,z)≥ d( x∧y,y∧z)+d(x∨y, y ∨z) . In the  next  sect ionit  will  be sh own t ha t the space oftheo ries T (L ) i ssti ll  a se c ond co unt abl e compa ct Haus d orf fs p a ce . Hence it can beem b e dd ed intothe Hi l be rt cube. Mo re ove r, d u e  to t he U r ys ohnCarru th met r iz at ion the or em thi scan  be  done in s uch away that  thes pace of theori e s inherits th e  r a d ia l ly c onv ex metric o f IN .  5 T he l o gi cal space  of t he o ri e s as a Vietoris spa ce  The f inalstep for cons tructing a m e tric sui tabl e  f o r truth approx imati on is to e n dow theset o f theori es, thati s , the se t C (ST  (L ))o f  c losed sets of S T (L )with an  ap propria temet ric , t ha t is, wit h a metr ic  t ha tres pects the exis ti ngto pol ogy a n d theorder  str uc tu r e o n C(ST( L) ) .  Rec al lthat  C( ST  (L))  can beconceiv ed as anord e r st ru ct ure 〈C( ST (L)),≤〉 wh er e ≤ is the  o ppo site o r d er to se t-theoretical inclusion ⊆.9 Inord er to  app ly the Ur yso hn-Car rut h  metri zation  Theo re m 2 . 4  oneh a stosh ow that 〈C ( S T ( L)),≤ 〉is a  pospac e. That is to say, ithas to be end owe d wi t h  a  to p ol o gy,  a n d t h i s topology hasto be comp at i bl e with the ord er  relati on ≤ in  thes ense th at the gr aph of ≤is  clo s e d w ith respec t to the  productt opolo g yof C( ST(L)) ×  C (ST  (L))  (seeD efi nitio n 2.5(i i)). T he re ci pe for e ndowing C(ST (L)) witha topo logic alstructure  ma y  be  traced b ackto Vietori s a ndHausd orf f  in t he t w en tie s  of t he l a st centur y .Itg o es  like this. T h eor em 5. 1 A s sume 〈 X,O( X)〉 to be a secon d  countable com pact H aus 
 prime_theories. This_is not very useful_in itself,_since_prime theories_are_rather elusive entities._What we are_interested in, however, is_measuring the distances_between_general theories in such a way that their order relations are respected. To cope_with_this task,_first_note_that IN is not just_a metric space but even_a pospace_with respect to the order relation ≤ defined_by_x ≤ y_: xi ≤ yi for all i ∈ N._Secondly, the metric of Definition 4.3_is radially convex_with_respect_to this order relation._Moreover, it satisfies the condition d(x,_z) ≥ d(x∧y, y∧z)+d(x∨y, y∨z). In_the next section it will be shown_that the space of theories T_(L) is still a second_countable compact_Hausdorff space. Hence it can_be embedded into_the Hilbert_cube. Moreover, due_to the UrysohnCarruth metrization theorem this_can be done_in such a way that the_space_of theories inherits_the_radially_convex metric_of IN. 5_The_logical space_of_theories as a Vietoris space The_final_step for constructing a metric suitable for_truth approximation is to_endow_the set of theories,_that is, the set C(ST_(L)) of closed sets of ST_(L) with_an appropriate_metric, that is, with a metric that respects the existing topology_and the order structure on C(ST_(L)). Recall that C(ST_(L)) can_be_conceived as an_order_structure 〈C(ST_(L)),≤〉 where ≤ is the opposite order_to set-theoretical_inclusion ⊆.9 In order to apply_the Urysohn-Carruth metrization Theorem_2.4_one has to show that 〈C(ST_(L)),≤〉 is a pospace. That is_to say, it has to_be_endowed_with a topology, and this_topology has to be compatible with_the order relation_≤ in the sense that the graph_of_≤ is closed with respect to_the_product topology of C(ST (L)) ×_C(ST_(L))_(see Definition 2.5 (ii)). The_recipe for endowing C(ST (L)) with_a topological structure may be traced back to Vietoris_and Hausdorff in_the twenties of the last_century._It_goes like this. Theorem 5.1 Assume 〈X,O(X)〉 to be a_second countable_compact Haus