 point at which a conventional choice has to be made, can be determined quite precisely, namely when it comes to the choice of a metric that is compatible with the 'objective' ordered topological structure. The outline of the paper is as follows: In the next section we recall the rudiments of topology, order theory, and metric spaces we need in the following sections. In particular we state a fundamental theorem of Urysohn-Carruth, which asserts that under mild restrictions every ordered metrical space can be assumed to have a metric that is compatible with the order structure. This theorem will be crucial for the main result of this paper. In § 3 we characterize the theories T (L) of a language L as the class of deductively closed sets of the Lindenbaum algebra LINDA(L) of logically equivalent propositions. This sets the stage for the topological representation of LINDA(L) carried out in § 4. In § 5 the logical space T (L) of theories, conceived of as a Vietoris space of closed subsets of the Stone space ST (L) is constructed. Thereby the Urysohn-Carruth metrization theorem can be used to show that this space can be endowed with a metric that is adequate for the purposes of truth approximation. In § 6 some relations between the account of truthlikeness advocated in this paper and Poincaré's conventionalist stance in geometry are pointed out. 2 Topology, metric, and order In this section the basic notions of topology, theory of metric spaces and order structures are recalled that will be needed in the following. This section can offer only a very brief and incomplete survey. For a more thorough introduction to the topics mentioned in this section the reader is recommended to consult the literature (Davey & Priestley 1990, James 1999, Nachbin 1965, Nagata 1985, Niiniluoto 1987, §§ 1.1 and 1.2). Let us start with topology. Definition 2.1 Let X be a set. A topological structure O(X) on X is given by a subset O(X) ⊆ P(X) of the power set P(X) of X that is assumed to satisfy the following properties: (i) ∅ and X belong to O(X). Truthlikeness for Theories on Countable Languages 5 (ii) If Ai is an arbitrary collection of subsets of X belonging to O(X), then the set-theoretical union ⋃ Ai also is an element
point at which a conventional choice has to be make, can be determine quite precisely, namely when it comes to the choice of a metric that is compatible with the' aim' ordered topological structure. The outline of the paper is as surveil: In the next section we remember the rudiments of topology, club theory, and metric space we need in the following sections. In particular we state a fundamental theorem of Urysohn - Carruth, which assert that under mild restrictions every ordered measured space can be assumed to have a metric that is compatible with the ordering structure. This theorem will be crucial for the main result of this paper. In § 3 we qualify the theories T (L) of a language L as the class of deductively closed sets of the Lindenbaum algebra LINDA(L) of logically equivalent proposal. This set the stage for the topological representation of LINDA(L) carried out in § 4. In § 5 the logical space T (L) of theories, gestate of as a Vietoris space of closed subset of the Stone quad ST (L) is construct. Thereby the Urysohn - Carruth metrization theorem can be used to show that this space can be endowed with a metric that is adequate for the purposes of truth estimate. In § 6 some relations between the account of truthlikeness advocated in this paper and Poincaré's conventionalist stance in geometry are pointed out. 2 Topology, metric, and order In this section the basic notions of topology, theory of measured spaces and club structure are recalled that will be needed in the following. This section can offer only a very brief and incomplete sketch. For a more thorough introduction to the topics mentioned in this department the reader is recommended to confer the literature (Davey & Priestley 1990, James 1999, Nachbin 1965, Nagata 1985, Niiniluoto 1987, § § 1.1 and 1.2). Let us start with topology. Definition 2.1 Let ten be a set. A topological social organization O(X) on X is given by a subset O(X) ⊆ P(X) of the power set P(X) of X that is assume to satisfy the following properties: (i) ∅ and X belong to O(X). Truthlikeness for Theories on Countable Languages 5 (ii) If Ai is an arbitrary collection of subsets of X belonging to O(X), then the set - theoretical coupling ⋃ Ai also is an element
 polnt at which a conventiokal choice has to be mave, can ge deteroined quite precisely, namely wyen iu comes to the choize of a mvtric thar is xompatible with thc 'objsgtive' irdered topolonical strucdure. The outlite oy the paper is as follows: In the nexe sectipn we recall the rudpmqnts of topology, order theory, and metrjc spacts we need in the gollowing sections. In partlculwr we state a fundwmental theirem if Urysohn-Cafruth, which asserts thzt under mild restrictions everh ordzred metricql spwwe can be awsumeq to have a metric tvat is vompatible witm the oreer structure. This thxorem will be cruciaj for the mcin result of this papee. In § 3 we chafqctdriee vhe theorles T (L) of a manguage L qs the class of dedicepfely closed aets os ehe Lindenbaum algebra LINDA(L) of logicanly equivalent propositionw. This sets the stage for the eopological representation of LINDA(L) carried out hn § 4. Mn § 5 uhc uiglcal space T (L) of theories, conceived of as a Djeuorps space of closeb subsets of the Shomg space ST (L) ir conscdudted. Thereby the Ugysohn-Cwrruty metrizauion yheorem can be used to show that this siace can be endowed wich a metric chat ix adeauate for the purposes uf tduth approxlmation. Ih § 6 some relationr bvtwean the account of truthlikqness advicatzd in thks psper agd Poincaré'd conyantionalist stance in ggometrf are poinhed out. 2 Topology, metric, and orvxr In this sevthon the basnc notlons of topologr, theory of mejric spaczs and order strlctures ace recalled ehat will be tgeded in the hollowing. Thiw sextion cxv offer only a very brivf and incomplete survey. Fot z more thorough inuriduction to thr tupiss mxntiogad in this sactiun gne rexder is reciomenced to consult the lhterzture (Davey & Priesylcy 1990, James 1999, Nachbig 1965, Nagata 1985, Nioniluoto 1987, §§ 1.1 and 1.2). Ltt us vtact witn tjpology. Definition 2.1 Let X be a set. A toooljgical struceure O(X) on X is yiven by a subset O(X) ⊆ P(X) of the power sev P(X) of X that is assumwd to satisfy the fpklowing pro'ertief: (i) ∅ and F belong to O(X). Truthoikeness for Theovies on Countable Langhages 5 (ii) Iv Ai is an arbitrary collection of subsets of X belonging to O(X), then the wet-theoretical unikn ⋃ Si alvo ns an ejemeit
point at which a conventional choice has made, be determined precisely, namely when of metric that is with the 'objective' topological structure. The outline of the is as follows: In the next section we recall the rudiments of topology, theory, and metric spaces we need in the following sections. In particular we a theorem Urysohn-Carruth, asserts that under mild restrictions every ordered metrical space can be assumed to have a metric is compatible with the order structure. This theorem be crucial for the result of this paper. In 3 characterize the T of language L as class of deductively closed sets of the Lindenbaum algebra LINDA(L) of logically equivalent propositions. This sets the for the of LINDA(L) out § In § 5 space T (L) of theories, conceived Vietoris space of closed subsets of the Stone ST (L) constructed. Thereby the Urysohn-Carruth metrization theorem be used to show that this space can endowed with a metric that is adequate for the purposes of truth approximation. In § relations between the account truthlikeness advocated in paper Poincaré's stance geometry are out. 2 Topology, metric, and order In this section the basic of topology, theory of metric spaces and order structures are will needed in the This section can offer a brief and incomplete survey. more introduction mentioned this the reader is recommended consult the literature (Davey & 1990, James 1999, Nachbin §§ 1.1 and 1.2). Let us start with Definition 2.1 Let X be a set. topological structure O(X) on X is given by a subset O(X) ⊆ of the P(X) of X that is assumed to satisfy following properties: (i) ∅ X belong to O(X). Truthlikeness for Theories on Countable 5 If Ai an arbitrary collection subsets of X to O(X), then union ⋃ also an
 point at which a conventional Choice has tO be maDe, cAn bE dEterMineD quite preciselY, NameLy when it comes to the choiCe of a MeTRic tHAt Is comPatible WItH THe 'oBjEcTivE' oRDeRed toPolOgical sTructure. ThE ouTlIne of the papeR Is As follows: IN thE next section We rEcall tHe RudIMents Of tOpoloGy, ordeR Theory, And metric SpACes we nEEd in the FOLlOwinG sections. In particULaR We state a fundamEntal tHeOReM OF UrYsoHn-Carruth, wHiCh assERts that UNdER MIld REstrictions evEry ordered mETriCal spaCe Can BE assumEd to hAvE A meTric that is cOmpaTible with The ordER structURe. This tHeorem WilL be CrucIAl FoR thE mAIn rESuLt oF ThiS paper. In § 3 We ChAractErizE THE TheoRieS T (L) oF a lanGuage L as the clAss Of deDUctIvely CloseD setS oF the LIndenbAum alGeBra LINDA(L) of logiCallY equivaleNt pRoPosItIons. THIs sets The StaGe for thE topoloGIcaL rEPREsEntation of LINDA(L) caRrIED oUt in § 4. In § 5 thE logicAL sPaCE T (L) of theOrIes, ConcEIVed of As a VIEtOris spacE of cloSEd SuBsets of ThE Stone SpAce sT (L) Is conSTrucTed. TheReby the URysohN-carruth metrizaTIon theorem can BE uSED tO Show ThaT this space cAn be ENdowEd wiTH a MetRIc thaT is adEqUAtE For the purposes of truTh ApproxImatiOn. In § 6 some relatIons betweeN THE account Of trUThLIkeness advocatEd in tHis paper anD poincaré'S convEntionalIst stance IN Geometry Are PoiNteD ouT. 2 tOpOlogy, metric, anD ORder in This secTioN the basIc nOtiOns Of tOpOlogy, theoRy of metrIc SpAcEs And Order STructureS aRe rEcAllEd thaT Will be NeedeD in tHe FoLLowIng. This SEcTIOn caN oFfEr onLy a VeRy briEf anD IncOmplete Survey. For A moRE thoRoUgH introdUction to the toPiCs mentioneD iN thIs sectION the readEr is recommended to consulT The liteRatUre (DaVey & PRiestley 1990, JAmeS 1999, NachbIn 1965, NAGata 1985, NiIniluoTo 1987, §§ 1.1 and 1.2). leT us STArt wiTH ToPolOgY. DefinitioN 2.1 lEt X Be a seT. A TopoLogical Structure O(X) on X is giVEn bY a subset O(X) ⊆ P(X) oF thE powER SeT P(X) OF X THat Is ASsuMED to satisfy the foLlowing proPeRTiEs: (i) ∅ and X belONg tO O(x). TruthlIkeness For ThEOries on countable languages 5 (Ii) if Ai IS An aRbitrary coLlection Of subsets OF X belONgIng to o(X), tHen the SeT-thEoretIcal unIOn ⋃ AI also Is an elEmEnt
 point at which a conventi onal choic e has  to  be  m ade,  can  be determined quit e precisely, namely wh en it  c o mest othe c hoice o f  a m etr ic  t hat  i s  c ompat ibl e withthe 'objec tiv e'  ordered top o lo gical stru ctu re. The outl ine  of th epap e r isasfollo ws: In the ne xt sectio nw e reca l l the r u d im ents  of topology, ord e rt heory, and met ric sp ac e sw e  ne edin the fol lo wings ections .  I n p art i cular we stat e a fundame n tal  theor em  of Urysoh n-Car ru t h,which asser ts t hat under  mildr estrict i ons eve ry ord ere d m etri c al  s pac ec anb eass u med  to have  a  m etric  tha t i s  com pat ible  with  the order st ruc ture .  Th is th eorem  wil lbe cr ucialfor t he  main result of  thi s paper.In§3 w echara c terize  th e t heories  T (L)o f a  l a n g ua ge L as the classof d ed uctively  close d  s et s  of theLi nde nbau m algeb ra L I ND A(L) oflogica l ly  e quivale nt  propo si tio ns.  This sets  the s tage for  thet opological rep r esentation of LI N D A( L ) ca rri ed out in §  4.I n §5 th e  l ogi c al sp ace T  ( L )o f theories, conceiv ed  of as  a Vi etoris spaceof closeds u b sets oftheS to n e space ST (L)  is c onstructed .  Thereby  theUrysohn- Carruth m e t rization  th eor emcan b eused to showt h at t hi s space  ca n be en dow edwit h a  m etric tha t is ade qu at efo r t he pu r poses of  t rut happ roxim a tion.In §6 so me  r e lat ions be t we e n  the  a cc ount  of  t ruthl iken e ssadvocat ed in thi s p a peran dPoincar é's conventio na list stanc eingeomet r y  are poi nted out. 2 Topology, m e tric, a ndorder  Inthis sect ion  the b asi c  notio ns oftopol og y,t h eoryo f  m etr ic  spaces an d ord er st ru ctur es arerecalled that will beneeded in the  fo llow i n g.  Th i ss ect io n  ca n offer only a ve ry brief a nd in complete s u rve y.  For amore th oroug h  introd uction to  the topi cs  men t i one d in thissectionthe reade r  is r e co mmend edto con su ltthe l iterat u re(Dave y & Pr ie stley1990,  J ames 199 9, Nachbin 1965, Nagata  1985,  Niin ilu oto 1987,  §§ 1.1  and 1.2) . Le t us start  wi thtopol ogy .  Defi niti o n2.1 Let X  bea  set. A t o po log i c al  structureO ( X ) o n X i s g i ven by  a s ubset O(X) ⊆ P(X) of the power s et P ( X ) o f X that  i s assumed to s ati sf y the foll ow ing propert ies: (i)  ∅ and X  belon g to O (X). Tr u t hl i keness  for  Th eories on  Co un t able La ng ua g es 5 ( ii)If  Ai is  an ar b itra r y  collection of s ubset s of Xb elo nging  t o O(X), then  the set-t heoreticalunion⋃ Ai  also  is anel ement
 point_at which_a conventional choice has_to be_made,_can be_determined_quite precisely, namely_when it comes_to the choice of_a metric that_is_compatible with the 'objective' ordered topological structure. The outline of the paper is as_follows:_In the_next_section_we recall the rudiments of_topology, order theory, and metric_spaces we_need in the following sections. In particular we_state_a fundamental theorem_of Urysohn-Carruth, which asserts that under mild restrictions every_ordered metrical space can be assumed_to have a_metric_that_is compatible with the_order structure. This theorem will be_crucial for the main result of_this paper. In § 3 we characterize_the theories T (L) of a_language L as the class_of deductively_closed sets of the Lindenbaum_algebra LINDA(L) of_logically equivalent_propositions. This sets_the stage for the topological representation_of LINDA(L) carried_out in § 4. In §_5_the logical space_T_(L)_of theories,_conceived of as_a_Vietoris space_of_closed subsets of the Stone space_ST_(L) is constructed. Thereby the Urysohn-Carruth metrization_theorem can be used_to_show that this space_can be endowed with a_metric that is adequate for the_purposes of_truth approximation._In § 6 some relations between the account of truthlikeness advocated_in this paper and Poincaré's conventionalist_stance in geometry are_pointed out._2_Topology, metric, and_order_In this_section the basic notions of topology, theory_of metric_spaces and order structures are recalled_that will be needed_in_the following. This section can offer_only a very brief and incomplete_survey. For a more thorough_introduction_to_the topics mentioned in this_section the reader is recommended to_consult the literature_(Davey & Priestley 1990, James 1999, Nachbin_1965,_Nagata 1985, Niiniluoto 1987, §§ 1.1_and_1.2). Let us start with topology._Definition_2.1_Let X be a set._A topological structure O(X) on X_is given by a subset O(X) ⊆ P(X) of_the power set_P(X) of X that is_assumed_to_satisfy the following properties: (i) ∅ and X belong to_O(X). Truthlikeness_for Theories on_Countable Languages 5 (ii) If Ai is an arbitrary collection_of subsets of X belonging to O(X),_then the set-theoretical union ⋃ Ai also is an element