Embed Size (px)
Citation preview
Basic Proof TheoryAristoteles
. syllogismus 384 -322 b. C.
Sokrates : There are infinity many prime
Frege : formal system 1848-1925mathematics via logic
Russell antinemy : foundation crisis
Brouwer, Wege ,
Poinore : Intuitionen, predicativism
Hilbert : lauter 's paradise
Programm: 1) Formalen mathematics2) Show onsistency by finitary means
Gödel : 1) Formal systems are in complete2) PAH EmpA
Gentner : natural deductiensegment calculus
(ut elimination
PRA + gf Tl ④ t SnpA-
PAt TI k) for all den Eo .
Februar Schütte : geredicatüe analysishas strength No .
Rathjen ,Arai
,..: ordinul analysis
Kreisel,Kallenbach : proof mining
Paris Harrington , Friedman :Concrete invmpletenen , boolean relation theory, . . .
Phase transition
⑨ ⑥ A finitetree Ist ) is a poets.
th
✓ 1) JRET HSET RETS\ 2) HtET { SETI Sept} linearlgardered .
°
V-s.tETI-infs.tt
TET'⇐,
.
7h : TUT'
ts.tt/hlnifs,t)kinf/hs.httDLetf:N-tN.FKTfisV-K7M#To , -An finiter trees
V-isnlFIEK-fitl-I-i.jsnlicjnY.at)Friedman : PAH FKT.aefrir-r.bg ni
Matussek Lolbl : PATFKTF,PAHFKTF
,-3kg c : = 0,6395576. .
.
W : r < c ⇒ PAI- FKTF,sie⇒ PAH Fktfr
Plan of the talk
→ 1) Gentner 's fundamental theorem• art elimination nsult• technique• applications : Herbrand
, interpolation , subfomula von .
→ 2) Classification of pwrably rennie function of PA
• boarding in temn of #da< so• V-XI-yHE.CH) is unprorable in PA• applications : Paris Harrington, Grabstein ,hydra games, Friedman
'
s feüte fern , phase transition .
• technique is very general and Seglerinproof theory of strong system
Plan of the Teeth : 1) Gesetzen machen 2) PA pro _ rechnen .
1) Glatzen lamentieren :
→ free variables an .az , . . .
bernd variables Xxxx - -
Signature with function und relativ symbols→ Jemen with free variablesPrime fernweh . <
,Pt. . ..tn
onyosed ferimlas with → and H
e. g .× not bound in gca, ⇒ ttxqx, femnla
www.ratienssg-q-sncfvy-zq-y9^4=-7 ④→ 74) 7×4=77×74
Segment : P :b with Mb finite sets of L- formlosIdee : There will be a proof free with anunntois in M
and vndusien VI .
The Gerten calculus
Asiens MP :P, b P primeP,< :b
Rubs T, 9 : is
→ sP : → 4. b
P:p , b 5,4 :b rote→
r, 94) :b HAM¥9# :b
P :X,b MY :b
euerP:b
%!} True if ET,T theory
Rule for equality
P,E- Z : A
=IP : b
r, FEI = f-⑤ : b
=pT, tja,
_ .
, tu> sn : I
P,RE ) : b
=pP
,R ⑤
, tiss , _ . , tu su : I
formula rankrk4) = 0 = rk CP) fer P primevk @→4) = mer frag ,
oh 4) tt , rh#4) = Ash¢)→ IFP:b F) there is a proof of heights u when all
Lemmaout herrenlos have rk < r .
oha - 2M, q : 9, b°
not ok④
Anume Cf = Hx 4 E)
an. FF P
, Nä : 4cal , b
a ¢ FUß, D , KVBÄfft P, ttxye : Yau ,b
5¥ P, 9 : es
Theorem t Mlb) : Alb)b # FV(TG) : Ica )) ,
Stern.
⇒ II PCs) : DG)
→ Theorem II P:b PEP'
DEI'
oha}
nen'rer
'⇒ II, M : b'
Proof of the substitution theorem
2nd on n :
Hs) I TG ) : Hb) , ttx 4# b)
is derivat from
µ Mb ) : Alb) ,d. b)
,tx 44 , b)
Ich. ⇒ µ Mb ) : blb)
,4 Cc , b), Fx 4K ,
b)
c fresh ⇒ µ PCs ) : ICs ) , 44 , s) , tx 44 ,s )
<9kV §,1h
.
ts ⇒ II MG : bfs ) , K x MED .
Theorem : a) ter : yay, b ⇒ Ea T.ee :X, b .
b) II ! yay: A ⇒ IT :b, pen II T.nl :b
c) II T: A, -44 ⇒ II T :b, 9E)
Proof : c) Hd on u .
Assuan that K) has been derivat -
I P :b ,Hx 999cal
Sieht. II P : b ,
HX9h , 9 ¢,chresh
.
in ⇒ % P:b, 94 ) , 941
euer ⇒ für :b ,9 E)
Reduction lemma
If IF T, 9 :b and IF T: 9.b and
rk (f)Er then MY " P : I
Proof : If rk@ 1=0 [email protected]
Cf is the principle formula .
1) 9=-4 → X they →7:b
FIT, 9 :X ,b and IF ! 9,7 :b
i. h.
⇒ ° P :p , I and IT" Mit:bwww.ien#II-P, 4 :X, b rhe ) < r
⇒ türk r: Is rk①ereuer
⇒ 1mF r :b .
eut
lut elimination theorem
If ¥ T:b then IF r :b
Proof : Indischer on u .
Simple :
Critical one is a out of arger .
III. 79 :b and M :b, 9
oh (9) < ru ⇒ rh (9)Er
+ I. h. 1¥79 :b and # T:b , 9
RL Ei r :b .
3"t 3"
. z <3"
.
HIER :b ⇒
Er:S ⇒3"
k r :br - 2
;
Ein:b
The formal system Z ZPA
function symbols fer prim rec functionsAseiems :Hx > Sx --0
, V-xyfsx.si/-x=y) , HE 0"
= 0,
Hä
PIEI-nxi.VE/fog)EI=fCgfEy...,gnED-VERedgiioCFt-gilt, Hä, Redginllsyiäthlyi, Reality,)
Fa →¥Eu → Fsa) → ttx Fa) FEL
hierarchie 4) a. < so
Eo = müde :S!} Hohn Hand - HAHA Hsv Haniel)
W"
1) [×] : htßtw""' fer 7 E Lim
✓"(ßtn) [x) : WK! ßtw - X,4) [×]=L
,0 O
Goal : If Ztttity 94M 9 prima
⇒ 7- aceottm In < Halm ) INF 9 [mir]
Ideen : Z - Zoo w - puh
+ F. gab beraten ⇒ 1- T : -49nA
Fhispnreesindnctein
%"" you.fr/qa-I9lsxD:9aZTalbnscuteleininatienAdditional feature : witness centre
F://V-sIN.mu ⇒ FCMIEFCN )(
Fln) 7MF- [m] - Fkmaselmi))NHN fortanN 0=0 Nfwdtß) > 1T NatNßLemma : LLß & NLEN ⇒ Hans Hßln )-
¥9944-au) :o)°-
es beraten
Indes : h -0
1-kqqesi99.ir0
„ →uxtetnume 1¥ 901,9 :
1-994 ) :c )
1¥" 90404 ) -441,994 )
901,4×(9%94,994)
Z! Definition Assuan P:b is closed.
F µ P :b holdes if Nas FCO) and
one of the follewiy apple.es
f- x) Ts False ¥01 er In True to
⇐ A) F TIP : befand FEI Mu :b
and do ,did and Cf→YEN
⇐ 5) FF Bg :X ,band a.<a and 9 -74GB
f- A) FIT, 9k ) :D andkEFCOIandgiaandV-xq.PHFEIER; 9C ) ,bandtiaieaandtxqc.beFLIP :p :b and FEM:p , band do ,d. < a and rk < r
.
Lemma
a) Ft M :b ,Per:bEÄ ,
nä,
Fl Fkk Gk) , Na'
EGCO)⇒ 6! p : jb) Ft Mg :b GE True ⇒ Fff P:b
c) Ft M :b, 9 q e- False ⇒ Ft P:b
d) FEM :b, ey
⇒ FK Mg :ab
e) ftp.q-y :b ⇒ Ft P : 9,b and
Ft My :b
f) Ft M :b ,ttxq ⇒ F-[ist P:b,di )Proof :
FIT T :b ,Hau
⇒ ti FEI IFM :b
Proof by indona :
Assuan that Fx go is the
principeformalen .
F- [j ] IE T :b , Kroos , 96) GrafI. h
.F) [ihn M:b , 94961F- i f- [ i ] [ i ] = F- [ i]
⇒ FE :3 1-9 M :b,(i !
Lemma :
a) IFF II I-xq.es 9 primethen there is k ? Flo) s
.
Ah NE9k )
b) Jh Ftf KXZY 9¥, 9 primethen for all m there is n < Fcm, s.tl/Nt-9-m,u]
.
Proof : a)
F- to 7×0, > Kx xD
IN Kx@*→a) → s
FII ttx @E)→ a ) : sIreland :
F ht : <
We find ke Flo)-2g
F II Fx@④ →D , 9kHz :Lleise 1: IN A 9h ) V.
lese2.INT/9Ch/zweninFtFttx4-s) : denkt⇒ F Ht@*→) : s
=
-
7h : Fch IN A Chl .
b) FIFAJy 94,4 )wären] II 7,9km)→ ⇒ 7ns FEY? @) N f- 94in)a)
Fcm)
Embedding :
If Zt ME) : BEI
then there seist k,r c-Nand a prim see Fate
for all ni F[ü] wir" MUßE ,
Now we study the effect ofout elimination on the bauchig operators .
CE,G) = F. G + Ft GReduction lemma
Suppe Gt Mg :b and Ftp:b,qand [email protected], 6)Är:bProof : 2nd on ß .
erst ase 9=7×44, isthe minimal formen .
GIF 79, Kk ) :b k£601
I. h. ( f-, 6) HÄ M
,Ch ) : b
Inversion FIKI 1-9 P :b, 49) erhellenderCE, 6) Für :b
Fats FEGCOBEFOGE (ff,6)
Ford <% dehine
r-4-rmaef-HYCFYFDHJY.IS?*?Lemma If ftp.bthenFT-IM :bProof : 2nd on a :
bertone out.
Na , EFCOI FIT Mal :b F / n:b, 9
" " Fate Me :S FAHR: sie
R ! CETA) n:b-
^' Fafnir :bF-d
Theorem If Z1- Kit y 9¥, and of prime
then there beim k,r EIN and
a prim see operator F such that ma
her all m there seit n ± ( f-wth) !!Im ,we have INF 94in) .Proof : Z 1- tx 719¥,⇒ 7 by ZF p.se. F- Hx 779¥⇒ F
"" HEY H> Zy 44
⇒ Guten)""
tun tity9¥'
=:#net ü""
t"
Kxz, 9¥"-G
⇒ Hm 7ns 6cm NE 9dm, n)
Lemma Ha HßQIE Hd ⑦ ßQI
Lemma FE Ha ⇒ FREIE Hwa#µ+¥ '
lesolbsy : Er k,r EIN there is a much
that
f. f-wie"!! ) : Ha .
9 pnie-So if ZttxZyq#, then there is a < %such thatfor all m there is nettµ , NEGTm.it .
E. [×] =Ü% eesoeeasy : PAH Fx Zy Hg⑦ =}
Note that Hefty is a prim rechnedicate ofX and y.
lerollary : PAH Ha7h amEBEN. . . [h]= o
PAH Hm 7h Wm④[13 . .. [h] = 0
.
PAH hpdm baute is always termin atingPAH all egordstein segnend ae terminähigPA H Paris Harrington asserton
PAH Friedmann binite fern .
