8/2/2019 Paris-Harrington tautologies

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Tlwip.Alwwico{hc {l}{hehoimp

Ehwmcrh @lwe}``i

`lwe}``iDki#}ciwhnl

8/2/2019 Paris-Harrington tautologies

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[aip ip ch ehcomw {w}m if hcm fh`}pmp hc p}i{lgey mlj icp{lc`mp hf {am p{whco twic`itemp%lp mqmnteimk ic {aip tltmw#

H}w wp{ wmp}e{ ip {al{ fhw {am jchc }ttmw gh}ck }!j) hc W!j% j) !k}m {h Mwkhp lckNieep Z=%

8/2/2019 Paris-Harrington tautologies

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|}lpi.theychnile }ttmw gh}ck fhw {am Tlwip.Alwwico{hc twic`item ic gh}ckmk.kmt{aFwmom pyp{mnp: ic Pm`{ihc ; m oivm l `hcki{ihcle ehmw gh}ck fhw {am Tlwip.Alwwico{hctwic`item ic Wmphe}{ihc# [am twhhf hf {am }ttmw gh}ck kmtmckp hc l {amhwmn ai atwhf ip {am `hc{mc{ hf {am ahem Pm`{ihc =#

7 Wlnpmy lck Tlwip.Alwwico{hc twic`itemp

_m ic{whk}`m {am `hngicl{hwile twic`itemp hf ic{mwmp{ fhw {am twmpmc{ tltmw lck {amiwtwhthpi{ihcle fhwnleirl{ihcp# _m `hcpikmw {am fheehico fhwn}el{ihc hf Wlnpmy [am.hwmn fhw gi.`hehwicop hf owltap# [am twic item p{l{mp {al{ lcy elwom mch}oa owlta`hc{licp mi{amw l `ei|}m hw l p{lgem pm{ hf lwgi{wlwy pirm# Hf `h}wpm elwom mch}oa ipwmel{ivm {h {am pirm hf {am `ei|}m lck hf {am p{lgem pm{#

[amhwmn !Wlnpmy [amhwmn)# [amwm mqip{p l c}ngmw w!j% p) ai a ip {am pnleemp{c}ngmw p}`a {al{ lcy owlta i{a l{ emlp{ w!j% p) vmw{i`mp `hc{licp mi{amw l `ei|}m hf pirmj hw l p{lgem pm{ hf pirm p#

[am `hc`e}pihc hf {am lghvm {amhwmn ip hgvih}pey pl{ipmk gy lcy c w!j% p)# _mh``lpihcleey ply {al{ p}`a lc c pl{ipmp {am Wlnpmy twic`item fhw tlwlnm{mwp j lck p#

Ic {aip tltmw m lwm nlicey `hc`mwcmk i{a Tlwip.Alwwico{hc twic`itemp# [am omc.mwle Tlwip.Alwwico{hc [amhwmn !fhw lwgi{wlwy `hehwicop hf aytmwowltap) lp ic{whk}`mkic Z1^ lp {am wp{ mqlntem hf l nl{amnl{i`leey cl{}wle i{cmpp hf {am ic`hntem{mcmpptamchnmchc fhw fhwnle {amhwimp hf lwi{anm{i # [am omcmwle vmwpihc ip jchc {h gm}ctwhvlgem ic wp{.hwkmw Tmlch lwi{anm{i` Z1^% lck {aip ip lewmlky {am `lpm if hcm `hc.pikmwp gi.`hehwicop hf aytmwowltap hcey Zzi%b~!Q7 )

Mi%b !

zi%b~!Q7 )

Mi%b !7)

0

8/2/2019 Paris-Harrington tautologies

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@el}pmp !

8/2/2019 Paris-Harrington tautologies

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twhvmk l w!j% j)(7 ik{a ehmw gh}ck ic Wmphe}{ihc lck lc mqthcmc{ile ehmw gh}ckfhw {am Klvip.T}{cln twh`mk}wm fhw Wln!w!j% j): j% j)# Wm`mc{ey Jwlbmj Z

8/2/2019 Paris-Harrington tautologies

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[amhwmn 7# Ta!j77j

: j% j) alp l |}lpi.theychnile pirm twhhfp ic gh}ckmk.kmt{a Fwmom#

Twhhf# Nieep Z 0 7w fhw

phnm w 4% lck !g) j ip ph elwom {al{ {am `hcki{ihc hf icm|}lei{y !?) ip nm{ i{a wmptm`{{h p}`a n# [amc {am twhthpi{ihcle fhwn}el Ta!jG!n): j% n) ip `hc{wlki`{hwy# Fhw mlpmhf ch{l{ihc m q C!j% n) > jG!n)# If w > 4 {amc C!j% n) > G!0) > 7# Fhw w 3 4%lck h}w `ahi`m hf n {am f}c`{ihc C!j% n) alp {am fheehico twhtmw{y#

C!j% n) > C

C

j%

n

7 n jG!n) > jG!07w) > j!07

w

8/2/2019 Paris-Harrington tautologies

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_m wm`lee {al{ Mi%b icki`l{mp if zi% b~ ip lc mkom ic {am owlta#[am wp{ p{mt hf {am twh`mk}wm ip {h wmlk lee mkomp gm{mmc ic{momwp fwhn j {h

W!j%n(7)# [aip `hp{p l{ nhp{ W!j%n(7)7 |}mwimp# If l wmel{ivmey elwom p{lgem pm{ ipfh}ck% {amc {am twh`mk}wm h}{t}{p p}`a l pm{ lck |}i{p# H{amwipm {am owlta mqtehwmkph flw `hc{licp l `ei|}m hf n(7 vmw{i`mp# Em{ {ampm vmw{i`mp gm T > zv ziMi%v< > 4~%

lck fhw { Z < lck Mi%v{*< > 4~% lck

Ln7

> ziMi%v< Mi%v7 # # # Mi%vn7

>

8/2/2019 Paris-Harrington tautologies

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{a}p {am owlta ick}`mk gy {am memnmc{p hf Ln(7 !twmpmwvico {amiw hwkmw) hc {am ic{mwvle

ZC!j%n(7)n 0% hw mepm !ii) n(7 lck C!j%n(7)n< > !jG!n(7))n< lwm p}`a {al{ {am`hcki{ihcp fhw {am vleiki{y hf icm|}lei{imp !6) lck !?) lwm nm{% i#m#% G!n(7) 7n(7lck jG!n(7) W!j: n(7)# Ic `lpm !i) m lttey [amhwmn

C# [aip `lcgm pmmc lp fheehp2 gy wmtml{mk lttei`l{ihc hf m|}l{ihc !1) m alvm {al{ fhw mvmwy i%C!ji% n(7

i) > C# Ic tlw{i`}elw C!j% 0) > C# Gy kmci{ihc hf C!j% 0) m om{ {al{

jG!0) > C# [a}p j >

C% pic`m G!0) > 7#

1

8/2/2019 Paris-Harrington tautologies

9/23

_m ch pah {al{ {am wm`}wpivm `lee lwipico fwhn !@lpm 7) lelyp khnicl{mp {am`hp{ hf {am twh`mk}wm ic !@lpm

8/2/2019 Paris-Harrington tautologies

10/23

Lppmppico {am |}lei{y hf {am wmf}{l{ihc ic [amhwmn 7 ip phnmah nhwm kiff`}e{ {alc}p}le# Fhw C > j7

j{am pirm hf {am hwp{ {wmm.eijm wmf}{l{ihc ip 7C

7ai`a ip flw owml{mw

{alc h}w }ttmw gh}ck# F}w{amwnhwm% p}`a elwom wmf}{l{ihcp lwm hcey |}lpi.theychnileic {am pirm hf {am fhwn}el i{pmef% ai`a ip 7!C)# _aiem {am pirm hf {am fhwn}el lck {amc}ngmw hf vlwilgemp lwm }p}leey theychnileey wmel{mk% i{ ip ch{ {am `lpm amwm% pic`m {am

c}ngmw hf vlwilgemp ic Ta!C: j% j) ip H!C7)# [a}p% aiem h}w wmf}{l{ihc ip ch{ n}`aehcomw {alc {am fhwn}el% {amwm nioa{ gm wmf}{l{ihcp {al{ lwm pnleemw {alc {am fhwn}eli{pmef !lp ic vmwy mlj Tiomhcahem twic`item fhwn}el{ihcp Z7

8/2/2019 Paris-Harrington tautologies

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[amhwmn 0# Em{ C j7j # [amwm mqip{p N > N!j) p}`a {al{i# 77

j(7 j * w!0% j) ci * w!0% ci) N!j) > cj(77# [am ic{mwvleZj% C ip kivikmk lp Zc4% c< 4% m alvm }!j) >j * w!0% j) 7 w!0% j) 5 7 j7# Fhw {am ick}`{ivm p{mt m alvm

}i*

77i*bZp^

!l%g)

ti%l tb%g# !

8/2/2019 Paris-Harrington tautologies

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! cj(77 lp ic {am twhhf hf [amh.wmn ;% lck E > w!0% N)

8/2/2019 Paris-Harrington tautologies

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Ic h}w `hcki{ihcle ehmw gh}ck E `h}ek gm vmwy pnlee amc `hntlwmk {h C# Ickmmk%m `h}ek alvm !if N ip `ehpm {h {am ehmw gh}ck ic 7H!!ehoE)

`)#

[a}p h}w `hcki{ihcle 7E 4% fhw { >

8/2/2019 Paris-Harrington tautologies

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nicinlei{y hf {am `}wwmc{ey icptm`{mk r% i#m#% hc {am lqihn O!r) O!r)# [a}p% malvm {al{ W L Mq%q > 4 inteimp

W q5r q

4Mqi%r > i Mq%r > 4

q5y5r

4Mqi%y > i Mq%y > 4

%

ai`a ip b}p{

q5rq @q!q4% # # # % q% r: 4% # # # % % 4) lp cmmkmk#If {am pm|}mc`m ip ch{ l ohhk pm|}mc`m% {amc {am `hcb}c`{ W!q% ) ic @q!q% )

ick}`mp l nhch`awhnl{i` {wilcoem hw l p{lgem pm{ hf pirm p qip inthppigem pic`m {am pm|}mc`m h}ek alvm gmmc mq{mckmk }pico {am lghvm kmp`wigmkicfmwmc`m twh`mpp#

Icbm`{ivi{y Lqihnp Kmk}`{ihc Fhw {am pin}el{ihc hf {am icfmwmc`mp i{ ip p}ff`imc{{h pah {al{ tq% ty%% i#m#% {al{ tq% ty% % fhw mvmwy q > y lck mvmwy #

Ml`a hf tq% !wmpt# ty%) ip l kipb}c`{ihc lppmw{ico {am mqip{mc`m hf l pm|}mc`m hf

vmw{i`mp hf emco{a b < {al{ `lc gm mq{mckmk gy q !wmpt# y) ph {al{ {am wmel{ihc W ippl{ipmk i{a wmptm`{ {h # Fhw ml`a p}`a tliw hf pm|}mc`mp% lck % {am mq{mcpihcp q lck y lwm kip{ic`{ !pic`m q > y)# @hcpikmw {am wp{ `hhwkicl{m ic ai`a {amykifflmw% lck em{ v% v gm {am `hwwmpthckico vmw{i`mp# P}tthpm i{ah}{ ehpp hf omcmwlei{y{al{ v 5 v# [amc W! q% ) `hc{licp l `el}pm lppmw{ico {am `hntl{igiei{y hf v aiemW! y% ) `hc{licp l `el}pm lppmw{ico {am chc.`hntl{igiei{y hf v 5 v# [ampm {h`el}pmp `lc gm picoemk h}{ }pico p{w}`{}wle w}emp lck {amc meinicl{mk gy l @}{#

[am `hp{ hf {am pin}el{ihc ip H!p7p p7p) > H!p;p) p{mtp#[am `hp{ hf {am ahem wmk}`{ihc ip {a}p H!p;p) {inmp {am pirm hf l wmf}{l{ihc hf {am

fhwn}el Tat!p*

8/2/2019 Paris-Harrington tautologies

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wmk}`m l twhhf hfTa!!j *

8/2/2019 Paris-Harrington tautologies

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[aip ip l `hc{wlki`{ihc {h h}w `ahi`m hf c#Ic {am hwp{ `lpm% {am lghvm twh`mk}wm {wlcpel{mp ic{h l gh}ckmk.kmt{a Fwmom twhhf

hf pirm l{ nhp{ 7cj 7H!j eho j) 77c 7;c > 70c*H!j eho j) > 7H!c)#

6 @hc`e}pihc lck f}{}wm hwj_m `hnnmc{ hc {am pioci`lc`m hf h}w wmp}e{p# Ic h}w `hcki{ihcle ehmw gh}ck E`h}ek gm vmwy pnlee amc `hntlwmk {h C# Ickmmk% m `h}ek alvm !if N ip `ehpm {h

{am ehmw gh}ck ic ! 7H!!ehoE)`)# [a}p h}w `hcki{ihcle 7E

j7

j{am pirm hf {am hwp{ {wmm.eijm wmf}{l{ihc ip 7C

7

ai`a ip flw owml{mw {alc h}w }ttmw gh}ck# F}w{amwnhwm% p}`a elwom wmf}{l{ihcp lwm

hcey |}lpi.theychnile ic {am pirm hf {am fhwn}el i{pmef% ai`a ip 7!C)# _aiem {am pirmhf {am fhwn}el lck {am c}ngmw hf vlwilgemp lwm }p}leey theychnileey wmel{mk% i{ ip ch{{am `lpm amwm% pic`m {am c}ngmw hf vlwilgemp ic Ta!C: j% j) ip H!C7)# [a}p% aiemh}w wmf}{l{ihc ip ch{ n}`a ehcomw {alc {am fhwn}el% {amwm nioa{ gm wmf}{l{ihcp {al{ lwmpnleemw {alc {am fhwn}el i{pmef !lp ic vmwy mlj Tiomhcahem twic`item fhwn}el{ihcp Z7

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Z7^ Lwchek Gm`jnlcc lck Pln}me W# G}pp# Pmtlwl{ihc wmp}e{p fhw {am pirm hf `hcp{lc{.kmt{a twhthpi{ihcle twhhfp# Lcclep hf T}wm lck Ltteimk Ehoi`% Z=^ Tl}e Mwkhp lck Omhwom Nieep# Phnm gh}ckp fhw {am Wlnpmy.Tlwip.Alwwico{hcc}ngmwp# Bh}wcle hf @hngicl{hwile [amhwy% Pmwimp L% 04!

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Z