Question

This is a question on mathematical logic, please provide full proof. Thank you!

of Theve's an nigue factorizs For Ratihn rotural number Prime that divicles PMIA but Pr tn Suppose by andd Sappose f exparent of pie We alefine is the Smallest n the ordler on 2 34 an Pins pum) ACm PCas-pem) asd Pn and NLn NCm or Nin) NLA) pens wert-orderig.canol he hot ths oroler wigue Show that Romorphz is ordlinal

Answer 1

A well ordesing on aset S is a ttal o de ons with (Ko s has a Subset Y least elimont fn tis Le lalsen is A a dutal orde on set X foall abrad cin x dthe Cellowig Sealiman old abadlbsa, hm nb (a) Tresiui Conneniy Q. n2 ,n isa natsal numbes, hne 15 m nne Suppose pin) is l smallst enponent of P phino factstis an on Paime thal divide n and Nin) is kT Cie PM ,but an osds on anine We Pime Pim) pin)=Pin) ad Ninenm) ox pin)Pim) ad Nin=Nim) and n)Lm)

we herma d show hod is auell -ondsing Frst cue her to show that it isa dabl Oxdeig let n,m,te2,3,9,3 Pin Pim)a Pim) d N (n)Ncm) Pin u=Pem and Nin) =Nim) and cn)im) GVD A ardl mn Pim) Zpin) = Pin) d N tm)Nin) o6 Pim) cn) Plm)-Pin) ad Nim)-N) evd (m) Pin)-Pm) Then Pem)pcm) ad Pm) pun) Pen) Pim)d PIMD=Pin) ad Nim)<Nin) PlnlPim Amr all 0te Cooe p(n)-Pim) ad m N -N which inmphin n =m) is antsymmdkic

. CM7 u Pen) Zpim) 9.0 Pini =pim) Pin Pim) nd NCn=N Im) ad n)(m) and mくも Pim pt Pim)=Pt) and N(m)くN(+) 0る PCm) PCtD ond nem)-NCH and im) +) wmih impuin Men PCn) <PCt Pen) -Put and Ni)=N ,n) t) Pin)<Pim)く PH)ラPin)<P4) 5ino Nen)Nim)NCt) =Nin)NC りく+ in)くて(m)く))てch)ぐr+) + > po Cu hくt is 1amsi ve m let m E 1,3,4, ithue nem

Crhor Sine Dun)くPLm) 03 Pem)くPiっ) 4 Pin)=Pim), eilau Nin) ヘ2 (3) 1in)Pim) d n nNm),4her tn) (m) o O2W 9 f 6n) Cennenity くis a O: C ob9 43,4,/3 iwery subset hoo a is adtatnl oader on CWh Nnt cwa howe o Show that leaot clemeu 1,3,91 ()) Sbetee snbsef o let on 5 t oadering 5ina 9S a welt lenantnbset ot 12,3,473 has a leashelimeua- 12.3,/ 1sauel-cadloinon