
prenex samanya swaroop

jarr teey parimani aanee baddh chalanchi eka string mhanoon lihile gele, jyala upsarg mhanatat, tyanantar parimani-mukt bhag, jyala matricks mhanatat, tarr vidhey cullon sootra prenex normal swarupat (pii. aine. eff.) asate. prastavanatmak tarkshastratil samanya swarupansah (udaa.jee. vichhedan samanya swaroop kinwa sanyog samanya swaroop), heey svayanchalit pramey siddha karnyasathi upayukt pramanit samanya swaroop pradan karate. 
shastriy tarkshastratil pratyek sootra tarkikdrushtya prenex samanya swarupatil sootrachya samatulya aahe. udaharanarth, jarr sootra _ 1, sootra _ 2 aanee sootra _ 3 heey darshavilelya mukt chalansah pariman-mukt sootra asateel tarr
matricks sootra _ 5 saha prenex samanya swarupat aahe, tarr
tarkikdrushtya samatulya aahe parantu prenex samanya swarupat nahee.
prenex swarupat rupantar.
pratyek pratham-kramankache sootra tarkikdrushtya (shastriy tarkshastrat) prenex samanya swarupatil kahee sootrachya samatulya asate. asey anek rupantaran niyam aahet jeey sutrala prenex samanya swarupat rupantarit karnyasathi punaravrutti mhanoon lagoo kele jaau shakatat. sutramadhye konate tarkik sanyojak disatat yavar niyam avalambun asatat. 
sanyojan aanee vichhedan.
sanyojan aanee vichhedan niyam mhanatat qee
aanee 
jevha sootra _ 18 heey sootra _ 19 chey mukt chull mhanoon disat nahee tevha samatulyata vaidh asate; jarr sootra _ 18 heey sootra _ 19 madhye mukt disat asel tarr sootra _ 23 madhye bandh sootra _ 18 chey naav badaloo shakato aanee samatulya sootra _ 24 midvu shakato.
udaharanarth, ringchya bhashet,
pann
kaaran jevha mukt chull "x" 0 chya barobariche asate tevha davikadil sootra konatyahi ringmadhye kharey asate, tarr ujavikadil sutrala mukt chull nasatat aanee konatyahi kshullak ringmadhye chukiche asate. tarr sootra _ 27 pratham sootra _ 30 mhanoon punarlikhit kele jaeel aanee nantar prenex normal formula _ 31 madhye thewale jaeel.
nakar.
nakarache niyam mhanatat qee 
aanee 
parinam.
parinamansathi chaar niyam aaheth done jeey purvavartitun parimanak kadhoon taktat aanee done jeey parinami parimanak kadhoon taktat. heey niyam arth punarlekhan karoon kadhale jaau shakatat
sootra 36 heey sootra 37 aahe aanee vareel vichhedan aanee nakarasathiche niyam lagoo karatat. vichhedan niyamanpramane, yaa niyamansathi aavashyak aahe qee eka upasutramadhye pramanit kelele chull dusya upasootrat mukt disat nahee. 
purvavartitun parimanak kadhoon takanyache niyam asey aahet (parimankatil badalachi nond ghya):
parinami parimanak kadhoon takanyache niyam pudhilpramane aaheth
udaharanarth, jevha parimananachi shreni hii nakaratmak nasaleli naisargik sankhya asate (udaa. sootra _ 48), vidhan
tarkikdrushtya vidhanachya samatulya aahe
aadhichya vidhanat mhatale aahe qee jarr 'x' konatyahi naisargik sankhyepeksha qamee asel tarr 'x' shoonyapeksha qamee asel. nantarche vidhan asey mhanate qee kahee naisargik sankhya "aine" astitvat aahe, mhanje jarr "ex" hii "aine" peksha qamee asel tarr "ex" hii shoonyapeksha qamee asel. donhi vidhane kharee aahet. pahile vidhan kharey aahe kaaran jarr 'x' konatyahi naisargik sankhyepeksha qamee asel tarr tee sarvat lahan naisargik sankhyepeksha (shoonya) qamee asali pahije. nantarche vidhan kharey aahe kaaran "n = 0" heey soochitarthala eka swarvidnyan banavate.
lakshat ghya qee kansanchi mandani hii parimanikaranachi vyapti soochit karate, jee sootrachya arthasathi khoop mahattvachi aahe. khaleel done vidhanancha vichar karah
aanee tyache tarkikdrushtya samatulya vidhan
aadhichya vidhanat mhatale aahe qee konatyahi naisargik sankhya 'aine' sathee, jarr 'ex' haah 'aine' peksha qamee asel tarr 'ex' haah shoonyapeksha qamee asel. nantarche vidhan asey mhanate qee jarr kahee naisargik sankhya "aine" astitvat asel, jee "ex" hii "aine" peksha qamee asel, tarr "ex" hii shoonyapeksha qamee asel. donhi vidhane khotee aahet. pahile vidhan "n = 2" sathee nahee, kaaran "x = 1" heey "n" peksha qamee aahe, parantu shoonyapeksha qamee nahee. nantarche vidhan "x = 1" sathee dharoon nahee, kaaran naisargik sankhya "n = 2" "x <n" chey samadhan karate, parantu "x = 1" shoonyapeksha qamee nahee.
udaharanadakhal.
samaja sootra _ 53, sootra _ 19 aanee sootra _ 55 hii pariman-mukt sutre aahet aanee yapaiki konatihi done sutre konatehi mukt chull samayik karat naheet. sootra vicharat ghya
sarvat aateel upasoochananpasun suru honare niyam varanvar lagoo karoon, tarkikdrushtya samatulya sutrancha khaleel kram prapt keela jaau shakatoh
mool sootrachya samatulya heey ekmev prenex swaroop nahee. udaharanarth, vareel udaharanatil purvavartichya aadhichya parinamashi vyavhar karoon, prenex swaroop
miloo shakateh 
samaan vyapti asalelya done sarvatrik parimankanchya kramane vidhanacha arth/satya moolya badalat nahee.
antardnyani tarkshastra.
sutrala prenex swarupat rupantarit karnyasathiche niyam shastriy tarkshastracha mothya pramanat vapar karatat. antardnyani tarkshastrat, pratyek sootra tarkikdrushtya prenex sootrachya samatulya aahe heey kharey nahee. nakaratmak sanyojak haah eka adthala aahe, parantu ekmev nahee. shastriy tarkshastrapeksha antardnyanvadi tarkshastrat implication operatorla dekhil vegalya paddhatine hatalale jaate; antardnyanvadi tarkshastrat, disjunction aanee negation vaparoon tyachi vyakhya kelee jaau shakat nahee. 
kahee sutranmadhye antardnyandrushtya-samatulya prenex swaroop qaa nasate heey bii. etch. keey. chya vyakhyevarun spasht hote. yaa arthghatanet, eka purava 
heey eka asey karya aahe, jeey thos 'x' aanee sootra _ 71 chaa purava dilyas, thos 'y' aanee sootra _ 72 chaa purava tayar karate. yaa prakaranat 'y' chey moolya 'x' chya dilelya mulyapasoon mojane manya aahe. purava 
dusarikade, 'y' chey ekach thos moolya aanee sootra _ 74 chya konatyahi puravyache sootra _ 72 chya puravyat rupantar karnare karya tayar karate. jarr pratyek 'x' samadhankarak sootra _ 53 chaa vapar 'y' samadhankarak sootra _ 19 tayar karnyasathi keela jaau shakato parantu asha 'x' chya dnyanashivay asey konatehi 'y' tayar kele jaau shakat nahee tarr sootra (1) heey sootra (2) chya samatulya nasel.
antardnyani tarkshastrat 'ayashasvi' honanya sutrala prenex swarupat rupantarit karnyache niyam pudhilpramane aaheth
("x" heey sootra _ 88 chey mukt chull mhanoon (1) aanee (3) madhye disat nahee; "x" heey sootra _ 89 chey mukt chull mhanoon (2) aanee (4) madhye disat nahee).
prenex formcha vapar.
kahee purava ganite keval tyaa siddhantashi sambandhit asateel jyanchi sutre prenex samanya swarupat lihili jaatat. ankganitiy padanukram aanee vishleshanatmak padanukram vikasit karnyasathi hii sankalpana aavashyak aahe.
pratham-kram tarkshastrasathichya tyachya paripoornata prameyabaddalachya godelchya puravyanusar serve sutranchi prenex samanya swarupat punarrachana kelee gelee aahe.
bhoomitisathi tarskyche svayamsiddha shabd hii eka tarkik pranali aahe, jichi vakye 'serve' sarvatrik-astitva swarupat lihili jaau shakatat, prenex samanya swarupache eka vishesh prakaran jyamadhye konatyahi astitvatmak parimankachya aadhi pratyek sarvatrik parimanak asato, jenekarun serve vakye _ 90 sootra _ 91 sootra _ 92 sootra _ 93 sootra _ 94 sootra _ 53 yaa swarupat punha lihili jaau shakatat, jethe sootra _ 53 heey eka vakya aahe jyamadhye konatehi parimanak nasate. yaa vastusthitimule euclidean bhoomiti nirnayak aahe heey siddha karnyasathi tarskeyla parvanagi midali.