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.smsst. 94

Cosmos and History: The Journal o Natural and Social Philosophy, vol. 2, no. 1-2, 2006

The Law o The SubjecT:aLain badiou, LuiTzen brouwer and The

KripKean anaLySeS o orcing and TheheyTing caLcuLuS

z s

abstract: o t tl tsks bsBeing and Events t lt t t s-t t k xmt tt tl t mtmts, , t s,t lssl zml-kl st t. T st, b, s sstll ttt ts vt, stts tsl m t q , s ll s t lst stm tss tt v t stt. T stv t s, tsl, vs t tml l tt. ot vt l tm, t stt, b, qtl st sttl tll, .. st-ttl, tms,s s t t vt tm v l lssl st t. b vtlss

sks t tlt t tll stt t st t st t, ts sts t cs ts t : t m vs s t st-ttstt t tt t t st ss, t ltt vs s t mmt l tstv , t l t st. T t t, t st s -l v vl sttmts m t ll stts t tt tt t ss. btts st-tt stts, t t vt tmlt, b vss tst s l l tt stt m l, st xsss l qt stt m tt vs t xmt lmt slssl tl. Ts vs t st s t q t bs k. w stklsml t t tt L.e.j. b, t ttst mtmts.b, t, ssts t ss stt tt m l, ts -l tml ss. Ts ss, t, s tl tt t tms tvt

t st. Mv, ts tvt, t t l ttst mtmts st, xsss ll stt tt, 1963, Sl Kk s t sm tt lt. i t ll ss, i tk q t t stt ts tts t st, sk t lt t t ts v t sts; t m, ll s, ml t st t st, l tltt ts m. T s t xmt t sqs tt ts stmls bs sltl lssl t tl, s tt tls tt .

Keywords: b; b; ittsm; ; gt; St


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zachary raSer 95

T t lts t m m: s tmst t tm, t t t t m, t s m t sm st t.1

g.w. L

o t l stks al bs Being and Event s t tlt t t st st t k t tss tt tl, t s qua, s t t xmt st t (sll, t zml-

kl xmtt z). i t ts tss, v stt tt sv situations l t tt ts tt s s ml tt stt s mtmtl set. T mlt t st,v, s smt tl. b ssts tt st t l ts mlt t t st, tt ts tsk s t sstll-mtmtl ts tm t vt. it s vtlss ssl, bmts, t tm t st-ttl m t sts tll st-tt m ts tt, t tm m St.3 iBeing and Event, tst-t stts l t t t ts tt pl j. cvls s k t tm tss:genericity orcing. i tsll,

generic subsets tt t s ml xssl t ml

t s sst. b mls t t sst t ml t t-ll stt t lls tts. a tt, , s mltlt tt s ll m-mt t t stt t s tt, t s t vt s lmt t stt, t s t lst stm ms t t stt qst. T t t t t tt vs m tt t ssts t t tt t lmts tt ll v t, t t k st, t event(i.e., tll st

1. gtt wllm v L, o m, g.h.r. pks (.), Philosophical Writings,ts. M Ms g.h.r. pks, L, j.M. dt & Ss, Lt., 1973, . 107.. . al b,Being and Event, ts. olv ltm, L, ctm, 005, . 130: St t,s s qt tk t mltl, t stt stt,ormalizes

stt tsv s s t ts t ltts s s; tt s, t mltl mltls mks stt. i, t ts mk, ts t ml atl stt,t t s st t s ast s tt ts tsts, t lst st, xssl tl t s l l, ml t tt t stt sttt st-t qst. t tts ts s ll vt be.3. hstll, t tm tt s st t t lsl t Mt hsl k. T ss i ml t , v, s sstll St. T tt t -tsl (l: t st) s t -tsl s s t is, s t s, stt m ts lt ts tvt. S j-pl St, Being and Nothingness, ts. hl bs, n yk, pl-sl L, 1956, pt ii, ct i, ii, . 79-84. Lks, t t bs st ssmt tl s t t sst, t t st mst vtlss tt sts ts tll s; t s ls -xstt xss v ts , be . 35.


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t mmt sst stt s tll ). T st sv s tmll l, t ls t, part s tt s, tsl, ls t (s ss t ts t). Forcing s l-t t t sts l t t ml t sst stlt, sttmts extension ts ml, stt tss t sst qst. b, xsss t l t s-

t, t stt m t tvt t t st tsms t stt t ss t tt s tts. Ts ts v b tt mtmtl mk vs t st s tll sttm l m kl, t s ts ll m t v tsmtv

t t t ts lt t t tt tt t s.b s t t st t v t st ts ; stklls t t st xss t k Lt ets j b-.4 b, t, vss t st s tml ss xss mtmt-l tts stt m l l, s mstts tt s t ms l t ts l ts Tt. 5 b,t tt t st s t t tml l mtmtstsl, t, mtmts s t oughtt st, t s ts ss s tms stv stt. Ts t sstts t tst xst mtmts, slts sl mtmtl tt b lls ttsm. as b mk,sm tm t t s ts t,

T ittst tvt s - sqs mtmts.i, t ittst sts vl, sl ts t mtmtlstt lt s t lls, stt ll v t t ll stl t; v t ts tt m ll qt stt.6

4. i m l t sss t t lts t t tts b b. it ssts t stl t t pt hlls Badiou: A Subject to Truth, st t ll:

bs vmt st t ttsm ss t svl ts s mm tbs l tt. Lk b, b ssts tt t -x tts(b, cssss, pls Mtmts, Collected Works, vl.1, . 488). Lk b,

b ml sts mtmts m mtmtl l, tl m t m- l s ttl l (hstl bk, . 509-10). Lk b-, b vs tt s stt m t tt tt m -tsts. h sks lt m tt tv t m ts ssplurality o mind (. 487, [m mss]). als lk b, b s t ll llt st t t tt, s tt ls ks st -t t t t l: i tl, [ls] sl t t t t stt (. 487).hll,Badiou: A Subject to Truth, Mls, uvst Mst pss, 003, . 379 ..

5. L.e.j. b, L, at Mstsm, Collected Works, Vol. 1, a ht (.), amstm,nt-hll pls c., 1975, . 7.6. b, Mtmts, S L, pl Ms (.), From Brouwer to Hilbert, ox,ox uvst pss, 1998, . 5.


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zachary raSer 97

a vtl slt ts tvt s tt, t l l t tt-st st, m t lssl ts b tlts t mtlt s t t st s sslv. nt t lssl lt t t ct t tl xtsll tm tssvvs tt. i ttst mtmts, mtmtl xst ms s-l s t stv stt, mtmtl vt ms tt msttv tt t ttst st.

bs vs t t mtmtl lt s, , stk stt bs t tt. i t lst st, v, ts st ls t sl t, t l t st. ws b xmtll

ls mtmtl lt beoret st, s s mtmtl lt s t vm stt l, t ttst stgeneratests lt t ts ts tml xst. ulk bs ssl mtlst t, bvs t mtmts ts st l tl lst ls: t t l mt l-xstt, mtmtll mlstt, t st s t t t stt t s ts tt.

w i s tt t t t t ts t st s ml tplace t st, i m ts qt ltll. as ll s,t tks vs t orm t st stkl sml tms. evts t s m t st s st t st t t l mtmtl tlllt, t s tt tt ts st vst t lsl tv tms t stt m l t l tt. w l m t k, v, ssss lttl m tt v tts t ts lt t t t ts, t st t t t st t s Sl Kks k k ttst smts. i s 1963 , ttl Smtl alss ittstL i, Kk vs ml-tt ttt ttst l. amt slts st tt s llstt cs -lt ssm t ttst tlmt s l s t ts t , s t lt vs lssll ( t L t exl Ml).T t t sq ts, s, s tt tm mlt, t st. Ts s t tk l t t t ltml t bs st tll mts ts tt.

T s t ll q s t lt t ttst t tst t ll vlt tt t ss mtmts, t s lt tmt lts tt t t t ttst t bs ts t st, tll t st t t ls, t mt smtt Kk sv t tm. i ll t b s s.

n t t ttt t, lssl mtmts tsl st


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t m tms, t mst ts t v lss t st mtmtl l l t sts. T s -tl ss m, bs s, tl event mtmts. a m stt tvts tk ss, s l k- mtmtl lt, tt s, t sl qmts ss - t sv t tt sst mtmtl tt. am t mmt sls tt tt lsm, ll ltm t rssll wt t Principia Mathematica, t mlst sl, s tst lt s ( ms) dv hlt. as Mldmmtt ts, t st t m t tl mls tt s

sl lssl mtmts, s t tl xst, t s, t slm-t, t. T lsts l ts ll stt mtmts, s tt t ltt l m t st s xts l tsl, s t m l s rssll q. 7 T mlsts stt slmt mtmts t stk ss xmt t xstmtmtl sls stll t slt xmts t mtmt-mtl sstt t sst l vlt.8 nt tstvts t s l t t tss mtmtl t s t t b.9 as dmmtt svs,

ittsm tk t t tt lssl mtmts t st stt, t s ll t stt s stt, t t,t s s tt smt s mss t lssl mtmts. m ttst stt, mtmts, tl , l t stt m tt, ttss m t s t m l:t l ts stt ts .10

7. Mtmts l, stll sk, v tl stt sts. Mtmts s t t s, l t gk. bt t v vl m tms: l s mm mtmtl mtmts s m m ll. T sq s tt t s mll mssl t l t t t; t, t t . T s m:l s t t mtmts mtmts s t m l. bt rssll, Introduction toMathematical Philosophy, L, g all, 1963, . 194.8. bt ts mms, tll, v s s. i t ss, dmmtt ts,

t lsl sstm, s s t t, lls t stv mtmtl

mms s t l lllmt: s s, rsslls sv tst-tt xs; hlts, gls s mltss tm. o s, s tmtmtl mms mlt v tms, s s l tst t tl -t ts svs s t sl t stt ; t t ss t vtllm t, s tt, m s t t hlt, t l mssl t l msl l-t ll t. Ml dmmtt, Elements o Intuition-ism, ox, cl pss, 1977, . .

9. tst stl t t ttst tvt t t ls t K t- s, s b p, ittsm s (l) K rvlt Mtmts, Studies inthe History and Philosophy o Science, vl. 31, ., 000, . 97-339. sss t lts tK vlts bs vts, s pt hll,Badiou: A Subject to Truth, . 10-14.10. dmmtt, Elements o Intuitionism, . .


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zachary raSer 99

wt s mss lssl mtmts, b t, s l tll -st t t t mtmtl tt xst. S st s sstm-tll s l-lst ts tt s stt m lmtmtl lts t t t t sstms, , t,m t q t tt st st a priorils.11 i tm, ts ls m t s mtmtl stt, m m t lvtt mtmtl tts l v t, mtmtl xsts sls, t Mll ll tl mlt l.1 clssl mtmts mstk t s t ; tt st tsl t ttll, t mtmtl st t sstms t ls -

t l t llts. S s bs t t l t txs.13

3

bs tvt t t st tt l tstvl ts t st at ittsm. T st at ittsm, bs s,

completely separates mathematics rom mathematical language, in particular rom the phenomena

o language which are described by theoretical logic, and recognizes that intuitionist mathematics

is an essentially languageless activity o the mind having its origin in the perception o a move o

time, i.e. o the alling apart o a lie moment into two distinct things, one o which gives way

to the other, but is retained by memory. I the two-ity thus born is divested o all quality, there

remains the empty orm o the common substratum o all two-ities. It is this common substratum,this empty orm, which is the basic intuition o mathematics.14

T st at sks t st mtmtl tt m t l, t t st, s ml m s v tmll tt-t. T m t t-t, s st s t mtv tt in-variance in change unity in multitude15 tt msts tsl tm. Ts ms sltl l t m mtv tms; t t o t st tsls t t. b s qt xlt ts t, tt t l v-l t st t o s m, tt ts t t tkst t l mt t ss sl tm mst stllts tm lt tt xs t.16 o t ss t tt ls t v -

11. . b, cssss, pls & Mtmts, Collected Works, . 49.1. as qt gttl , The Foundations o Arithmetic, ts. j.L. ast, n yk, h bs.,1960, 16, . .13. S b, hstl bk, pls Mts ittsm, Collected Works, .508-515.14. b, hstl bk, pls Mts ittsm, Collected Works, . 508.bs tls.15. b, Collected Works, . 96.16. T st t stt s two st ts tt tt (ls t cantor,Vt ntsvsmml Kssl 1903); . M (Vl. tt. Mt. c. hl


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t st, s tl tt ssts t tlt t tl sq tl ms, t ls v ts sq.it s tt tmt s ts , s mvmt ll t t t tt t st.

T sltl ml t tmt s sstl b. atm-t, s b sts t, s vtll ssttl t t sts v xst-. i s m sltv txts, b ssts tt t t -s t t-t tmt s , t st s s s t t tkm.17 b ltmts t stt mtmtl tt m l l t t sltpriority stv mtmtl stt t t stl-

lt t st l. L tsl s v s smt tls t t xst sl sts; ts ss, t s vst ts ll-tsmss, t tts stlt s ttl ls ml t ss, l mtmtl lbss l stllt t sl , stl m l.18 is smtmts sts tsl s s l stt m tt tt s t ls t ll ts l, t s, mv, v tt sl s.19 w t tlttstllts s s sstllsttv, s s t sts s m t tt tt tssss tsttkt l.0 w mtmtl t s lsl st t l-st t, v, t ss t svs m t tss , mk ts tss ll t m . al ts ls, bsks t vst ttsm t tl mts.

4

a ml t Being and Eventll mmtl t svl ts s t bs tvt t bs t t st.gt tt sl sttmt tvt t ml tms st stt m l, l T, sl tm. it ll ssl t s m lt ts smlts v mlt lss. , lt s mk t k ts.

st ll, t ss t ttst sts stt m l ls

1904, . 678) ss onet s sft, s t mst tt i tk t s st; ts s ls, xtl ts adding(.. stt t l t m s t)presupposes the intuition otwo-ity; l ts ts smlst mtmtl sstm s t t st t t ego whichthinks the thing, Collected Works, . 96, .1.17. . cssss, pls Mtmts, Collected Works, . 480-494.18. Vlm Ts, The Mathematical Roots o Postmodern Thought, ox, ox uvst pss, 001, .48. Ts vs ll-m l t bs t l t 4 ts txt.19. b, S dls, Collected Works, . 450.0. Ts, The Mathematical Roots o Postmodern Thought, . 47.


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zachary raSer 101

t t stv mtmtl stts t lst t. Ll ms t mtmts t t t, ls t t t ft, tv lll xt, tq mm mtmtl stts, sst tm t ts; s tt mtmtl l tsl v t mtmtl sstms.1 b tst, b vs t stt m l tl vt tms, t t st l m t t mst l x-stt lst ts. Ts st, mv, stts tsl m l, t lst t, ms l, ss lt ss t l lsttmt. t bs st, t stt m l s tl st. ll s st t t t tt l ls

bs tt, hlls ss t bs st s slt ttts t mm ts xst22 s ts smt xt. Ts ttl s t-t sv t b st, t tv st ttst mtmt-s, , s ll s, ts t mm mtmtl xst ssmst t pt sm, t it d, ltt t o, vs s t t t vs nm. ulk t pt -t, s, t ttst sm s mmtl, stvl t, sssss t l sttl l t d t t o.

bs v m b, t st t t stt m l t ttst, s tmtl t t l mts bs t: ml, t -vs t t t st m m-s t t ms tt t s tk ,23 b ls s t t t St (be 3). T st tt s st t s ts, lms b,

s l t st, t xv, s tm s mdsts t hl ms ll Mx ( t, hssl St). T tm St s v, lv, -ssttl, -xv. Mv, l ss ts xst t txt tl sss s ts s. (be 3)

it s t flt t l t b t t st sql t tst , s sstll t tt dsts. T b sts sstll t t sss ts tv xst s fm, t ms t xv t s t t t vlt xst (i.e., tsttlt). T ts t t bs st s t m s-stll beoret st. T t v t sk t st, s t s-

t v s t t t. h t s mtl stt

1. b, hstl bk, pls Mts ittsm, Collected Works, . 141.. pt hll, al b t l ls sl, cls rm (.), Alain Badiou: Penser lemultiple: Actes du Colloque de Bourdeaux 21-23 octobre 1999, ps, Lhmtt, 00, . 96.3. . be, . : [i]t ll tt tl ts s qt lss t s ms tt tt-tl tts t m t t st, tsl tl t tl -sss (ll ltl).


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b t t tml l tt tt stv t xmtll st tll txt t t ttsts. Ts txt s v classically, s -v ll tl ks tll stt s xss xmt st t. i tst t t s-

tv tt s tt xl t tks, ts xmtll stl s tmsl stt st tt t t s xstt. o ts t b ls st t t sttl t-cts t-Kt t s tt v s t, tt stlss -sl t s m t tst. o s t sttt t m st. o st t.4 nt l t m t tlst tss tt mtmt-

s s sstll llss tvt t m v ts t t mv tm.5 i, t s t sl t sst tt t s t b ttmtmts s ts cts x. nv s t s tt t v t t mtmts m stv tst.

ct ss m ls t t t stv ls l tm-lt t t T t t ts t st. bt b b st t st s tmll l xst tt l t T. h b vss t b-s tms: t T ts t t -stv sstt stt t m event, vs s t t t st t t tmlt t st tlts s t tt tt t s.6 T sstl vt t t tts t qst t tt tt tts t st,v, s t ml t ts. it m s t l t Ts . b, t vt t t st s t mst tt, mm t ll x. it s tll sttt tks l t mtst s t t stts. m t tt mtmts, s s t vt s tll st, t sxtl t t l mtmtl tl (be 184), mtmtl tl , b, lssl mtmts tsl ( tll st t). is s v

4. al b, Le Concept de modle, ps, Ms, 197, . 4.5. b, hstl bk, pls Mts ittsm, Collected Works, . 510.6. T stt t vt tsl, lk ts xt-ttv lmt -stv mtl lt,

s lttl sml t t t t ittst l. mll, t s v s -ll mltlt, mltl s lmts sst ts ts st X tsl(b tls ts m t st e

x= {x X, e

x}). S mltl, b svs, s mstl

-sttl, t sl-mms tt ts t vt qs t t tsl tt s sttvl mssl (be 304); stt t vtl mltl l ttt s l s. ht xltl ls t t sslt -ll sts ( ss) ttsm, v tt []l ts xl t t tt t mms s-s Smst l tl t t S; ts t s vs m t sttvt v. it ssts t ss smls t ts, aht, Intuitionism: An Introduction, 3 ., amstm, nt-hll pls, 1971, . 38.M s, , v, s t s m t ()tll sstt t vt, t t blttvl lls ts ss ts st, ms t T (. be, 06).


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zachary raSer 103

st s tt vt, t vts xtt t mtmts sts bst t st m s mtmtl tl. T bs st s tsnotml mtmtl stt, m t mtmts s stv -stt. o s, s i v l stt, b s sk t tm t smtmtll stt in t st, t ts t t t bsst tsl.

5

b t ms t t ts t st, mst xm t ttst t sttv mtmts t tl, t s t tt t l t stt t b st, t t-

tst t tt, l ll. Lt s xm t bms mathematical construction. T smlst ts s t tst ttstmtmts t ts lssl tt.

am t tms lssl mtmts, t ttst s s vll ts m vt t xlt tl v ltms. it s sst l l t t s v t-tvl tl , tt s, t k mentalstt.7 pvlt, tt, xst , ttst mtmts, sl s tt. T t-tst s t tt mtmtl s, v s, mtt stvstt s m t l t mtsl ts t lssl mtmts (s ttsm sts t) s m s. Ts t-

s, mv, l t ssl t ttv st t tmstt v m t t t mtmtl tt. as a ht s t,

[] t xst s t m t stt, t mst v sm mtslm. it t t tsk mtmts t vstt ts m t t t s tl t. w v t st mtmtvtl mtt mtsl t lks, t bs mtls tt st mtmts s smt sml, m mmt tmtss. i t st mtl mtmtl stts t xst mst sms t t stt.8

wt t ttst mss stll mtmtl xst t t s s-tvl sttl s mtmts lt, t lss s, ll

, t mtmt ss t tst t s t sslv .9

T ttst, tt t lk t t tt s m, ts t s m-tlt mtmts s tt mst mtmts s t m tlt t tt t. it ls s, ts ht, s t xs xs mts, tl m, t ll sst.30

7.dmmtt, Elements o Intuitionism, . 7.8. ht, Intuitionism, . .9. wl, o t tl ss mtmts, Ms, . 13630. ht, Intuitionism, . 11.


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t ttst, t tt xst stv stt -ls l mtmtl tt xms m lls.31 w -t sml xmtt s ttst mtmts, t s l t -

v stll sl, t sstll s ltl tsks hts195 xmtt ttst tv mt ( s, tll, ssstt t s tt bs svs). T ttst s-xmtt mtmts mks t st v m bs tt.wl b, lk b, ssts lt mtmts m t ssttssst tt t s ts tt xtl t mtmts, tmtmtl tt t t v mvmt ts tt, t m, t xm s

sl t t t mtmtl tt tt. Tk st ss msl s s t t gl plt, b ssts tt t mst st tt s sl s vtv tt t t t tlllt xms.3 b, t sl st mtmtl t-t s m tsl qs, s

t s tlss t t t t t tls stt xtll lts. o t t, t stts stt (t sl t s ttst stts, s t v tt t vt tt s t pltst) sl st t t ms tk s.33

bs tt s st s pltsm svs sm mmt .

bs plt s t qt st m t ltsm tt ts ttxtks t ls mtmts (s ll s m t ttsts s-ssdmmtts txt s xml34). Ts txtk ltsm, s b ss t,sml ts plts tt ,

s t sss tt t pltst sss stt t ttl t xtl, k st k t; stt s ttl t t plt mk. [] plts mtl s s t l t mmt tt, t -l tk m t k, t sstl tll mmslt.35

i m st sts ( sm t v s l t b s -sl ttsm s t v t dmmtt s sl pltsm 36),

31. b, T et ittsm clssl al L, Collected Works, . 551.3.b, pltsm Mtmtl otl, Theoretical Writings, . ts. r bss alt Ts, L, ctm, 003, . 5.33. b, pltsm Mtmtl otl, . 5.34. l xml t txtk ltsm tt b s s, . dmmtts sss ltsm 7 Elements o Intuitionism: [T] ltst t s lm mtmtl lt,xst tvl tl kl, s sttmts t ls.35. dmmtt, Elements o Intuitionism, . 49.36. i sss ttst l tt i l l xm lt ts ss, b ts tt tt-sm s s t mst lls stt mtmtl objects, be, . 49.


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zachary raSer 105

bs vs pltsm ms t ttst vs mtmts. it s v-tlss l tt smt qt t s t stk bs st ttm tt. i t lst lss, ts s vt t t t -mtl t ttst mtmts, ts tss s t ml tt t tt. T mvmt tt t ttsm ssts tl lt tt t tt, tt s, tm. dss t ll m ts ltt t ll ; ss sstt, t s t m tllxss ts l tll vt. cvlls s ts ms ttsmqt ll, t tt t ttst,

mtmts s tms m, m t t m,

t t t s mssl t s mmts t sst t l ss. m t t t ltts, t s tt tlt.37

ws t s t t xms t , s s s ssl, sl m t, ts sl s st ss, v mt ms t-tst mtmts s s lt t ts l ss. it s ts t lsttll qt t t t ats ttsm s xms t lsslss. T sstl t tt t ttst sks t sv t t xmt- mt s t tl t bs pltst, v. b ssmsl s ll plts tsts ts tmtmtl tt q t t t t s t t t ll x -t-

tt. ast t ttst lvl tt lss tt , tt -sttlt, b ss t lssl ( tt, hlt) xm tt ts, tt consistency.38 T xmt ls s tt stt-lt t tk lv ts s, t pltst, b ts, sv t mtv sttv t tt mxml xt-s t ssttl tt.39

o t mst mmt mt sqs t ttst st s t t ct t tll t mltlt, s s t xst- s mltlts l v t tss xm. 40 wt tt-st mtmts, v xst s stvl stt t ms, llt s ttl t: t s mlt t.41 Ts tss ms, sml,tt t s t stt s t s t ss ts t, tt t t s stt s t t tt ss, tt t t stt s

37. j cvlls, o L t T S, js j. Kklms T j. Ksl(s.), Phenomenology and the Natural Sciences, ts. T j. Ksl, evst, ntst uvstpss, 1970, . 367.38. b, pltsm Mtmtl otl, . 54. emss .39. b, pltsm Mtmtl otl, . 54.40. S Mtt 14 be.41. dmmtt, Elements o Intuitionism, . 55.


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t s t tt t ss ll t tmt.4 S tss sts tt-sm t l st m t st-ttl s bs ts, t m xtslt t qt tl ts sm.it s tst t sv, v, tt t ttsts qtl ts tss t sm s b s t st st, ml, ts tt mst t t t t s ts ss t tt tt. t ttst, t ct sts t l ss t, ls t t stt s ls t, sl s tss stt s v mlt; sk tl ts, t c-t sks t ss s t ml tll l t .43

b, t s t vts sttl ttl t t tt t v t l t t ls t. is s t tms tt t t o, s t l lt stt t,44 tttst sst, bs lts, ms slv t t rmtst t-t,45 tt tt mst vm tt s t skl tsl m t lt t.

wt s t stk ts st s, , t tlt t st t l mtmtl xst. S l s t (t) st s v s t tl -t v mtmtl xst, t ss s xsts mst vt st t t lt t t st. m ts stv, t t is ttst stv stt. t ct, t t s l tsss stv t sts st. T st, , s t tt t stt t t, t l ts tvsl.46

6

T ttst tt mtmtl xst t stt, tt t mstt, s sqs tt tt t t t llstt mtmtl s tsl. bs t ttst ts t tt

4. dmmtt, Elements o Intuitionism, . 56.43. dmmtt, Elements o Intuitionism, . 5.44. b, pls Mtmts: it t e rmtsm, Theoretical Writings, .

5.45. xt lss t lts t bs ttsm t rmt tt,s t 4 Vlm Tss The Mathematical Roots o Postmodern Thought.46. . b, Le Nombre et les nombres, ps, ts Sl, 1990, 3.17: ev l traversetm m t ls ss, sss s t mst mm (t t tl , m t), mst t ll tt ts ls stttv t m? it s s ts v t ss m m t t xt, m ss ms t ts lmt. bt t s tt v lst mt t t l tt m s sttt s sss. [sc. .nn, 3.18: ctl, t ttsts t ts mvs stv.] it m ll ( ts s mtss) tt m tsldoes not pass, tt t s mmmll l sm xtsv t ts -. [] t m m s t tll st mmsl t st, mms t t ts.


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zachary raSer 107

sttmt t t stt tt vlts t, t lst sttmt t tstt tt mstts ts st, l s s m-t tt v sttmt A s t t ls t t tt tlvt stt. T l t t, l v t mt tt tstt sttpriorto its having been constructed, s ssl.

T mst mt sl t ts tt tt s ttsms ll-k t t L t exl Ml (LeM). LeM stts tt, v sttmtA, tA s t ls ~A s t, tertium non datur. T st A or ~As ts lssll vl A tsv. wt ttst txt sttmt mst v t s t tt l sst, A or ~A, ms

l mt t slv v lm, m xltl, l mt st p ls slt t p ~p. as t ssss s mt stt, v t t sst t l.47T ttst t LeM tls t t ts ll, t l l t. Ts l stts tt ~~A s t l A s, ltmts mt mt (qt mm lssl mtmts) k s apogogicindirect, tks t mstt t tt A mstt tst t st A (i.e. mstt ~~A). wl ttsm tst vsl vlt tt lssl mtmts vs t LeM ts sqs, ts, vtlss, mt t ltm s t sl msts. stl, LeMls A vA s l tv st (s, A = ~B): t ~B~~Bmst t, ~~~Bmls ~B. Ts s t t t tt t ttstts s sl-vt t l ttBmls ~~Bs ll s t l tt v A Bt ls v ~B ~A. o t ttl tst s t t tt LeMs ls l t vl ss s t sttl t m. Ts ts s tt

v stt t t t mtmtl sstm l ttmt t m s, ttmt t t mlt, t t tl t ss s mssl.it lls tt v sst sslt stt tt . S, ts xtl s, lt t l t xl t s mssl.48

b s tt t vslt t tt v l ttt t

LeM sl t ts q m s t t s tstts.49 o mtmts ts t t vstt t t

47. ht, Intuitionism, . 101.48. b, hstl bk, pls Mts ittsm, Collected Works, . 510.49. i ittst St T, b lms tt LeM t xm slvlt (tt v lms slt),

ms tt v t t t st stt t sstm lssl l mt mtmts ssts t t st, t ttt t ts sstm xst- t mtmts, ll l t l t ss ts t tt t mtmts t sts. (ittst St T, Ms, . 7, .4)


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b ls ms, tt s v mltLeM mmtl lss tsttv .

b vts s Mtt 4 t st t ttstt LeM t lt l l-t. bs st ts mtt s (s mt xt) sltl lssl. hs mt s stssm t xms st t s t mm t t. S vst-ttl st s sstll l t sttmt tt s t t m st xxsts, s tt t m st x, s tt s t xst, bs tt t ss tt t sttmt s t t ls s t sstt t, tmt mltlt s t xstt -xstt. S s-

t s stl, b ss, s s l t tm, t-xst xst, s tm t, l v t t t t t -xst xst (be 50).T s stl t st . it ls tk ttst l t -tmtl tvlt, tt s, t l t t tmt tt vls. ol ts t s l t s t ttst l t tll t t t l t sst. Ts st mstk t v t t ttst tt tt t mstt xst t stv stt. t ttst, t (~P) s sttv mstt st, fmt (P) s sttv mstt vt. w t ttst ssts ~~Pt -t sst P, t s s s stt mstt t st stt mstt t st P, t s t (t) sttvlmstt P. wt v s xsttl t ~~Pls lack o oundation P. T xsttl lt t ll t ~~P Ps t tmt tm t xst -xst, t smlindetermination xst, t ttst ls ms stv st-t. b ls ts t stt t mt m t tst t txt xmt st t, xst -xst stt vsll vltl.

Ts ls m mssl t t txt bs ts, tks s ts tll k t tt lssl st t. it s ssl t bs mks ttst l s xlt t L t

exl Ml, sqtl t tv mt , s vl

els, cssss, pls Mtmts, ts ttT l l t vsl vlt t l t xl t mtmts s s- ttsm s m st vlt t sm k s t l-tm l t tlt t tt t mmt xs ss t t t. attsm ts t xl t l sst ts m t ts: stl t vs --ttt t l t sl sst; sl t tl vlt t l lssl l xtsv simple every day phenomena. T ltt t tl m s st mss tt t l tt tt lssl l ll s, m -tt tt s s t l s sl t v st. Collected Works, . 49


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zachary raSer 109

t t (mt)tll mk Being and Event. wt s lss tl s st vs lm tt t t LeM,

ttsm s mstk t t t t l k t tl t t come rom elsewhere, sll m t mtlltv ts. i tl, ttsm s s t mst lls stt mtmtl ts. (be 49)

T lm tt ttsm s ts ls m t st mtll tv -ts s ; , ts t b s sss t mst tv -ttsts (l, t sms, t b ht). T lm tt ts lsm m ls t t m tl, v, sml ssts bs

xmt tss tt tl islssl st t; ts st, t lm s tvl, s s tt ttst l tll ms m lssl stt. bt lt s t mv t qkl t ll, t tl tss Being andEvents tt mathematics, tt t tt ts stl m, st s xssl q (be 8). wt ts mtmts s lssl ttst ms s s; t s t v bsmts ( tmslvs m lk xms) tt t stt stlll l tms mltlt. ittst mtmts, t, ss tl v xst s l s mltlt, t st l-st pt sm tt o (t t T). i s,t t tt ttsm s t v ts ls m mtll tv -

ts s t l t tss tt ts ls tl s t s expressible ; t ttst tlst l v flt t t tls , s s ls ttl t tt tt t lssl tlst ls, t , ls m m t mentally eective operations regarding fnite collections, -st s t s m t ts t tl t mltls t stk. bs sst, s t tk t t t s qt ,tt ttst l m l t t mst lls stt mtmtl ts s t q. T t ttst mm tks ts t t sk t vm t svtl stt tt m tsts ls mtmts, tt mtmtl mtss t mts objects.50 T l ttst t s t mtl t vs mtmts s tms tvt t st, tt t xtl t. it s st tt b lts t mt ts; ss sstll t sm t.51

7

T ll sqs t ttst st t -tm-

50. b, hstl bk, pls Mts ittsm, Collected Works, . 508.51. Ts s t t bs pltsm, s sss 5 v. (. b, pltsm Mt-mtl otl, . 49-58.


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tt mst sl s llstt ml ts l, s sKk s s 1963 , Smtl alss ittst L i. its txt, Kk vls ml-tt ttmt hts mlt ttst l.5 i ll st, t, mst t ttt Kk hts vs , t ss, xtl t ttsm. T ml stts m t ss ttst mtmts, , t t ttst ts, t tk s xss t ss t-tst mtmts tsl. T mks tt ht mks t ts t t t - s ml rls ittst L t t . ittstmtmts, ts,

s mtl tvt [Denkttigkeit], t v l, l tmlst , s l tl mmt. it s l mssl tst sstm mls tt l qvlt t ttst mtmts, t sslts tt t t t m ls st v. [] t stt mtmts t s t ss t st ll ls l vlt; ts ls sv sl s t mtmtl sstm st.53

Mtvt t s t ltt t tt ttst ts ts ts ts s,54 ht vtlss s t stt t -l tv stt m ttst mtmts. T slt s stl t lls, st t ml sml stl, t llsqs t ttst st sstmtll l. Kks ts s t t v ml t hts ll ll, ts v v t m t tx t ttsm stt sml t lssl mtmts. bt v t mss Kks ml lssl mk s t t sstl lm . Kk msl vs svlts t stt t ml m t ttstll, s s t ml lssl s s ml mtt x (t slmst ls sml t lssl mstt t ttst ).it s t tt, t ttst, t t t models ltt s.bt t s ts t?

b t, t l ml ( ml smts) s t tm t vt sss mtmtl ll sstm ml stt

t sts t sstm s t t t tts t t ts t m. a mtmtl ml ts ssts -mll s m ts (sll sts, ssts lts) t s, ll t interpretation t ml, stls t ts ts

5. Kk vs ml t t stl t t lls ttst l. it lls, v, ll stt ttt t t stl lls t sk smlt vt.53. ht, Intuitionism, . 311.54. ht, Intuitionism, . 311.


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zachary raSer 111

t stt lmts t sstm qst. as b s Le Conceptde modle, t ml-tt smtt tt s l s mstvl ls t t mst svtl m, mks tt s t xtl ts.55 Ts s sl t tt tttt ttsm sks t vm t tt t t stv mvmt msttv stt, tt t xtl t. nvtlss,b, s t, sks t s t t t ml m ts mst -t, ll s tt t s t ss t mtmtl mlmt mls s ss t t svtl mst tmt sts ts. Ml t s t t lt t

mtmts ts xt. esstll, ts s s t t ml stt t ttt t ml sstm qst s vlt tmslvs tl within mathematics. Smts, l,

s intramathematicallt t t xmtl tss(ml sstms) t mtmtl ts, s t s,ts t, tk t mstt, tt v smtt t llt xs st l t v stts t ts.56

T s mls, ts v, s t t t m mtmts stlxvt, s ll mmt t mtmtl tt; s mtmtsll xtl ts t t mtmtl kl.

is s t s Kks smtl lss vtlss vts smt

m t st ttsm, ts vt l ltts q. it llss t stls mm mtmtl t t ml sts tbs bs ts t st t. T stt tt ll v Kks smts ll, ssl, vt .

Lk ml, Kks ssts t stt mts. st, v tmodel structure, s s st K, st lmt G K, xv,tstv lt R v K. G s ql s s t t t ltR, s tt t xsts H K s tt HRG (G s R-mml K). S, v t interpretationunction (P, H), Ps v sts t htlls H s v lmts K. T vls ts t v tst {T, F} (mk st ssmts !). w ls ms t t tt, v t lmts H H s tt HRH, (P,H) = T mls (P,H) = T. Tt st s, t lt R svs tt-vls.

w ll ssm tt s ss vl m {T, F} t tm st t l. i ts, v, t s l t t tt l t vl T svs tst ttst tt (mstlt), Fdoes not immediately represent intuitionistic

alsity (mstl st). it ss l t absence stt vt st qst (ll t P), s ll l stll t t

55. S al b, Le Concept de modle, ps, Ms, 197, ts 4 & 5.56. b, Le Concept de modle, . 53.


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kl tt Ps intuitionistically alse t s st tt H xsts stt (P,H) = T t ts ms lt. T t t m s tt t xs-tv ssmt tt-vls t t tm ml t l s t ttt ttst t t lssl vs -tmt tt LeMsts. T ml v t ssmt F sl ts s tt s t

t tm s t t ls.T smt vls mlx sts t v t lt

ml, t t ll ls t tvs t l. Ts s lls:

. (A & B, H) = T (A, H) = (B, H) = T; ts, (A & B, H) = F.

. (Aor B, H) = T (A, H) = T (B, H) = T; ts (Aor B, H) = F.. (A B, H) = T ll H K s tt HRH, (A, H) = F (B, H)= T; ts, (A B, H) = F.

. (~A, H) = T ll H K s tt HRH, (A, H) = F; ts,(~A,H) = F.57

as Kk ts, t ts t (&) st (or) xtls t s ts lssl t st (94).T ts mlt t, v, stl m tlssl tts. xml, t sst t t A t st ts s stt, t s ss t st tt ssl xts tsstt s l vA. Ts tl t sl m; llt t ls. T t ms mlt svs t vt i then lt t t ttv tss , s -t stt ls ll t tst, s lk lssl l. ittst-ll, m l fm sts t m A t B t s ssl tsttvl tsm stt vA t vB. i Kkssmts, ts t s xss llA Bt v stt Hl xts H H svs ts mlt.

it s ssl t llstt ts ll stts, s Kk s, ms m. T t-lk stt fgure 1 s ttst ml mlA m-s t v tvs t tm s-ml P, Q , R.58

57. i qt ts ts lmst vtm m Kks txt, lt l t tv sm-ls t m t t st ts t ll tt s b.58. Figure 1 s tk m Kk, Smtl alss ittst L i, . 98.


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zachary raSer 113

P,Q

P,Q ,R

P

P,R

P,QG

H2

H1

H4

H3

figure

i t ml mm v, v tk G, H

1, H

, H

3, H

4t t l-

mts K. h, t t s t. T lt R s st ls sss t t, s tt HRH v t m H tH. nt tt t R-mml lmt G s t t t t t. i t m, tltt tm ml s tt v H

v (, H

) = T;

(, H) = F, s t v H

. as Kk s t,

w t t s H t st ts tm ( vtl stts), t m v vs s mt. i, t tl t H tm, v mt t v st A, s tt (A,H)= T; lk s mt, s tt (A,H) = F. i (A,H) = T

s ttA s verifedt t t H tm; (A,H) = F, tA s not beenverifedt H. [] iH s stt, s HRH , s s k, t ttm H, m lt t mt t v t H.59

Kks ts ss t t tml ms tt, ttsmssts, mst t ll s qt t stvl stt tts.Tt, , , s mt l t x t xst sttv mst-ts, s t s tt t s mmmll v ssl st; -sts v tt l t ss sttv vt ms t ss.

a tst t t ttst t ll tm ( m ll t tt), stt l tt s ls smt mst t t tvt st tm, , t tt s l t tll vl. T l

ttst mtmts, s s t mtl t m st ls t, s t tslss, st t l tmlt t stt sttt ts tts. dmmtt vs ll xml ts t. 60 t

59. Kk, Smtl alss ittst L i, . 98. Kk s t m t tt v m v. it vtlss s t:

nt, t, tt T F t t ttst tt lst; (A,H) = T, tA s v t t t t tm H; t (A,H) = F s t m ttA s valset H. itsml s t (t) v t H, t m stls lt. (. 98)

60. T ll xml s s dmmtt, Elements o Intuitionism, . 337.


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ttst, 188, t t k Lm, t sttmt s ts-tl m t. p t 188, s tt xst; t s vtlss mssl tlm tt 1881, s, s nottstl, t s mls -st-tv m t: xst 1881 tt s l msttt -tstl t , ms xst mstt tt couldxst tt l stls tt m tstl. i 1881, tt sttmt s tstl t sttmt s t tstl t,t t t ls. as t sttmt t s tmt t s t-stl tts s sml t mtmtl sttmt.61 it s sttmt tmtl. b mtt s mtmtl sttmts l ts l t

xst sttv , ttsm vs t ttst tslss sts tmll t vts. i ts ,ttsm s l tts tt t tl t.

dmmtts xml svs ls t llstt t v t t Kkml, ttst l l. as i v t, t s , 1881,t s t ltmt t fm t -tstl t s tt xst tt s l s tt Lms ( sm t t t smt) s t tm. Ts stt s s xss qt ll t smtttt t Kks t-ml. T sst ~A ttstll tstt H, Kk ts,

t k t H t l ttA s t v t H, t tt t tssl v t lt tm, mtt m mt s ;s s tt (~A,H) = T (A,H) = F v H K s tt HRH.(99)

T ttst sst tv st s ts t ml sttmt t s t tll t s (t s t t stt t s -stt); t s sttmt t cannot t s. Ts mlt t tv stst t ttst st tt ts stv ss. wll t t ls.

8

i Mtt 7 Being and Event, L mtmtq st -

s,6 b tls t t t tts tt, sts tm scsttvst, g Tst Tt, stvl. T ml

61. . ht, Intuitionism: An Introduction, . 19: ev mtmtl sst xss tm: i v t t sttA m m. T mtmtl t ts sst xss s i v t sttB m m, s tt m t ss-t tt t sttA t t , s t sm m. o t t, ttl t t st sst s: i v t t t sttA m m; ts sttmts t t m mtmtl sst.6. al b, Court trait dontologie transitoire, ps, ts Sl, 1998, . 39-54.


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zachary raSer 115

t ts txm, t s mmtl tmt t l ttsm t csttvst Tt. o l mtmtl st, t s m t m-m stt ttsm t t sttvst tt, t s mm -t t ltt t s t xsss sttv mtmts ttstmtmts m lss tl.63 nt t ttst t sttv-st ( bs ss t tm) t xst stts t stt t ss t ltm, t sls tt sst tstt ll qtt t ms l-stt tts.64 bt t t bs xst sttvst tt t l tt ts t- s smt ll-st t b ttsm. csttvst tt, s b

sts t, s ts ss [] ll mm. o, t xt, t ss ttl vls s t m t m tl s xsttmltlt (be 87). nt l m tmt t bs tt. as v s, bs st (t st at ittsm) s t mm sss mtmtl tvt m l.65 Ts st t t t mtt, v. T flts tt mst vm l ttsm t any bs t tts.

all t m tts tt tt b sss, s s t xt mtmts, mt t t ttmt ctstm lm. Ts lm s t qtttv lt t t st tl ms

0 t l m tm, , m ll, t

v tst st

t st ts ssts (

). T qst tt t ss s, t s, qt sml: m ts l? o, qvltl, mssts l t st tl ms? i 1963, c s ts lmt l t ss t xms st t. T qtttv ssts v lmts t st t v ms tsv, t t sl v; cts tm l tlls s tt t tllm ssts v st t t lmts. b, t st-tt

63. . ett bs, Foundations o Constructive Analysis, n yk, Mg-hll, 1967.64. t sttvst, b ts, ss t xsts, ts mst st s st xsts tm m t stt; ss ll, ts mst st s, llm tms t stt (be 87). i ts txt, m s tk t m t ltm

t mltl qst stt.65.Mv, t t sttst l tt b sks t t t t sttvst tt tt s qt t b, l v l t stt t tst ss stlt, sst ss st t l t t tt t stt. . cssss,pls Mtmts, b, Collected Works, . 487: o s t ls t-ll llstt s sm t tt t t, sl t mmt tt, tl sl t mmt t t stt. St t stt, t ll lst t (Collected Works, . 487). T s mtmts s t l ts st s l m bs vs mks t mtt: mtmts, tsv t, stts tsl m t ll s t stt. b ts v t, t s tt mtmts s lt t l. Ts l s t t t xt l, t ttss sslt, Collected Works, . 484.


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xms ss t plt is tl, tts ts mtmtl mss s l l s s, l vsll smt ms , stv s. T t t tts tt s ms sl ts ss, , t t v lst, ms t ts (be 83). T Tst tt ss t x st t t - t tt mltl s tt t s vt s t (be83); t txt st t, ts t s xt t vt xmssttt t xst l ls, tst ms vstl tst -t tt ms t st-tt xms. as bsts t, ts t s mtmtl l t v s t t

g. b t s lssl ts t t st tt xmt, st l t ss tt ss v xt ms (0).

T csttv mm tt ss slt t t sm lmt t mt stt t xstt t t tvl sl. ist t, ts tt s mst gls sttl ml t xms.Ts ml s l stt, tks t mt st s ts mtvsttm ts ssqt sttm tk s lmts ll ssts tvs sttm tt s ml stt t tt sttm.66 Tslt s st ml st t tt vlts t qt |(

)| =

+1, stt-

tt t st tst l

s sl t xt lst ts-t l. T t tt, b ms g Tt, s ts m sk t sl t s m s t sks t sls t tmst xss (be 83). T t tl t ts tt s t ss mtm t sl, s t tt t ml tstt t m s st m t ts t m s l sltl tl t tm (be 83). T g ott s tsmtmtl xss cs k t tm lm, st s tt mt t ll lt sl sts t ml st t, mls t t tmxs tt t tl ms s m s lks, s tt t st

0

m ss lt t ll tt s t t 0

(t t sl xttt t l slt t l t

0; tt s, t t

0).67

66. S Mtt 9 be m msv ttmt gls . gls st-t s slts vlm s Collected Works.67. b sts cs slts Mtts 33, 34 36 be. cs mst ssl s-tt s k s t : pl c, Set Theory and the Continuum Hypothesis, n yk, w.a.bm, 1966.

o t t gt otts, t g ott ms lsst t bs t, ss cs t ssts t v t st t tt tt t x-sss t ts tll stt. nvtlss, b ss t st msl smt mt g ott s s, ss s k s s t , tt s tsvslt t t ts,

ls tt t truth t tll mss t s tt mm t tl


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zachary raSer 117

T flt tt ts s l ttsm bst ( ) s s tt t mss t t s, st s ctstm lm, s sttl sk, s invisiblet ttst mtmts.68 Tss s t lm s ms t tss tt t s ltmt t tt ttm s mlt, tst st st tts, t ts ssts tl ms. T ttsts l ts tss t mssl.69 T s-t ts mtt s ts m t sst t l t ts t t t t t , t t st t ts t t.70 t ttst, t lssl m t tms tl tmt, t st ts ( sts tl ms) s

s t v. T qst t qtt s st s t vs t t ttst l. it l vtlss t ssm m tstt ttsm s t tm ltt, ls t ts t ltmtl mtmtl tt. o t t, t s t st t t l ml lm t s t lt t t tm t tl ms tt ttsm s m ts mst st v-ts. a, t t t st bs txt, t s ts l tt ttsm s tsl

tsl, t sltv mttl. it sss t -ms t stt [sc. t st ssts, tl m tm t st st s

0] t t stl lmtt [].

its tss ssts s tt l render justicet st m t l t vt tvt. T s ts t - , s t s tl sm - tt v tt ts, ltt tt s stks l tt v -. (be . 84-5)

68. p, . 317.69. b. Ms Mt s tst ss bs t sttvsm slt ts t t Ls tt. Mt vs stt s tt s t lk t :

T tm, L, s m ts: ms, , s b ts, sm-tms qt t mtsl ts, t t ssttm t sttml xts tt xstl llsl. bt ts s t m, s b lms, tt t m s tt mltlv tstl t t tm, st t stts t t mmsl l.

ist [] t tm ssts ts mmslt; ts lt s t sml t t l, t stv qlt in actu vts t q lst stt[]. i t s ss t sss L m ls mtmts, t m

lss t sttvst tt t t ttsm b ht, m ttm s xll st s s dis-continuity, m mt mtmtl sm ll tt t, s t xstl t ts-t ts t v tt s m llt ts. b,ittsm mlsm, pl b hll ptm (s.), Philosophy o Mathematics:Selected Readings, . cm, cm uvst pss, 1983, . 80. b. Ms Mt, Tct rvlt: al b t pls St T, Polygraph, vl. 17, 005, . 87.

o s lt t s t t sttvst s ts.70. o t s ts t s, . wl, o t tl ss mtmts, Ms,. 95: T qst t t tm s ml t ssl s ts t, , - t t, t s l l t t tm st st lmts, l ttxls ms.


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v t lt ts t t mtmtl st s s t tstt mm ts, ts, st ll tts, t kl mtts.

9

i t v m s t s t stk , lt s xm l smlml ml t l tm, lssll st. cs t t , s s mk t 0s 1s, s lvls mt t tl ms (t lmts

0). w ll ll branch sq s

m t t t t (< >) tl. Ssts 0

t t s b

, t t ll vt: n l b s 0 ts n

t l (.. b(n) = 0). S l s t t lsslmtmts, m s t actually infnitestt, s m-ltl tvs ll t tl ms n. e b

ts mltl s

sst 0(i.e., sst t tl ms). n, sst

0 m t

s t sq tl ms l (ct s s ttt tls ml, s ssm s mt s tk l lt sst tl ms t sst t tls). S t t stt sslts xts t v st vlmt t tml, b

(n) = 1 b

(n) = 0t m ssts

0mst ql t

* * * ad infnitum, 0, lssl mlt t t tm.71cts lt tm tt t st S s ss lss t t

t st ssts S tlls s tt 0 t qtttvl ql t 0 tsl, t ts, lssl mtmts s t tm. ev-t s t s tk t ltmt sst

0, , t t t t ,

ltmt t t . it s tt t mm-sttvst tt st t l ms ts stt t xstt t t lstll stt-l, mtt l ssts v tv t t st tt s stt ts .

T l ttststs k t t l l t st ats ll s m -ttsts7 lk bl p, m t lm tm tt s tvl sml t t mm ,st v t ttl mtvt. wl t t tt t ttv

tm m ll t mtmtl tlllt, t -mtll ms t lmts t t ll t t tltm, sst st ts l sttv ms. Ts lmtts, t, ms t t tt t ttsts l l tt mltls (e.g.,

71. i ts stt m M Tls, The Philosophy o Set Theory: An Introduction to Cantors Paradise,ox, bsl blkll, 1989, . 66-67.7. p-ttst s tm v b t -mtl mst mtmtsl p, bl ts, b s m smts, sll t tvlmt s mt ttst mm. S hstl bk, pls Mts ittsm, Collected Works, . 509.


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zachary raSer 119

ssts 0

ls) s tvl t sttvl kl lxss t l t. T t-st tt t l ttsts t- s sttl s v smll t t tm tl t mmtt; ts t x t ml s t l t t,v vsll, t t st t tl ms.73 it s, v, sf lmt t svl xtt mtmts.

T vt ttv q t tl tm s vtlsstl t t ttsts. T slt t tm s, b, stl tt mst vm. hs ss, tl ll vlt-,74 s t vl t t tl ts t lm t

tm s s. T t t tm xlts t t m-tl ls tt kt ttsm t st m t lssl st-ttlttmts t tmt sl t tt t tm s t-st any, t sst t l potentiality t tms xst-l t t. i t ts, b s, t s ss t sss ttl sst s t st at. a S at itt s tl. Ts at

recognizes the possibility o generating new mathematical entities:

frstly in the orm o infnitely proceeding sequences p1, p

2, , whose terms are chosen more or

less reely rom mathematical entities previously acquired; in such a way that the reedom o choice

existing perhaps or the frst element p1may be subjected to a lasting restriction at some ollowing

pv, and again and again to sharper lasting restrictions or even abolition at urther subsequent pvs,while all these restricting interventions, as well as the choices o the p

vs themselves, at any stage

may be made to depend on possible uture mathematical experiences o the creating subject;

secondly, in the orm o mathematical species, i.e. properties supposable or mathematical entitiespreviously acquired, and satisying the condition that, i they hold or a certain mathematical

entity, they also hold or all mathematical entities that have been defned to be equal to it75

T S at mtll ss t ttst mtmts, vs t k t b lls ittst St T, sl, lk ts ct tt, sts tsl t tsk t s t tlt t tm. bt t t t t b tt tl t t ts s ct, v. i t l ll

tt ct slt s t sts s tt v xt qtttvms t l t st t st t t tl ms, t tsk

73. Mk v att, b, s nv r hssl, Synthese, vl. 137, . 1-, . 3-19, . 3.74. hm wl, gm ls mtmt s bs m s-t lls, msl xlm: btt s t vlt!, Vlm Ts, The MathematicalRoots o Postmodern Thought, . 54. i s tl ttsm ml, Mk v att ts, sml v: a 1917, t vlts tk l, k, t. T t mtmts, sst t tt sqs b, b, s nvr hssl, . .75. b, hstl bk Collected Works, . 511.


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t ttst st t s tt mtmtll tk t tm ts vindeterminacy vis--visst mt, t ts tt lttt tm sslv t tlll mst. T t tm,sll t mm tt, ms ls mtmtl vstt ttsm, s xss t l s slss sqs.

i t s ts s ssl, t s ss t s t tstt t S at qts s. T t stts t at xltlts t choice sequences species. a ss s sstll lss, , t-ll s lttl m t lssl t, sv t s t ss lsss

ttstll mssl stts. o sqs, t t sstlts. Lawlikesequences tl sqs tl ms t sttl mtmtl stts tmt ltm l. T t s ls t t b lls discerniblestst ll m t tslt. Free choice sequences tl sqs tt t tm l ltm. bt t t, vt tmt ms ssl, ls m ms mv m sqs t st t vlmt s t st s ss. a spreads ss sqsssss mm t st tm, s v t ls ( ll t lls t t sk vt): st, t s t spread law, t

X

X s t s qst. Ts l tms t msslt fnite initialsegments sqs t t s. ev s l mst: (1) mt t mtsq < > s t t t s, () t mt sq ssss mssl tl smt, (3) mssl tl smt, mt t lst ssl xts ts smt t t s, s tt v mtt smtm tl l t lst t. t stts m mst ss t s m. T s l s ll t complementary lawt s, s s st

X. Ts l mts s t ss mtmtl

tts t t t tl ms ss, t v mtt sq ts, sm t ttstll stt stt. T l stt tsl s tt t tvl l v ssmt.

w st t t ttst stt t t-m. ws t lssl tm s v s tmt st l ms,

t ttst tm s ms real number generators. T m sslms ts; ll s tl sqs tl m-s {r

} s tt |r

r

+1| < -. rl m ts, ttst mtmts,

ls t t lssl ts l ms. T l ls t t tt t t v s mlt t sts, t s tsll t-m, tl sqs. T ttst tm s stts lls: ssm mt t tl ms r

1, r

, ;

t s tl ms t s l C: ev tl m

ms mssl -mm sq; a1, , a

s mssl sq, t


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zachary raSer 11

a1, , a

, a

+1s mssl sq l |ra

ra

+1| < -.76 T complemen-

tary law t s C, t C, sss t tl m ra

t v mssl

sq a1, , a

. T s Cts mss v ssl l m -

t tt m v llk ltms. b ts llk ts, v,t xsts ml llt s tm s-qs t t tm, t v s t , l ms.it s ts tt v t t t tm, , lt tvl l ms, s t sslv t st l ms s s s; t v medium o ree Becoming.77 Lt s t tt ts s mm t t v st tvss t, st l ll sl.78

it s t S at tt stt l k t mm-sttvsm ttsm, t t mtmtl mk t ltt. a sq s ttstll sttl tt tt s notsttl t mm ss. b t, sq s tm sttl ltm tv ml. wt st t t t q,ts stts st t t tt, v t st ll vsttst mtmts, t t s t m xs t bs t sqs. ev ls, t lst mt sttlt s llt sm ts s l k st mm tt. h l v sttv m tt t only stv ss. it st t ttsm s ts stv ss t ts tll m-m tt.

o ts t s mm tt, t t s-qs l sm t t s, st, ts t g ott, s st s t sss sqs t t t m lmt s, lslt llk mts ts tt l t stsl. T m s, m s tt t t ts sqs ls m lsl s bs st-tt mlmt t tt tt t tsl. i t t t st st Being and Event,t s ttmt t mt t mtmtl t t v t, stv tt s tl tm, t s t mmt t. it s, t, t t t ll tt vs t mst t mt-mtl sm t stt stv tt qua. ts

mst lk ls.

10

a sts xst, s b s t, s ls tml, , t t tvt tt ms msl t vt, ssts tvs-

76. ht, Intuitionism, . 36.77. hm wl, o t tl ss mtmts, Ms, . 94.78. . hll, al b t l ls sl, . 96.


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coSMoS and hiSTory1

tl mlx stt t xstl ss tt b llsfdelity. T sss lt ssts m ss enquiriest ssl ts tt m m t t t s s l-mt t stt (smt s st) t vt t t st sks tm tl. a lt s s t truth procedure t t t sst t stt sst ll t lmts stvl t t t vt ll vgeneric. b t, sst s t st s ml stt t t stt, m sl, stt t t mlstt S t stts tll m s xss. Ts ms tt tt stt, t xsts l tt l ss sft t

l t t tt. n, v tt t ls s in time, t vmmt, vtl lt s vsl slt s ms tv qs t s s t t mltls t t tvt (be 34), ts vsl slt s ls fnite. nt t t t lt (ts tll ttlss) t s s slt, s l s st s t t s ls ssl t ms stt ml tt l ss sft t mms tt st, v ts mls s mt s [

1 a & &

n a] n s t m lmts a.

ol t st s , v tt ml l t lt ( s, t allt sts , stt). i s tt t mtmtl ss t bs tt- in the act, t t sl tt sttl xtsl s t sts lt s sft, tt [] qt slss (be 35). i t qtl tk t ss l-t, mst tt t ts tmlt, ts t ts -xstt xss v ts t t vtl stt ts t ts tsl.

T sstll tsl tml t sqs, s vl b s sl, s t t t s t t t tml xtsll tm ls lssl st t. S l s m t tms v st mstt s ssl t t t v-m-lt l t tms b sq. i s t st ts.

Lt s ms t m t st ll t. a- t bs xst, ts ssts st conditions, t , tt s t

lmt sst t mtl stt S t t st. t sk smlt, lt s ll bs tl xml tk t sst tmt st , sts tl, t ssl t, sqs 1s 0s(lt m t mlxt sm ss). Ts sts ll tts . ll bs tt, ll t s sts t ltt ,tt tm t ml ssts ss. T tt ttxsss t mlt stv s m m l t sst v , ll t sst. T t ts ssts s v t ls, t Rd

1 Rd

2. Rd

1qs tt t

1ls t t


30/40

zachary raSer 13

sst , t t tt s sst

1(tt s mt

1, s b

ts t) s ls lmt . h, v {, } , t mst lsv {} . Rd

2qs tt t lmts t sts sts l-

t compatibility mst t. T ts 1

s t compat-

ible l t 1

s sst

s sst 1. xml, {,

} s mtl t {, , } t {, , }, tt t {, , }. i t s tt ll t lmts tsst mtl t t, Rd

2qs tt v ts l

t t xsts t, ls l t , t st t t ssts.mll, ts t ls tt:

Rd1: [1 & 1] Rd

2: [(

1 ) & (

)] (

3)[(

3 ) & (

1

3) & (

3)]

S v t t t s t bs m xst.79 w ll ts , spread t ssts v , ll ll .

i t t t mtl vlmt t t ssts, m ts l t tt s ll , lt s t smtl tt t bs ts. w ll t (n) t t t sstt n lmts. Ts n ll ls st t st t sq qstm t t t s. (m) ll s tl sq (n) (m) (n) m < n. i t sqs t mtl ( s t tl sq t t), ll tt t t ssts ( e.g.

1

(n)

(m)). as sttv, s s v t s ts spread law ts complementary law. h, t sk s, ll t t t, sk t stt tl ms tl t t mt s v ; t lll t

. wt s t s t s s sqs

ll ll t ts . its l mst, t, ml t t ls tss,Rd

1 Rd

2. Ts l tks t m t, s m s t st ssts

(i.e. ()) s s t st {1,0}. Ts t s stt t t 0 ts mt s mssl, 1 t s t. T l

s mlt

s lls:

((n)) = 0 i [(n) = {}] or

[((n-1) = 0 & (n-1) (n) & (1(n-1))((n))(1 )]

((n)) = 1 otherwise

it s l sml xs t st tt sq (n) mtt ts ls t t ls tss stt v. a smll t t slt ss mm fgure.

79. T t t t ssts s t st 3 Mtt 33 be.


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< >

figure

T xt st stt ssts lt potentially genericsq . it mst st tt t t t sq mst ls mpoten-tial, s l s t t ttst s, l tll tsq tl . Ts stt s s t t tt t tt , t t mt v Being and Event.T t mltlt t t xs lt tt-sls l vtl (be 36) ttl. Tt ts t s ll tl ls t stt tsl, s b sts t, s t t t tt ttt- tsl s tll t potentially t ss,

lss t ttstll mssl sq. ev s, st tl - t ttst sq bs tt- s ll; ml, tt t mm t bs st s t ts t.

it s l tt lawlikesq s t t xss t t ttll - , s llk sq s sl s lmts xttm t s t sttl l, tt s, ml sttt t (-stt) vs t s ls. Llk sqs , bs l, sstll discerniblesqs. o m t sst ttll tt s t t smt ttll s t tt s ree choice sequence, sq s sssv s tlstt, s l s t m t t s st t s l. Ts

t, v, s sft, t ts tt s t sq llt vttl (i.e. xtsll) ll llk tmt, m s-l t t stt, ll st t. nt t t llstms t ll sft tsk. T s, v,m smt ls t t t . T t lm ls l tss stts t m t sq, sl t sq tt s t t ts m, t ks t t st m tL. M bs t t qst lt l mts lmt. hll s qt t sv tt b ss m s


32/40

zachary raSer 15

xtll l vmt, qt lk t ts , m Kt tt St b, s t s ss smt.80 it s t qst, , ml m, t t t ; bs tss s tt t lt tst t tll ts-tll ss mtsl: t, s t qst m tsl tks l l xtl vts, lt, s t m t st l ssts l s t st mts t tt t stt tsl m t l.Mv, ts t ts st t t , t vt s lssl sl s sv t vtl t ts sqs (be 11).ol t t t t tt m vt s mk ts

mk . n s lt ts t b st, v t l-lk sqs v tms sqs st sl-msstts.

i t tll stl t stv ss t sqm t s-mm ss tt ss t, xt stt tt x-sss t l sl m tt ts t stv tt -, mst l t stts ts sq. Tsmst sft t sl t sq s tt t s t ll tslt (mtl) t xst llk sq, tt s tst t l vt llk sq ts . T l tt ll msll t ll: llk sq , (n) = (n) t t mst xst smm s tt (m) (m). gv tt s ml (v v s t-ll t sst) ls tvl mt t ss t tl m t ll t ssts, t lls tt v s m slt sst, t t t t s x t l. T xm ts t, mv, ll ls sttl, s t ms ts tm-t l sttvl v t s t ms ltmllt llk sq stvl stt sq.

a sttl tl sq ts st, sftl tst ttl t t , s tt ttll - , ssttl, m llk indefnitely: t s ls ssl t sttts v m v llk sq. it s, s t sk, ll t tm t l t m . it s t mssl t , s ml

v, t s s t . Sttl sk, t truthulness s t sls tsl xtsll tmt v; t tst t l t tt t sq, t st t ts t ts.

a lt t xst tt mst, t, ls m -ttl tt, tt t sft v v v. s l s s v s sts tt st lmts stt, t s l tt v ll ts v stl mlt. T

80. pt hll,Badiou, . 167.


33/40


t t, lk t lss t t Kt st ss, s -t tl t ssl l s l t mmt s xst. S, vtlss, t s q s tll ss, t s t fm s truth, t l progress in infnitum ts ttt , t tt discordance, t t L.81 n lss t Kt,b s t stlt m mmtlt t st. b s tlk t ts x, ssts tt ts ss, stvt is mmtl.8 Tmmtlt v , v, s t tt t m ml s ttt qst, t tprogress in infnitum t stv tsl s, l, l, t tt tt t svs ms t. Ts llmts

st t t bs st tt v t t mt:t st s t tl t t vl s s, t t t t vl s . T m t lltv sts, st s t m l sts, s s xs tt sl tt. all ts sqt t m bs sll qt slsst ts. n s b-s t t st l t xst tl t, tts s qt t s ll mmtst vs t st tsmtmtl tsk.

11

T tt t t s t vt t st m

lss t stlt l tt l m t t ttt . Ts s t t orcingms t l. b t s s-sl, v, t s ss t lt t tl stt t t m-lx ts ms t lmts t l, t xts S() t tl stt S. Ts ms , t t xt tmt t ts, s sts t tl stt t kml, s ssst ts t, vsl stt ms. i t tst vt, i ll tl t ts m ; s mt sllstt Mtt 34 Being and Event; t s v ct iV, 3 c-s Set Theory and the Continuum Hypothesis, stll ts vll t xstltt t t. as c ts, t s mt s t lt

ms s mt s l s v t lt st t x-ts (c, 113). i ml stts, mv, t s tl t xt ttt mt sl m s tt t t. i s, ts sstl s tt t tl vl ts ms s tm t mst ; m sl, t tl vl m s tm t mm-

81. S imml Kt, Critique o Practical Reason, ts. T.K. att, amst, pmts bks, 1996,. 148.8. b, Ethics: An Essay on the Understanding o Evil, ts. pt hll, L, Vs, 001, . 1.emss m.


34/40

zachary raSer 17

s t ts t t t mst t m qst.T stllt ms s t t st t b lls

t st-l, mlm t tv l t stt t mss s tt t mst t tt . Ts ls tll mt ssl tts t tl stt S, s tms t mls, l, t v t S; t stt t t, mv, s t t ll v, v t tl t s l-t t t, ts S, s tl sl.83 ot t ts l, t st s l mk t tss t m: Ii sstt t sl tt ts sts s s tm smtt t t

q , then s sttmt t st-l [sc. t stt, t xts S()] ll v s m ll ( t)v vl (be 400). T ttl t ts sttmts s -ll, t v mltl, slv tt t s t , st lmts t tt m k t t st qst (i.e., s t x n (n) ss). o t t mst , t st sll tlsst s st t mt m t st stt t qs. all t st, tl, s mtt , k l (be 400).

T tl ms t st t mks s ss-ts tss s v torcing relation, b ms s t -mtl l t st. T tll m ts lt vs m csk t tm lm, s s t mstt t xst mls st t t t tm m x 1 vtll t ll (t l stt

0< |(

0)|

0). i t txt b-

s t t st,[t]t tm t sttorces sttmt t st l ms ttt vt ts sttmt t stt t m s qvlt t t l ts tm t t sl t slts m t .it ts ms tt ts tm, t t sttmt t lt ,

83. as b ss t, ts stt s s l s bs k. as j v pltsvs, bs 190 s ttst mtmts lt t st t st

tml tt. Ts sl t mtmtl l, v plt ms s,s ts ttl mtvs t mm t S cl, lsl btt. T l, ts, m t ml ttmt mk t s-lstm. b msl lv tt l s t ml tts tt l t vl tts. m, l s t st l ms tt v ts. Ts t st s-ll ttst vl ( tt, t stll ll sm ttsts) s t tm l vs. (j v plt, rv dk v dl,Mystic, Geometer, and Intuitionist,The Lie o L.E.J. Brouwer vol. 1. The Dawning Revolution, Bulletin o Symbolic Logic, vl.7, .1 M, 001,. 64) i, b s s ttst mvmt t , stl t st , tstl ltl mt, t s tsk t vl mts ls tstl ts m l t st t vs sl t (S- dls, Collected Works, . 448).


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ls t t tt. (be 403)84

T lt s tmtl lt t t ll t implication entailment,s c ts t (c, 111). it tms t stts s tt ll s tt tt ts tt st ls t t t ss S() sstt. as b s t, t s t stk s t mmt l stvtt . it s, svl sts, ls t t l tt s, tlssl ll lls st t ts. w mll vsm lssl l, t s s s s t s mll t v ts vts m t sq s ttl mst s ssl t ltm tm-t. it s t tt ts sl t xs t ttst

st, t m tt t tl s.a t t lt t st t tm ml t -

qtl st t t lmts ts . T s m t- ttmt cs txt, qt lss t tm ml ax 7 Being and Event. wt s m st s, s, s t llstt t lt tks t st t m ml. h, tsttl v m lssl tlmt s ll ll. wt stt t stl tvs,85 t t s s lls:

. s P& Q s P s Q.. s Por Q s P s Q.. s P Q t s P s ~Q.

. s ~P ll mt , s t P.86

as c mks, ts ts t ml tt P mst v t s P s ~P. als, s t sm sml ls t -stl lls. Ts, m ~ ~P t t P.87 T m s,t ts v t sm sml ls t classicalstllls; s l tlmt, t lt s, t, lsstv t intuitionisticlls. cs t t t. as ctlls s, t s l ssl t t ~Ps l s t t

84. i m tl tms:- t s sttmt t ms, t, t s tt

, t sm sttmt, ts tm t tl vl t ms, s vl t xts S();

- ll, sttmt s vl xts S(), t xsts t s tt s t sttmt l t t ms s vls tvl sttmt qst. (be . 41)

85. as t v xst Kks ttst smts, i lv t t ts t qt-s . a, ts s t tst vt. T tst m slt ct iV cs txt.86. Ts ts st c, . 117-8. i v lt sm t tt t m tbs. Ts, s, s t t t ms ml qst.87. c, . 118. ntt lt; s vs tt.


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zachary raSer 19

tt t sq s P. i , b svs, t -t t s smt ml t t: t s ssl t s tst t fm; t tt -stt ls tl tsq s mlt. Ts mlt t tv, b ts, s t-st stv st-vtl t (be 415)st s t s tst ttmll l stts t ttst st (. 7). it s t ml sl sttl smlt tt s t ss ; t ml t tt stv ls s t t sstll tl qmts. Ts q-mts stm m t t tt t sts tt t tlt tt s ts ll tmt l tm. w v s tt bs tml-

t t stv tt s t t tslt t sst t st tts t t ttstll ll m potentially genericchoice sequence; s tt ts sm tmlt sms t t l tst-vtl st t mt t t l ttsm.

i t sm 1963 m l t smtl lss t-tst l, Kk ms sss (. 118-10). h ss tt ml st-ts t st st 7 , t sttl, tt s ml-l t lt st t ht lls ttst l. rlsk, ts vlvs ssorcing conditionst t s t ml stt s tt R v v . ts lmts , v (P,) = T (P,) = F, t t s P ls t s. i tsml, ts s s tl l t t t t ( t lssl txt Kks ml, m s ts t s mlt stt).

Kk sts s t st tm ts ml: s tt s Qv t xsts s Q, t ll Q, s t Q ~Q, i and only i is generic. Ts tm ltl s tt t sstl l lssl lt t t ts m ttst l s -tt t lssl, -ttst tpar excellence: t tl mlt xtsll tm, tsll -sttl, t st.

n, mst ll tt tt t tt (n)s stl xst, n, mks t t (n) ts lt, ms t. S l s (n) st, t s not yet a generic sequence; t s mlpotentially , t xtsll s-, t s t t t t st ts st. T L t exl

Ml s t notll vl t st tt , s s tsst ms t. T l t st s t lssl. it s ttst.

1

it s ssl t t t ttst lt t tl [i.e.mtmts] ls t come rom elsewhere (be 49) tms m -s m sl vl t t v tts msm -tvsm t b smsss ttst l t 4 t Mtt. w m


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t t l t ttst st tms tl t t tlts st t Being and Event: t ls ttst l sl tss t l t st, t l tl t tt . i tt-st mtmts s st l ts ls k t mtmts, t s sttsm ss mtmts as a truth procedure. cvsl, mtmts is tt, t ts ls t s t v m ls; t - m t v stvt s tl ts tt.

T x, , s tt ttBeing and Event, mtmts s t l tsk. it s tl smm t l t v t tll lmts t l, t ls t st s xml tt , s t m

t ss tt s (t st). yt mtmts s stl t tt , t ts l s t lssl. a mtmts s tl-, t t ts l t t ttst l s t l tst, ls ts tl t ml st-ttl.

Lt s tkl lm t tm: s t ssl t l t mtmt-l tt t lssl, ts stv l s ttst? i tt, tlm s t t b ts m, s t mk lssl -t t law o the ontologist subject. ist, t s v s tls operatoro fdelity, t l t tlst st tts t lmts ttt t t s tl (be m. 4). bt lssl ttst t v ts , s t t ms lt (be 49). is ts tt-t t ltmt? u t tss tt tt s , t l smtt t s t. a sq lmts tt s tt s deduciblem t ss t t ll t m . is s lssl mtmts sl t xss, ts xms, t ls qua, t ls tt tml stt oany presentation whatsoever, ts ls ssl tv ttll stt t mtmt xss lt. a sqt tlt t ts ls l discernible lawlikesq, -. o s, t sm lm l t s s tslt intuitionistic deducibility s l t, t ts s t t ss .dt as sucht t l t . Tl t mtmtl tt s ts ms s. os, ts s st t t t sqs: t, t -

t lt t llk. it ms qst t t mt-mtl lt s t tt. as t, t m ssttl vs t stv l s t t mtmtl tt , tt s, s mstt t lt. a t, ts s , t t l ttst l, tl, t s tt , l t lssl.

i mt, st ll flts, tt mtmts s tt tss tl Being and Event, t xt qst tt s ts stts s -tl. Ts s qst tt s m flt tt qtllt t . a , tttv mks m m t ts t, v.


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zachary raSer 131

st ll, t slts t, t mtmts s t tt tl, t t l mll t ttst l. Tss t t s tt t mst ttst mtmts s ss s mtl, t l t ttsm t ts stt ll m. nv-tlss, ts s sml vl tss. i tk ttst mtmtst tt xsss t sl , v, t t mmtsq v t t l t ml ts tt s ts t ts t. w ls t t t mlt v s v vs stll, ls t -ll mltlt tt b llst vtv s stt s l ls m t lssl tl.

Tm, t t , ts t m m stl lt tttst tl t t s t ts lssl ss, t s s smt k slmt. T qst ls ss s t t t ttst tl. T s t ssm v tt t l ml lst mttl, s s t t mtlst t tt b st t t lssl st t; b msl, t lst l, ss t sts -t tsl t s sayable o being.88 it s ssl t l ts stt mt t s sttv tt, stt t , s t, s t.

Lst ls t t tk t t ts ss, lt s tk stk t ll ts: wt t mm t s, t ttst t st-vtl st sl m t. it s t sts . T st-vtl st s tll stt m t ttst st tt ts m s tlt t mm tt t t t, tt t sm lt vt tt s t t t t tll ts tt lvs tsmm (lk t b t-t), t xtsmt lss t s-ltt ms tt ts v ts. yt t tll ts sitsel t v s stv , s v t tk t m t bs tt-st t t l l t st tt. it st tl s t mll t t tll sm t s-

t, t st t lm, t st smt tl sincongruentt t stv m tl s s, s s ts tl s ls-sl. i sst , l m lssl tl ts

ttsm, t t tk ts t l q mlt t lm t s.

at ts t, t ssl sltv slts t ts flts ss s t s l. it sms tt t l t m tl m tst mlt t l qsts tt t s . T t:

88. . be, . 8: T tss tt i st s t l tt s mtmtl, s t s ms mtmtl tvts. it s t tss t t l t t ss. itfms tt mtmts, tt t tt ts stl m, s t s xssl q .


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wt l t m tl t tt ?wt l t m ts t t t s?

it ms t s t t s t t txt t s.89

z suvst Ks cll

hlx, nS

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