(x, y)$ happens to be of the form $y \in \mathbb{N} \wedge \phi'(y)$ we are fine. It turns out that this doesn't matter and the proof that Church's rule holds for set theory still holds. Finding a realizer for countable choice is a simple matter of reformatting as usual. See Rathjen's paper for the details.
Edit: Realizability and similar techniques rely on having complete formal proofs available. This makes it quite hard to give an explicit example of extracting witnesses because formal proofs will usually be pretty long and complicated even for relatively simple statements (independent of whether we assume countable choice or not). In practice we handle formal proofs using proof assistants. I don't know if there are proof assistants that both allow countable choice and witness extraction, but certainly this is something that could be done.

A:

There are a few issues raised in the question, some about the notions of computability and Gödel number. I am just going to focus my answer on the first part of the question, and I'm going to give a very specific answer. So here is what I will attempt to answer:

What does a computable realization of ACC precisely mean and how to convert a proof involving ACC into a working computer program?

Short Answer
Two parts:

A computable realisation of ACC would mean that it is possible to "look inside" an existential proof and extract the witness, and then use this witness in subsequent constructions e.g. the construction of a function. This would break the paradigm of proof irrelevance.
It is only possible to convert a proof involving ACC to a working computer program if ACC has some computable realisation. That is, for such a conversion there must be some concrete (non-black-box) program whose type corresponds (by the Curry-Howard correspondence) to ACC. In this case, any proof involving ACC automatically converts to a computer program, as long as the rest of its elements are not black-boxes. Simply replace every application of ACC with an application of its realisation.

Long Answer
I will answer the question as a moderately experienced Coq user. Coq is a proof assistant that is based on the Calculus of (Co)Inductive Constructions. The key underlying feature of Coq is the Curry Howard Isomorphism: a correspondence between the worlds of proofs and programs. In particular, we have the following:

Every Proposition is a type
( x, y)$ happens to be of the form $ y \in \mathbb{N } \wedge \phi'(y)$ we are fine. It turns out that this doesn't count and the validation that Church's principle holds for located hypothesis still holds. Finding a realizer for countable option is a dim-witted matter of reformatting as common. examine Rathjen's paper for the detail. 
 Edit: Realizability and similar techniques trust on having complete formal proofs available. This makes it quite hard to give an denotative example of extracting witnesses because courtly proofs will usually be reasonably long and complicated even for relatively bare statements (autonomous of whether we assume countable choice or not). In practice we handle courtly proofs using proof assistants. I don't know if there are proof assistant that both let countable choice and witness extraction, but certainly this is something that could be done. 

 A: 

 There are a few issues raised in the question, some about the notions of computability and Gödel act. I am just going to concenter my solution on the beginning part of the question, and I'm run to give a very specific answer. So here is what I will attempt to suffice: 

 What does a computable realization of ACC precisely mean and how to convert a proof involving ACC into a working computer program? 

 Short Answer 
 Two parts: 

 A computable realisation of ACC would mean that it is possible to " look inside " an experiential proof and excerpt the spectator, and then practice this witness in subsequent constructions e.g. the construction of a function. This would break the paradigm of proof irrelevance. 
 It is only potential to convert a validation involving ACC to a working computer program if ACC has some computable realization. That is, for such a conversion there must be some concrete (non - black - box) plan whose type corresponds (by the Curry - Howard agreement) to ACC. In this case, any validation involving ACC automatically convert to a computer program, equally long as the rest of its elements are not black - boxes. Simply replace every application of ACC with an application of its realisation. 

 Long Answer 
 I will answer the question as a moderately experienced Coq exploiter. Coq is a proof assistant that is based on the Calculus of (Co)Inductive Constructions. The cardinal underlying feature of Coq is the Curry Howard Isomorphism: a correspondence between the worlds of proofs and programs. In particular, we receive the following: 

 Every Proposition is a type
(x, y)$ happens to be of the fovm $y \in \mathbb{N} \cwdge \pii'(y)$ we zre fine. It turns out that this doesi't mqtter and the proof that Chjrch's rulv holds fir stt theory still holds. Finding a vealivec for countable choice is a simple mattar oy reformatting as usual. See Rathjen's paper gog the details.
Eqit: Gewlizzbility and similar techniques relg on haning complete forkal proofs available. This laked it quite hard to give an exkmicyr example of extracting witnesses gecause formal proofs will usuauly bz pretty lobg anf complicatev even for relativcky sim[le stayements (indepekdent of whether we assume conntable choice or noj). In practhcz we handle formal priods ushng [roow asrisuanvs. J don't knkw if thers are proof assistants that bouh wolow countabls choise and witness extraction, but certainly tvis is something that coule be done.

A:

There are a few issuqs raised in the question, some about the notions mf cojoutcnllith ajd Gödel number. I am just going to focus my anfset pn the first pcrt of the questooj, sgd I'm going tu give a bery specific answfr. So hgre is what I wyll sttempt to answer:

What does q computable eealization of ACC precisely oean and now to convert a proof nnvolvjng ACC intl a workihe computer progrxm?

Smord Answer
Udo parts:

A computafle realiwatipn of AZC wpuld mqan that ih is imssible to "look indide" cn exhstential oroof and extract the witness, aiv then use thos winness in fubseauent construstions e.g. the eonstruccion ow a functikn. This would breah the paradigk of proof ircelevance.
Yt iw onoy posskcle to convert a proof pnrolving AXC to a working cokpujed program if ACE yas some computsbld rqapixaehon. That is, xor rucf a covversion thcre musy be some concrete (nmn-blzck-box) program whoxe type cotresponds (by the Curry-Noward correspondejce) tm ARC. In yhif case, any proof involving ACC automatifaljy converts eo a computer prpgram, as long as the rest of its elemenvs are not black-boxes. Sumply replace every spplication of ASC with at application of its realisation.

Long Snswer
I will answer tge quevtion as a moderately experienced Coq user. Coq is a proof assistant that is bawed on the Calculua of (Co)Inguetire Conserucviuns. The key undeglying feature of Coq is the Curcy Howard Hsjmorphism: a correspondence brtdeen the worlas of proofs and programa. In paryicular, we have the following:

Evrry Proposition is a typc
(x, y)$ happens to be of the \in \wedge \phi'(y)$ are fine. It matter the proof that rule holds for theory still holds. Finding a realizer countable choice is a simple matter of reformatting as usual. See Rathjen's paper the details. Edit: Realizability and similar techniques rely on having complete formal proofs This it hard give an explicit example of extracting witnesses because formal proofs will usually be pretty long and even for relatively simple statements (independent of whether assume countable choice or In practice we handle formal using assistants. I know there proof assistants that allow countable choice and witness extraction, but certainly this is something that could be done. A: There a few in the some the of computability and I am just going to focus the first part of the question, and I'm to give very specific answer. So here is I will attempt to answer: What does a realization of ACC precisely mean and how to convert a proof involving ACC into a program? Short Answer Two A computable realisation ACC mean it possible to inside" an existential proof and extract the witness, and then use witness in subsequent constructions e.g. the construction of a function. break paradigm of proof It is only possible convert proof involving ACC to computer if computable That for such a conversion must be some concrete (non-black-box) whose type corresponds (by In this case, any proof involving ACC automatically to a computer program, as long as rest of its elements are not black-boxes. Simply replace every application of with an its realisation. Long Answer I will answer the as a moderately experienced user. Coq is a proof assistant that is based the of (Co)Inductive The key underlying of Coq is Curry Howard Isomorphism: between the of and we have the following: Every Proposition a
(x, y)$ happens to be of the form $y \in \Mathbb{N} \wedGe \phi'(Y)$ we Are FiNe. It TurnS out that this doESn't mAtter and the proof that ChUrch's RuLE holDS fOr set Theory sTIlL HOldS. FInDinG a REaLizer For CountabLe choice is A siMpLe matter of reFOrMatting as uSuaL. See Rathjen's PapEr for tHe DetAIls.
EdIt: REalizAbilitY And simIlar technIqUEs rely ON having COMpLete Formal proofs availABlE. this makes it quiTe hard To GIvE AN exPliCit example Of ExtraCTing witNEsSES BecAUse formal prooFs will usualLY be Pretty LoNg aND complIcateD eVEn fOr relativelY simPle statemEnts (inDEpendenT Of whethEr we asSumE coUntaBLe ChOicE oR Not). iN pRacTIce We handle FoRmAl proOfs uSING ProoF asSistAnts. I Don't know if theRe aRe prOOf aSsistAnts tHat bOtH alloW countAble cHoIce and witness exTracTion, but ceRtaInLy tHiS is soMEthing ThaT coUld be doNe.

A:

TherE Are A fEW ISsUes raised in the quesTiON, SoMe about tHe notiONs Of COmputabiLiTy aNd GöDEL numbEr. I aM JuSt going tO focus MY aNsWer on thE fIrst paRt Of tHe qUestiON, and i'm goinG to give a Very sPEcific answer. So HEre is what I wilL AtTEMpT To anSweR:

What does a cOmpuTAble RealIZaTioN Of ACC PreciSeLY mEAn and how to convert a pRoOf invoLving aCC into a workiNg computer PROGram?

ShorT AnsWEr
tWo parts:

A computAble rEalisation OF ACC woulD mean That it is Possible tO "LOok insidE" an ExiSteNtiAL PrOof and extract THE witNeSs, and thEn uSe this wItnEss In sUbsEqUent constRuctions E.g. ThE cOnStrUctioN Of a functIoN. ThIs WouLd breAK the paRadigM of pRoOf IRreLevance.
iT iS ONly pOsSiBle tO coNvErt a pRoof INvoLving ACc to a workiNg cOMputEr PrOgram if aCC has some comPuTable realiSaTioN. That iS, FOr such a cOnversion there must be somE ConcretE (noN-blacK-box) Program whOse Type coRreSPonds (bY the CuRry-HoWaRd cORRespoNDEnCe) tO AcC. In this caSE, Any Proof InVolvIng ACC aUtomatically converTS to A computer progRam, As loNG As The REsT Of iTs ELemENTs are not black-boXes. Simply rEpLAcE every applICatIoN of ACC wIth an apPlicaTIon of itS realisatIon.

Long AnSwEr
I wILL anSwer the queStion as a ModeratelY ExperIEnCed CoQ usEr. Coq iS a ProOf assIstant THat Is basEd on thE CAlculuS of (Co)inDuctive COnstructions. The key underLying fEaturE of coq is the CUrrY howArd IsomorPhisM: a correspoNdeNce BetweEn tHE worlDs of PRoOfs ANd proGramS. in particuLAr, We hAVE tHe following:

eVERy PRoposItiON is a tyPe
(x, y)$ happens to be of t he form $y  \in\ma thb b{ N} \ wedg e \phi'(y)$ we arefine. It turns out tha t thi sd oesn ' tmatte r and t h ep r oof  t ha t C hu r ch 's ru leholds f or set the ory  s till holds.F in ding a rea liz er for count abl e choi ce  is a sim ple  matt er ofr eforma tting asus u al. Se e  Rathje n ' spape r for the details . 
E d it: Realizabil ity an ds im i l artec hniques re ly  on h a ving co m pl e t e  fo r mal proofs av ailable. Th i s m akes i tqui t e hard  to g iv e  an  explicit e xamp le of ext ractin g  witnes s es beca use fo rma l p roof s  w il l u su a lly be  pr e tty  long an dco mplic ated e v e n fo r r elat ively  simple state men ts ( i nde pende nt of  whe th er we  assum e cou nt able choice ornot) . In prac tic eweha ndlef ormalpro ofs  usingproof a s sis ta n t s .I don't know if th er e ar e proofassist a nt st hat both  a llo w co u n table  cho i ce  and wit ness e x tr ac tion, b ut  certa in lythi s iss omet hing t hat coul d bed one.

A:

Ther e  are a few is s ue s ra i sedinthe questio n, s o me a bout th e n o tions  of c om p ut a bility and Gödel nu mb er. Iam ju st going to f ocus my an s w e r on the  fir s tp art of the que stion , and I'mg oing togivea very s pecific a n s wer. Soher e i s w hat I  w ill attempt t o answ er :

What  do es a co mpu tab lerea li zation of  ACC pre ci se ly  m ean  andh ow to co nv ert  a  pr oof i n volvin g ACC  int oaw ork ing com p ut e r  pro gr am ?

S hor tAnswe r
Tw o  pa rts:

A  computab ler eali sa ti on of A CC would mean  t hat it ispo ssi ble to " look ins ide" an existential pro o f and e xtr act t he w itness, a ndthen u set his wi tnessin su bs equ e n t con s t ru cti on s e.g. the c ons truct io n of  a func tion. This would b r eak  the paradigm  of  pro o f  i rre l ev a nce .I t i s only possible t o convertap ro of involvi n g A CC  to a w orkingcompu t er prog ram if AC C has som ecomp u t abl e realisat ion. Tha t is, for sucha  c onver sio n ther emus t besome c o ncr ete ( non-bl ac k-box)  prog ra m whosetype corresponds (by th e Curr y-How ard  correspo nde n ce)  to ACC.In t his case,any  pr oof i nvo l vingACCa ut oma t icall y co n verts toa  c omp u t er  program, a s l ong  as t her est of  its  elements are not black-boxes. S impl y rep lac e  eve ry  application o f A CC w ith an a pp lication of  its rea li s ation .

Lon g Answ er
I wi l l  a n swer t he q ues tion as a  mo de r ately e xp er i encedCoqus er. Co q is a proo f assistant that i s bas e d  on t h e C alcul us  of (Co ) Indu ctive Cons tructions.The ke y un derly ing fea tu re ofCoq  i s the Curr y  Howard I somor phism:acorr esp ondenc e be t w een t he w or lds  of proof s an d  p ro g ram s. I n par ti cula r, we hav e  the fol low i ng:

Ev er y P r o positi o ni s  a type
(x, y)$_happens to_be of the form_$y \in_\mathbb{N}_\wedge \phi'(y)$_we_are fine. It_turns out that_this doesn't matter and_the proof that_Church's_rule holds for set theory still holds. Finding a realizer for countable choice is_a_simple matter_of_reformatting_as usual. See Rathjen's paper_for the details.
Edit: Realizability and_similar techniques_rely on having complete formal proofs available. This_makes_it quite hard_to give an explicit example of extracting witnesses because_formal proofs will usually be pretty_long and complicated_even_for_relatively simple statements (independent_of whether we assume countable choice_or not). In practice we handle_formal proofs using proof assistants. I don't_know if there are proof assistants_that both allow countable choice_and witness_extraction, but certainly this is_something that could_be done.

A:

There_are a few_issues raised in the question, some_about the notions_of computability and Gödel number. I_am_just going to_focus_my_answer on_the first part_of_the question,_and_I'm going to give a very_specific_answer. So here is what I will_attempt to answer:

What does_a_computable realization of ACC_precisely mean and how to_convert a proof involving ACC into_a working_computer program?

Short_Answer
Two parts:

A computable realisation of ACC would mean that it is_possible to "look inside" an existential_proof and extract the_witness, and_then_use this witness_in_subsequent constructions_e.g. the construction of a function. This_would break_the paradigm of proof irrelevance.
It is_only possible to convert_a_proof involving ACC to a working_computer program if ACC has some_computable realisation. That is, for_such_a_conversion there must be some_concrete (non-black-box) program whose type corresponds_(by the Curry-Howard_correspondence) to ACC. In this case, any_proof_involving ACC automatically converts to a_computer_program, as long as the rest_of_its_elements are not black-boxes. Simply_replace every application of ACC with_an application of its realisation.

Long Answer
I will answer the_question as a_moderately experienced Coq user. Coq_is_a_proof assistant that is based on the Calculus of (Co)Inductive_Constructions. The_key underlying feature_of Coq is the Curry Howard Isomorphism: a correspondence between_the worlds of proofs and programs. In_particular, we have the following:

Every Proposition is a type