A Model Theoretic Analysis of the Church-Turing Thesis

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ZFC, a dierent on out face completely would turned a we ZFC or is adjustment that If minor but a shift, paradigm. with on current get the either explore could we to inconsistent, is want we what paradigm, 5 a ehv eiinpoeuefrsm rbe i h av es) tcnas be also can It sense). naive the (in problem some for procedure decision a have we Say suce will it now for diculties, these attack to how about idea an have do we While edntma htarfrnet h os kthb hc earvdt ntl eemndformlulas determined nitely to arrived we which by sketch loose the to reference a that mean don't We V ω sas-re oml;bcuetewtessaeiay this nitary, are witnesses the because formula; rst-order a as V ω hsi nuht sals nagmn o the for argument an establish to enough is This . 5 13 n earayko htadiinb a by denition a that know already we And CEU eTD Collection hsi oeuigaihei.Tecoc fhrdtrl nt esi oiae ythe by motivated is sets nite hereditarily of choice The considerations: following arithmetic. Classically using sets. done nite hereditarily is of this context in theory recursion of elements expose will We sets nite hereditarily among theory recursion of Elements 2 Chapter • • • e hoy( oano ahmtclojcsi general). in objects general mathematical to of extends domain objects) (a the nitary theory way of natural set domain very (the the on sets rely nite we hereditarily 1.4, of in universe mentioned topics the discuss to able be algebraic To an using than structure convenient relational more a is using substructure) structure. so a universe, is our subset of every subuniverses (where nite on rely will We hereditarily was use one so.) to this doing and chosen for theorems, also motif limitation main has and their [‘wi03] theory [Fit07], recursion after  discussing for, for idea created sets novel was nite a theory set not reason is primary mathematical (This the represent was all. to this process  theory natural set a in It's objects theory. set of expressiveness The 14 CEU eTD Collection then if is, that neighbours, ingoing its of set v of the for denitions two given have manner. we bottom-up this: in like it put can to we of order So identity in it and stated existence we the and that constructions), demonstrate to (co)limits categorical such turning for tices DAGs tensional 2.3 Theorem graph. target the of 2.2 Denition sets. nite hereditarily all of set the 2.1 Denition relation. membership set-theoretic the denotes pronounce language rst-order the use will We 2 G 1 G ∈ . sacntutv ento for denition constructive a is 2.3 digraph A hr sawl-nw orsodnebetween correspondence well-known a is There A tnsfrietdaylcgraph. acyclic directed for stands DAG uuaiehierarchy cumulative 1 2 u V or , (ie. 1 = rp medn san is embedding graph A . G v V . 2 1 e n extends end ⊆ G sebrhp,adwe edsusstter,w ilasm the assume will we theory, set discuss we when and membership, as V ntectgr fdgah,tk h iga hc ossso nt ex- nite of consists which diagram the take digraphs, of category the In 1 = If V 2 n nta mednsbtenthem. between embeddings initial and , ω G h E ,E V, stes-re tutr flanguage of structure rst-order the is 1 1 = = i h G : E V ∀ ssi obe to said is 1 V 2 1 w or , E , 0 ∩ = ∈ ( 1 V G i G ∅ , 1 1 G ( n o each for and , × h sa nta emn of segment initial an is 2 ,u w, nta embedding initial V = h 1 e extensional V ) h ,v u, ,mroe if moreover ), ∈ i ω V i where , 2 ndsus ta s hr r elkonprac- well-known are there is, (that disguise in E , E 15 ∈ 2 G i ⇒h ⇐⇒ r irps esay we digraphs, are and , V n ω e fec etxo ti eemndb the by determined is it of vertex each if V ∈ os' eedo eea e theory. set general on depend doesn't ω fisiaei niiilsgetof segment initial an is image its if , sabnr eainsmo.W will We symbol. relation binary a is N ,v w, and v , 1 V V ∈ ∈ i n G ω N +1 V 2 saclmto hsdiagram. this of colimit a is u o[c3] e srecall us Let [Ack37]. to due , h 1 if , E V , = e ω ) v n ntpdw n one and top-down in one , , i G 2 P ∈ h nvreo hc is which of universe the V 1 n sa nue subgraph induced an is V G hs r transitive are These . 2 2 , sa n extension end an is h v 2 v , 1 ∈ i E 2 then , e   CEU eTD Collection reig ihasals lmn nacutbyiit e,adtu hr saunique a is there thus is and h This set, inite countably them. a between on isomorphism element smallest a with orderings ordering an getting thus n for and in theory recursion of notions of representation binary the from claim We number ack that see to easy It's set we of ordering lexicographic ucso prtos,moreover operations), successor So representation. binary in for index lowest the nd rule: following h follows: as on ordering) Ackermann's of (as function ento 2.4 Denition e ssi ob of be to said and is subsets) set their also are elements (their sets hrfr,if Therefore, . ∈ ( cemn' orsodnecnb sdfrgvn ipediin o h usual the for denitions simple giving for used be can correspondence Ackermann's ohtentrlodrn on ordering natural the Both i < j i ) V V e ω ω < h ∆ osdrtecaatrsi function characteristic the consider ; h n to hnetevleto value the change 1 htis, that , b and n ...rcriest.W ol utsythat say just could We sets. recursive a.k.a. N +1 h steivreof inverse the is hsas ie hrceiainof characterization a gives also This . { ∪ g 0 g o each For ftemxmmo hi ymti ieec acrigto (according dierence symmetric their of maximum the if m otherwise. stescesrof successor the is n rank { \ } = < n P x n < and V { n fi sin is it if : n 2 of edea ordering an dene we , +1 i m < x n 0 V χ : b < ack V hsi xcl h aerl eices ubrb one by number a increase we rule same the exactly is This . ω h ( ack s1. is ihrsetto respect with ω n succ otipratyfor importantly most , b ewl ee oti reigas ordering this to refer will We . +1 a etogto stebnr ersnaino some of representation binary the as of thought be can oseti,s fal e' netgt h successor the investigate let's all, of rst this, see To . ( ( ∅ N i } ge on agree ( ) V 0 = ) h where , i e n n cemn' reigon ordering Ackermann's and h )= )) +1 o which for cemn' correspondence Ackermann's h sequence the , h } \ hc means which , Set . V V succ 16 n ω hsgvsrs otefloigordering: following the to rise gives This . V m h ucso of successor The . n < χ b ( hrfr ecnuiythe unify can we therefore , V ( b stemnmlstin set minimal the is χ h n h ( 0 I te od,if words, other (In . h h = ) : ( ⊂ )) < i ack N 0 = ) n V Σ (where χ n X 1 on { → : 1 g hswyw dened we way This . b ⊂ ...rcrieyeueal sets enumerable recursively a.k.a. ack a eotie from obtained be can ⊆ hnetevleat value the change , steuiu order-isomorphism unique the is V V n 0 V ( 2 , i ω nutvl:let inductively: succ . . . ) 1 srcrieyeueal / enumerable recursively is e } o which for , while , h ack cemn' ordering Ackermann's eest h respective the to refers ∈ ack V ( n V ω h V ) ω ω V r iceetotal discrete are 6= the i : ω a eobtained be can hc sntin not is which N g = < χ ∈ < < → S b h n n i V n ( i n<ω χ : sin is ) i +1 t i in bit -th relations, n to V 1 = ) V h +1 ω ω ythe by ethe be 1 Say . V then , → and , n A . g N . .)  if . CEU eTD Collection ¡ 2.5 Denition in denition their of to context the in notions these dene enumerable recursively of a subset is recursive correspondence Ackermann's / by image inverse its if recursive, l h eain ftelnug uls ttdotherwise). stated (unless language the of relations the all wrt. . . . like ( annotation extra an with notions these use hc unie sivle) oml hc sbitu sn h ola connectives Boolean the using up built is which ∧ formula A involved). is quantier which tication constructs these call will We and Σ , 1 1 ∨ , . . . , • • If of denition the see let's all, of First fterelations the If omlso agaecnitn fbnr eain,w ilma tas it mean will we relations, binary of consisting language a of formulas ) , ¬ ϕ en of means formula A formula. formula A , → ¡ safruaof formula a is epciey(ecnsekof speak can (we respectively , n . n one unicto scle a called is quantication bounded and ∧ ϕ Let , ϕ ∨ sa is ¡ sa is xseta unicto n one nvra quantication. universal bounded and quantication existential , L N 1 N , . . . , Σ . eas-re agaewihcnan h iayrlto symbols relation binary the contains which language rst-order a be ( L ( oee,w r oeabtosta on hs eitn to intend we this: doing than ambitious more are we However, . 1 Σ ∀ ∃ ecnuetefloigabbreviations: following the use can we , x oml fi so h form the of is it if formula x oml fi a eotie rmqatrfe omlsby formulas free quantier from obtained be can it if formula ¡ ¡ ¡ one nvra quantication universal bounded n i i y y antb eie nmiuul rmcnet emight we context, from unambiguously derived be cannot ) ) ϕ ϕ for for V ∀ ∃ ω x x one quantication bounded nasl-otie anr aign reference no making manner, self-contained a in , Σ ( ( 1 x x 17 formulas. ¡ ¡ i i y y → ∧ one formula bounded ∃ ϕ ϕ x ) ) 1 ( ∃ ( 1 1 x ¡ ≤ 2 ≤ 1 . . . , . . . , i i and ≤ ≤ ∃ fw o' att specify to want don't we if x n ¡ n k ) one xseta quan- existential bounded ) . ψ n . .We esekof speak we When . where , ψ sabounded a is Σ ( Σ 1 wrt. ) Σ  CEU eTD Collection hoe 2.7 Theorem is, That formulas. original of validity the with interfere doesn't in Γ Teter of) theory (The with language then symbol, is there xrse ntrsof terms a universal in like concerning expressed something least as (at appear bounded they are which so ideas, quantiers), express to theory set in constucts of replace terms can in given we are bounds that where so formula, complex 2.11), in condition exact the give 3.5). (see on later show will we as general, in true not is reverse The with equivalent is example above the (e.g., equivalence Σ if example, For other. 2.6 Remark express e | n in and ) oml.Hwvr ti ayt e that see to easy is it However, formula. = A h olwn lse ffrua r qiaetin equivalent are formulas of classes following The utemr,let Furthermore, ie theories Given h rbe sta ti o biu fteepesv oe of power expressive the if obvious not is it that is problem The ϑ fte r qiaetwrt. equivalent are they if 1 < ϑ Tr ↔ 2 ( in ∈ ,b a, ϑ N 2 Θ h If . V (wrt. ieal h e of set the Literally A ) R ω 2 aetefloigsto iayrltoson relations binary of set following the Take od teeaesets are there i holds R , h uhthat such A ymaso a of means by teitrrtto of interpretation (the Γ A i R , , < ilsadfrtes-re tutr egtfrom get we structure rst-order the for stand will Tr Θ sas re tutr,te esythat say we then structure, order rst a is i stesm.T e htteeaeatal h ae eshould we same, the actually are these that see To same. the is ) e 1 {h sa is ψ etefloigrlto on relation following the be , u,say, but, , Θ ,y x, sqatrfe,then free, quantier is 2 Γ esythat say we , osraieextension conservative i | = : x ϑ Thm Σ Σ 1 si h rniiecoueof closure transitive the in is 1 ⊂ 1 ↔ oml atog simply (although formula omlsadtestof set the and formulas h hoe eo st e vrteeconcerns. these over get to is below theorem The . A a ϑ If . 1 2 a , . . . , R Θ n o each for and 1 ilflo rmtecneti u s cases). use our in context the from follow will A 18 Σ is sas-re tutr,and structure, rst-order a is 1 qiaetwith equivalent n omlsaealso are formulas ∈ Σ ¬ V f(h hoyof) theory (the of ( or ω V ∀ e x ω uhthat such Σ h : ( ϑ loteeaesvrlvr natural very several are there Also . V ∃ 1 ¡ 2 ω x oml,bttebud r not are bounds the but formula, 1 ∈ <, , < ¡ y Θ buddqatr ysome by quantiers -bounded Σ ) 1 V ψ Θ y 2 ⊂ ω omlsdnticueeach include don't formulas ) 2 : y hr is there a Σ , ¬ sa is Tr wrt. } B Θ 1 ψ Σ = . Σ i hc sa is which , 1 snteog,w will we enough, not is = : a omlsu ological to up formulas Σ 1 seuvln with equivalent is 1 Γ A 1 { omlsin formulas e ffreach for if , .. h extension the i.e., , e A oml u o a not but formula ϑ a 1 <, , 2 yetnigits extending by ∈ e R Θ ⊂ . . . sarelation a is Σ 1 , Tr uhthat such formula). e V ϑ } a ω 1 . n (wrt. ∈ = Θ Θ b  2 1 . CEU eTD Collection n oiieeauto of evaluation positive any em 2.8 Lemma eoeof one be h 2.9 Lemma Set members is show to have actually this claim We Otherwise ψ rnfraincnb oeidciey codn otesrcueo omls h only The formulas. of when structure is the case to non-trivial according inductively, done be can transformation ϕ V ro (2.8). Proof readability. degrade seriously would time the all e ω 0 ssome is . uhthat such , • • hstermcnb eopsdt h olwn lemmas: following the to decomposed be can theorem This Then osdra vlainof evaluation an Consider rcial ewl osdrol the only consider will we Practically u ¡ Σ Σ = 1 , . . . , 1 omlso h language the of formulas { omlso h language the of formulas Σ x z <, Σ ψ 1 1 z , . . . , omlsof formulas x , . . . , ¡ 1 V ⊂ so h form the of is Σ Let ω n oml.If formula. ϕ , ehv oso htif that show to have We i 1 Tr | 0 = r qiaetwt the with equivalent are and ssial o us. for suitable is ¡ . k k ϕ } 1 ftevleof value the of This . , . . . , ↔ Σ ϕ h omlsaeeuvln in equivalent are formulas ϕ V 0 e ¡ ω a entd hsi laspsil nteohrdrcin.The direction). other the in possible always is this noted, we (as ψ ϕ so h form the of is , n u ∃ ehave we , | = ¡ sbudd then bounded, is ebnr eain on relations binary be ϑ z veries 1 ϕ ϕ , . . . , i.e. , in → S h where , V y ϕ e ϕ ¡ o each For . ω 0 0 ϕ ϕ . i V n ssurely is ; hc vlae ore.Teeaeieymany nitely are There true. to evaluates which 0 h , ω ϕ is ¡ o hseauto.Bcueti a edn for done be can this Because evaluation. this for e Σ ( | = is ∀ i ( i S x ∀ omlso h language the of formulas r loeuvln with equivalent also are 19 Σ ϕ vs. x ⊆ e hni a etasomdt a to transformed be can it then , → e ϕ h Σ y B x e ) y ϕ 1 i sbudd o,teeoei is it therefore too, bounded, is ψ and , ) and , 0 hr utb a be must there V ∃ V . u oteidcin ecnassume can we induction, the to Due . ¡ ω ϑ z ω . i uhthat such ae aigmninof mention making  cases e Let . ϕ 0 ∈ → S ϕ ϕ . 0 Σ satuooy owa we what so tautology, a is be omlso h language the of formulas z ∃ Σ i h u e ihwhich with ( omlsof formulas ∀ i x in e V y ω )( n let and ; Σ ∃ 1 ¡ Σ z ϑ formula 1 h 1 , . . . , e holds. e sis. as u i ) ϑ in ¡  ¡ . n CEU eTD Collection f2.11. of also is qiaetwith equivalent ede by dened be hswyw o formula a got we way This form ede by dened be if is, (that formulas. of structure the to according inductively Proof. h then em 2.11 Lemma of both ecnsml a  say simply can we A 2.10 Denition eue necagal. esay We interchangeably.) used be because e | = i If If ot e h eto .,w aet hwta h relations the that show to have we 2.9, of rest the see to So If edene We ie ls fformulas of class a Given ϕ in ¬ Σ ϕ ϕ ϕ [ Σ x ~x/~u V so h form the of is omlso h language the of formulas sqatrfe,w a suengto sapidol oaoi omlsi it in formulas atomic to only applied is negation assume may we free, quantier is so h form the of is X Let Σ ¡ ω wrt. 1 . y and ] ¬ . and ϕ ξ with Π ϑ τ sasboml of subformula a is e ea be Let V where , where , 1 Σ ϕ sis. as , ω n = τ esay We in r qiaet ec hnol qiaec sconcerned, is equivalence only when hence equivalent; are \ ( ¡ X ,y x, {¬ X Σ V ⊆ ( ∃ ω is ∀ ϕ oml of formula are ψ x ϑ ) . σ x n hnrpaealrmiigocrecsof occurrences remaining all replace then and , V Θ , : is ω ϕ ¡ ϕ τ , nta fhr sa is there of instead  ϕ Σ × ( Σ ψ x Θ y 0 are 1 ∈ 1 ). 1 ) hc seuvln with equivalent is which V wrt. x , . . . , ψ o h language the (of ∨ ω Σ where , If . Σ 1 ψ ϕ h } 1 X 2 h then , and , e e e omlsof formulas or ¡ n , ⊆ Then . , ) ¡ ¡ ψ is V denes i ψ 1 i = Π ω ξ ∆ n ewl transform will We . r qiaetwith equivalent are ∧ is saoi) Let atomic). is 1 is 20 ψ Σ and ∃ 2 ∆ {¬ where , u wrt. h X 1 h ( e fi sboth is it if , ϕ ϑ e {h ϑ ⊆ ( i : i ,u y, y, ∈ e rsm xeso fi) and it), of extension some or elc l ufrua of subformulas all Replace . ϕ A { Θ hnlet then , n ψ ϕ ∈ ) x , (in ∧ in uhthat such : Σ ψ ( x 1 } V ¡ ∀ A Te r qiaetin equivalent are (They . ¡ and ω x ffrany for if ) n sa is and , ⊂ ede by dened be Σ Σ y e ϕ , }i 1 {h omlso h language the of formulas u ψ Tr and oa to y, ) 2 : ϑ ψ , are y { < ) denes x ∈ Π is ultecondition the fulll Σ x : Σ Σ 1 V Σ x ~u ¡ i te words, other (in ω oml of formula oml of formula wrt. ¡ } wrt. ∈ y Π σ X y is with A 1 }i and : Σ in and e e ~u : X ϕ then , wrt. V y n is and , σ ∈ V ω ⊆ ( ∈ fthe of Π ω . ,y x, 2 h X h V e V \ can V e e ω ω i ω ¡ n i ) ϕ   } i . . , , , CEU eTD Collection oaoi ufrua,adte s h bv omlztost e i of rid get to formalizations above the use then and subformulas, atomic to bounded allow language not do we form i.e. the quantiers, of bounded in quantiers bounds the be to variables only allow ( for atomic as binary formally only written have be to can happen we notation case the our so (in formulas, formula atomic negated a or formula function the that shorthand the use to like would we it. discuss.) discuss to don't necessary we it and see might see we to cases straightforward some be the in will be (Yet with fact to interfere this claimed doesn't is Usually to / relations care shorthands). looks the the take it of will (if resolution formula we formal the shorthands, the of such that status use way we a When in them which formula). variables use the choosing by in (e.g., occur that formula yet the so variables in doesn't variables auxiliary other the with choose interfere to don't care they taking with question, in shorthand formula ee si rciew o' s ucinsotad sqatrbudr.Se22 famr formal more well). a as of quantiers 2.21 bounded See function (including bounders. transformation quantier formula as of shorthands kind function this on use take won't we practice in as here, cases above the in (e.g., rela- meaning dened formal include straightforward we sometimes a hand,  one like the formulas, On in tions formulas. writing when use to like 2.12 Remark ∀ z 2 hr sas ain hnsm variable some when variant a also is There nte brvaini ucinapiain fw en function a dene we if application: function is abbreviation Another R nfc,w a swl lo uhfnto one unies u ti o ot ogtit this into get to worth not is it but quantiers, bounded function such allow well as can we fact, In e ( F x ) ( z ~x h e ) e F , y i epciey,s n uthv osbttt hs omldiin othe to denitions formal these substitute to have just one so respectively), , ( iei h ro f21:s rnfr ts htngto sapidonly applied is negation that so it transform rst 2.11: of proof the in like ~x eew irs ntelgtmc fsm rcia hrhnsw would we shorthands practical some of legitimacy the on digress we Here 0 )) .S fw aea have we if So ). F ( ~x ) x ∃ sde ythe by dened is y R stastv  transitive, is ∃ e ( y ,v u, F ( ψ ( ) ~x ( ~x, y ) sefru) n aeteshorthand the take and us), for ne is 2 F ,te tcnb rnfre oa to transformed be can it then ), Σ ( ) x ∧ oml ftelanguage the of formula ) hsnesabtmr icsin e sassume us Let discussion. more bit a needs This . R x ( Σ ,v y, ⊆ 21 formula y )) .Teeaesml orsle hyhave they resolve: to simple are These . hc sas a also is which , f ftelnug sgaate ob a be to guaranteed is language the of Σ ψ twl ereally be will it , ( ~x, y ) Let . ( ∀ y h e Σ e F , oml d hstwice this (do formula x R )( ( i ,v u, ∀ wrt. z Σ R Σ F ) ( oml fthe of formula F ∈ fe resolving after F ea atomic an be yaformula, a by e . ( y ~x ) ) weewe (where z v , e ) This . x and CEU eTD Collection h omroe eas norcs ti o reta h atrapoc os' c h mathematical the them. aect on doesn't rely approach we latter and with the substructures, stick that changes we true symbols the not precise, function is of more introducing it and conservativeness context: case cleaner our the is in to approach because latter one, due the former context the that extended think mathematical the might the to one switching aecting Although say where we without extension. language, approach, done extended formal some be more in can a just language taking formulas, resolved or proper appropriately write in; be always can occur we they they that that context herself convince mathematical to the reader interfering the without to it leave we discussion some after formula bounded the by ordinals dene can we so comparability, In ordering. Proof. 2.13 Lemma Σ in) pair ordered (an in by (as quantier bounded existential be outer can the and formula, bounded a with Moreover, otherwise. Apply entries second unique with of pairs shorthand ordered use we of and set entry), a rst each i.e., for sense, theoretic set the (in function oml hc denes which formula V ω tef n ecnuefnto hrhnso h form the of shorthands function use can we and itself, f 3 ewl utilize will We npriua,w a tt h olwn Lemma: following the state can we particular, In in shorthands function use can we Therefore hr r w sa prahst uhhrhns:ete etk hmral ssotad,and shorthands, as really them take we either shorthands: such to approaches usual two are There | .Ti a etae akt h bv aeb replacing by case above the to back traced be can This ). = stefnto hc gives which function the is ∀ Let x 1 . . . ( F ∀ V x ω ∀ If ede ythe by dened be x e o ngnrl nfuddsttere) h odrnpr olw from follows part ordering the theories), set founded in general, in (or n F − n ordinals 1 ⊆ )( ∃ f ∀ V V ! ( y ω x ω x n n ) e n \ safnto n is and function a is ψ a ersle yabuddformula: bounded a by resolved be can ealodnl r rniiest ihnwhich within sets transitive are ordinals Recall . x F ( ) x . y 1 x , . . . , e n f ∧ Σ ( x ( n ∀ formula ) ) x n therefore and , if f e . f f n ( 22 x safnto and function a is )( ) Σ ∀ ψ (where hni salso is it then , y ( Σ x e 1 x , . . . , omlsi h ucini usinis question in function the if formulas n )( x x ∃ sgaate ob ntedomain the in be to guaranteed is e f ψ x n ( ) y x hn as Then, . ) ( ∨ f x nbuddformulas. bounded in ( y 1 x ∆ x x , . . . , e Apply ) 1 Ord si t oan and domain, its in is . x with ∃ ∨ y ( n ) y tefcnb dened be can itself ( ) ψ Apply ∧ = F ( ~x, y x x safunction, a is ) 6= . ( ) e ,x f, ∧ x satotal a is R n ) where , ( sa is 3 ,v y, )) ∅ Σ   ) CEU eTD Collection aetefloigpoete of properties following the Take . 2.14 Lemma works, formula: operation bounded successor Ackermann the how rule the Utilizing eciethe isomorphism order describe unique a is there thus and enum set, innite countably a on element smallest reig hscntuto a euldut esohrta rias(sw ilseit see will we (as ordinals than other sets to 2.17). in up pulled be can construction this ordering, a ii ni.Hwvr fw aesm property has which some ordinal have we if However, it. on on quantication limit universal boundless a have we Here of superset closure transitive The closure. transitive e.g., Take, manner. 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Tr ⊆ n {h hs nfc,masthat means fact, in This, b h bv osrcinrsmlsthe resembles construction above the Nb. ∃ ( t ,y x, e .Frany For . ,y x, ϕ ( Enum ie that Given seuvln iha with equivalent is ) i a edrcl xrse ythe by expressed directly be can : Tr ( ,e y e, n, ( hsflosfo . n h trtdapiaino 2.9. of application iterated the and 2.8 from follows This shown already have we that check and back look to reader the ask kindly We ,y x, h rtmtcoperations arithmetic The Let Tr x Tr h minimal the , ( ϕ ) ,y x, ). ψ } ) h ψ in ( ∧ egtta h relation the that get we , is ,y x, ⊂ aetebuddformula bounded the Take ) ψ V ∆ g yoeeitnilqataino rniiecoue oi is it so closure, transitive on quantication existential one by ω Σ ( ) 1 ,y x, . implies nturn, In . ea be ∃ ain nquestion. in variant t Σ ψ Σ ) ( ( t ∧ 1 oml in formula ,y x, t Σ stransitive is omlsin formulas ( uhthat such ∀ g < h oml,adlet and formula, ) q ∀ ∧ e Tr n y the , ustecniin f21,wihte provides then which 2.11, of conditions the fulls )( 0 ( V + ψ ψ Σ ω ψ ( 25 , ( V . ,e x, Σ · ,y x, ( ∧ µ formula ω ,t x, ple from (pulled formula y r sepesv sin as expressive as are prto faihei euso theory. recursion arithmetic of operation ( {h 0 q ) ⊆ ) )) ,t x, → ψ stetastv lsr of closure transitive the is ϕ t ( → ¬ ∧ ,t x, i eteformula the be y : q ⊂ ϕ ) x t 0 stepeeesrodnlof ordinal predecessor the is ( y hc as says which stetastv lsr of closure transitive the is e ,y x, 0 N ) t . ) via . ) : ack are ) N t . stastv and transitive is ∆ 1 x in ec by Hence . V ω . x n } )) Σ is     . CEU eTD Collection structs 2.20 Denition in functions recursive of denion [Fit07]. in 4.4 4.3, 4.2, topic. the on textbook any in details is in exponentiation discussed based is 2 that showing to down boils number in operations arithmetic the use also can we formulas. 2.18 2.12, by and ordering), Ackermann's Proof. hc is which hoy ohsrslsaentado-nrpaeetfroroe,hwvr hs at ewr o just now were we facts those however, him. ones, by our details for in replacement covered drop-in are to a referring not are results his so theory, 2.19 Theorem reader. the to left is the yields arithmetic multiplication cardinal and of produces formalization union, straightforward disjoint The the product. Cartesian of of cardinality cardinality arithmetic, the cardinal yields of terms addition in i.e., denitions natural have multiplication and addition there Proof. 4 o ewl ieadrc,ped-ytci ento for denition pseudo-syntactic direct, a give will we Now sections see questions, cross-denability these on account self-contained compact a For pulling that show should we direction other the For ntestp[i0]hscoe,teoeao feeetijcini ato h omllnug fset of language formal the of part is injection element of operator the chosen, has [Fit07] setup the In y27w a s aiheissye onsin bounds style arithmetics use can we 2.7 By are they that show to enough is it 2.13, By n ∆ Σ the i 1 omls apn hsto this Mapping formulas. in N Σ i If t i of bit -th wrt. 1 , ψ Π ( 4 1 ~x, ~y < , ∆ sw oe,anumber a noted, we As . ) 1 safrua by formula, a is n in nisbnr ersnaini .S aial h problem the basically So 1. is representation binary its in V ω N wrt. . ∃ ~y V e ω ( ψ with ( orsodto correspond ~x, ~y 26 ψ enum ) buddquantication -bounded ∧ Σ ϕ is aeaihei ihnordinals: within arithmetic take First . ) i n uni to it turn and ilb nlmno oeother some of element an be will e ∆ 1 V Σ vrto over hsi eycasclfc,it fact, classical very a is This . ω 1 , ie,weetebudi wrt. is bound the where (i.e., Π ∆ 1 , 1 N ∆ es nlgul othe to analogously sets, via Σ 1 in omlsusing formulas ack N ema h con- the mean we wrt. ilsarelation a yields < by Enum ack . Σ   CEU eTD Collection to Σ Proof. 2.21 Theorem between switch ucinlk in function-like oetefloigitrsigpoet ffnto-iebuddqatr:if quantiers: bounded function-like of property interesting following the Note respectively. and as these abbreviate will we and omls(as formulas Π • • • • • efr h rnfrainof transformation the Perform ewl a that say will We edetefloigset following the dene We formulas). if RecHF ( if if in are formulas bounded ∀ ϕ ,ψ ϕ, ψ ~y osee To ( sbudd hnreturn then bounded, is : ~x, ~y ψ ∈ stesals e ffrua hc a h bv w property. two above the has which formulas of set smallest the is ( ) ~x, ~y RecHF RecHF RecHF and ∃ ~y h esdal by denable sets The then , )) and ϕ ϕ ψ then , r loin also are → scoe ongto,ti loipista hycnb transformed be can they that implies also this negation, to closed is r in are ∀ sfnto-ie(in function-like is fte cu ne ucinlk bound. function-like a under occur they if , V ∆ ω 1 | ( = ¬ tse oso that show to suces it , RecHF ϕ RecHF , RecHF ∃ ϕ ~y RecHF ∧ ϕ ϕ : and , ; ∀ ψ ∈ ψ ( ; ( ~y ∀ , ∃ ( ffrua ftelanguage the of formulas of RecHF RecHF ; ~x, ~y ~y ( ϕ ~y ψ ψ ∨ : ( : ~y ~x, ~y )) ψ ψ ψ ) sfnto-iein function-like is 27 ( ϕ ( r loin also are , ~x, ~y if ~x, ~y ytefloigprocedure following the by ) omlsaeeatythe exactly are formulas ↔ V → ω )) )) ( ϕ ϕ, | ∀ = ϕ RecHF ); ~y ∀ ~x : ψ ∃ RecHF ! 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(also tried shown sketch I be a postings can subsequent given it also recursivity thinks has of he he arguments how and standard [Ena04b]) by science, seen computer be theoretical can / statement theory above the that claimed He the took was: conclusion has his who Enayat Finally Ali result. was this It evaluate to 2004. eort June-July in list mailing Mathematics dations correspondence) Enayat (The 3.10 Remark determined. semidetermined. nitely negatively and semidetermined nitely positively are ( sets which decidable ones same; the the also are are sets semidetermined nitely negatively and sets Σ [Ena04a] in Enayat Ali was by It xation positive witnesses). that nite out of pointed who encodings the involved it as complex → 3.9 Remark 1 mlcto smwa eebigte2 the resembling (somewhat implication 3 esadpstvl ntl eieemndst r h ae n hn fcourse, of then, and same, the are sets semidetermined nitely positively and sets eeItyt eosrc nytsagmn n aeaohratmtt on out point to attempt another make and argument Enayat's reconstruct to try I Here nitely is it i decidable is set a that showing by strengthened be can this However, o,i at ehv led ueymdltertccaatrzto fdecidability: of characterization theoretic model purely a already have we fact, in Now, eiv hti olw rmsadr ruet n hudteeoeb considered be therefore should and [Ena04b] the arguments also folklore. judgment standard I my but from before, follows In print it in appeared not. that not or believe has it new that sense is the characterization in new, his is characterization whether asked Henk ] . . 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D omlzto rsre neso,a M4sy.Teeoeo the on Therefore says. LM:4 as intension, preserves Formalization . fteManifesto. the of δ ilb one oml,thus formula, bounded a be will ∃ x δ w has ( ,x w, P teei opudiayobject nitary compound a is there i ) where , 51 ϕ h ls fhrdtrl nt eswhich sets nite hereditarily of class the  w δ stefraiaino h description the of formalization the is n t eainto relation its and ϕ h omlcounterpart formal the , x a edescribed be can w V ω which The . 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Thesis Church-Turing capture of the theoretic train deducing same for the use by to wanted existence I to argument come have express I ideas mathematical read be can Both explicit. mathematics modern texts. of self-contained paradigm as folklore a Thesis Church-Turing make the to to tries related and and issues methodological decidability discusses (Chapters of which part equivalence 4), metamathematical Chapters the the 1, and of formulas, proof determined sets the nitely nite with and hereditarily denability statement of context the the in in culminate theory recursion which of exposition an 3), 2, (Chapters dierently. now provide see to I place which this Introduction use the also of I parts those here). to expressed amendments ideas the (of background historical some n r reo h dhceso hoei ahn oesadqaisnatcgenerative quasi-syntactic and models machine theoretic of hocness ad the of free are and 3.10. in demonstrated clearly fairly discourse is of universe This a sane. context, reasonably a on looks express rely we which won't way does The just  algorithms walk. you including seaside   after-lunch ideas an natural mathematical having being for suit from diving far armored an are in formulas dress character, determined nitistic nitely strongly a that have realized problems I algorithmical that reassured us make might 1.4.) of end the at to referred idea the is This rgnlyIpandt aeti hssmr once.Temteaia n meta- and mathematical The connected. more Thesis this have to planned I Originally part mathematical the parts, independent fairly two of consists now document This nteohrhn,Iraie that realized I hand, other the On form determined nitely a to denitions algorithmic converting of possibility While V ω a swa o e hnyufraieagrtmcdenitions. algorithmic formalize you when get you what is can Σ 1 52 omlsaeide eyntrlconstructs, natural very indeed are formulas CEU eTD Collection hm twsawl rnirweetikr fe a otrwetr ok ntedustbin the in books entire throw to had often thinkers where regarding frontier consensus wild established a an not was was It there foun- least them. formal at when or down, Noosphere, laid the yet not of were homesteading of dations nature of true kind perceived formal a the a as of in as concepts, exploration was not mathematical analytic tone then an a of back such part down, when as laid time but were theory, a logic speakers mathematical was mathematical Hungarian there of as guess basics I or the quackery wood. when imagine is of fashion, can made it I ferrule think a it!). will it it, in vein proof, put used technical would mathematical are more a variables a not (even of is proof readers it mathematical that that a fact of the tone a on It have put does Manifesto. I Long emphasis I the much deduction of how the terms of matter the style from No the Thesis Church-Turing be the the might to are thing get it such to of One use aspects frown. which readers idea my an make have which I ones that. from Thesis Church-Turing the of duction the to belong not do but concept standard either). a curriculum, standard are semideterminacy extensions nite (end positive determinacy xation, nite extensions, and end grok of to necessary concepts not non-established is It the basic science. the computer theoretical with and familiar logic is mathematical who of background concepts necessary the has anyone accessible, Thesis. more my much of again, title yet the So from formula. fun the made of and me structure with the ironic by intension was reected is fate is it Because intension and notion. syntactic matters, the what but is witnesses, corresponds nitary what of notion concept theoretic naive model the the to not while is And, it  semideterminacy so. nitely doing positively in characterization upon theoretic preconception model much their have too also formulas apply you of without procedures that terms decision worry in for in to properties trade having can expressing you of concept naive idea a is naive which the witnesses, nitary to correspond clearly They schemes. ial,afwwrsaotta htIdbe steogMnfso,adtede- the and Manifesto, Long the as dubbed I what that about words few a Finally, became Thesis Church-Turing on thoughts my way this way: a in news, good is This 53 V ω ymaso h oinof notion the of means by  is weird. 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Manifesto frameworks Church-Turing Long all the that of fact puppet Deduction the this The is in entitled and admit Manifesto, drama Long get a the to of stage aegis needed I the I theatre under which out it premises rolled those aiming, was collected I then where and concepts, purposes, naive my on pseudo-semantics for articial custom-made an imposed I that impression the have might rather deduction, leaky a arbitrary day. for another any vote yet induction, I pleasant promote Thing. a to Real than attempts the it those personal as  than my model quackery issue Regarding machine a this abstract this. of contrived about about less care nothing it really do feel not can I do I stance: I thing, take-it-or-leave-it point of this sort at this a curious!), is how about even discussions too, open, to unsure, just open am (not am I it I while reasonings. Nevertheless, pseudo-formal methodologically. via qualies operate deduction philosophers not that do suspect days I enough but familiar these trends, not of contemporary am of I about know, statements midden make don't the to I on formal that  philosophy with up a regarding ended in And and survived science. vague either of regarded history reasonings were the similar or and theorems base, mathematical formal as solid form considered a is upon mathematics then built Since be paradoxes. to to leads do they what that realizing upon mmr ocre bu h atclrtrsIue nta euto.Tereader The deduction. that in used I terms particular the about concerned more am I Σ 1 omls tmgtse ob uldcs fnepotism. of case qualied a be to seem might it formulas, 54 CEU eTD Collection aeawr eei t w ih ni tcmie otemciecd fstter.(n fsm day interfere some doesn't if which (And paradigm separate theory. a set be of shall code theory machine theory.) category the set then to with anymore, compiles this it like until be right won't own it its in here word a have pitch. a win can who one only the am I sure made is it friends. where and bids machines Turing mastering with of as here matter arbitrariness a such just no is is picture There in getting concepts. and the with procedure, problems of have concept might the topic to the attached long with as familiar it sentence, less read the is the just with who problem you objectifying one No or but like you, concerned; this to am to answer I sense an as make imagine can it I have it? to grasp sentence and this to explanation any add procedure a we turn can Remember: that procedures. suggests here to used rewind notation just convinced, procedure not I'm a are turn again: you can over it If thinking After no. with machines? sticking abstract of expressions harmless perpetrators elaborate the a as using is arbitrariness when it object  think nitary Still I compound So a subobjects. like mention subobjects. also besides I stucture, that inner customization mention their did sentence about previous see the spoke typing I upon I Manifesto and if the to general back looked as just I acknowledged Well, being structure? inner for successfully more apply could . 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(1968), 20:2952 , ayit fthought of Labyrinth nopeeesi h ado sets of land the in Incompleteness 1:01,1937. 114:305315, , 59 ul yblcLogic Symbolic Bull. ikäsrVra,Bsl eodedition, second Basel, Verlag, Birkhäuser . u 04 onain fMathe- of Foundations 2004. Jul 1 , Mathe- of Foundations 2004. Jun 2 , oue5of 5 volume , http://www.cs.nyu.edu/ http://www.cs.nyu.edu/ 43:95,2008. 14(3):299350, , tde nLogic in Studies Mathe- Com- CEU eTD Collection Sr8 psoo Syropoulos. Apostolos [Syr08] theorems. incompleteness Gödel's and sets Finite ‘wierczkowski. S. [‘wi03] In analysis. conceptual machine: and man by Calculations Sieg. Wilfried [Sie02] some and subcategories Cone-implicational Sain. Ildikó and Németi István [NS82] In theory. category for foundation a as universe One Lane. Mac Saunders theorem. preservation [ML69] Feferman's of proof theoretic model A Marker. David [Mar84] eodteCuc-uigbarrier. Church-Turing the beyond Mat.) (Rozprawy Math. tiones 1998) Log. CA, Notes (Stanford, Lect. mathematics of foundations the on ections In theorems. Birkho-type III Seminar. Category Midwest the of Logic Formal J. Dame Notre 1982. Bolyai János Soc. Math. Colloq. ae 90.Asc ybl oi,Ubn,I,2002. IL, Urbana, Logic, Symbol. Assoc. 390409. pages , Hypercomputation nvra ler Ezegm 1977) (Esztergom, algebra Universal 53:11,1984. 25(3):213216, , 2:8 2003. 422:58, , ae 37.NrhHlad Amsterdam, North-Holland, 535578. pages , 60 ae 90.Srne,Bri,1969. Berlin, Springer, 192200. pages , pigr e ok 08 Computing 2008. York, New Springer, . oue2 of 29 volume , oue1 of 15 volume , Disserta- Reports Re-