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ACTUAL CROSSOV] HAPP5S NE> A 8M]ELY0 A/RONOMICAL NUMB] >.D #AJ@5_<#CAF_> ,SKEWES' B.D 0 J ! F/ "T ANY"O PROV$ "! 0 "O4 #F4 ,GRAHAM'S NUMB] ! NUMB] T BROKE >I?METIC ;G;1 .K #3 $<33O $<33O $<33O $<33O #3 ,9 #AIGA1 ,RONALD ,GRAHAM NE$$ AN UPP] B.D = A PROBLEM 9 ,RAMSEY !ORY ! MA!MATICS ( UNAVOIDABLE PATT]NS4 ,! B.D HE PUBLI%$ REQUIR$ A NEW NOTA;N 9V5T$ = ! O3A.N3 ,KNU?'S UP->R[S1 ": EA* >R[ IS ! PREVI\S OP]A;N IT]AT$ TO 9SAN;Y4 ,"O >R[ IS EXPON5TIA;N4 ,TWO IS REP1T$ EXPON5TIA;N "4 ,?REE >R[S IS REP1T$ T[]S4 ,F\R >R[S ! DEF9I;N ( ;G AL"O ALR PRODUCES A NUMB] ^: DIGITS C'T 2 DESCRIB$ BY ANY T[] ( EXPON5TS WRITABLE 9 ! UNIV]SE4 ,!N ;G USES ;G >R[S4 ,GRAHAM'S NUMB] IS ;G SIXTY-F\R LAY]S1 EA* LAY]'S >R[-C.T 2+ ! 5TIRE PREVI\S NUMB]4 ,9 #AIGG ,M>T9 ,G>DN] 9TRODUC$ X TO ! PUBLIC Z 8A B.D S VA/ T X DEFIES COMPREH5.N10 & ,GU9;S LI/$ X Z ! L>GE/ NUMB] "E US$ 9 A MA!MATICAL PRO(4 ,TO FEEL "O RUNG3 #3 $<33O $<33O #3 IS A P[] T[] ( ?REES #3^3^3 : IS ALR #G4F TRILLION4 #3 $<33O $<33O $<33O #3 IS A T[] ( ?REES #G4F TRILLION LAY]S TALL A NUMB] TOO L>GE = ! UNIV]SE TO HOLD EV5 XS DIGIT C.T4 ,& #3 $<33O $<33O $<33O $<33O #3 ! HUMBLE ;G IS A *A9 ( ^? T XF DEFIES IT]A;N4 ,!N COME SIXTY-?REE M LAY]S4 ,PHYSICI/ ,JOHN ,BAEZ'S DESCRIP;N IS CANONICAL3 EV5 ! NUMB] ( DIGITS 9 ! NUMB] ( DIGITS ( ,GRAHAM'S NUMB]1 REP1T$ ANY 3CEIVABLE NUMB] ( "TS1 IS 9EXPRESSIBLE4 ,& "H'S ! B1T T MAKES ! RE/ ( ? LI/ "W3 ,GRAHAM'S NUMB] IS N[ R\T9ELY DESCRIB$ Z 8E6ECTIVELY Z]O0 COMP>$ TO :AT'S NEXT4 #E4 ,,TREE"<#C"> ": COMPREH5.N 5DS ,KRUSKAL'S TREE !OREM "<#AIFJ"> PROVES T C]TA9 F9ITE SEQU;ES ( 8TREES0 M/ EV5TUALLY T]M9ATE & ,H>VEY ,FRI$MAN TURN$ T 9TO ! F9ITE FUNC;N ,,TREE"<;N">3 ! 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NUMB] NO COMPUT] C R1* ,! ,BUSY ,B1V] FUNC;N ,,BB"<;N">1 DEF9$ BY ,TIBOR ,RAD 9 #AIFB1 ASKS3 AM;G ALL ;N-/ATE ,TUR+ MA*9ES T HALT1 : RUNS L;GE/8 ,,BB"<;N"> IS T *AMPION'S /EP C.T4 ,"H'S ! TRAP3 ,,BB"<;N"> IS UNCOMPUTABLE PROVABLY4 ,IF ANY ALGORI?M CD COMPUTE ,,BB"<;N"> = E ;N1 X CD SOLVE ! HALT+ PROBLEM1 : ,TUR+ PROV$ IMPOSSIBLE4 ,S ,,BB"<;N"> ISN'T J BIG X GR[S FA/] ?AN E COMPUTABLE FUNC;N T EXI/S OR CD "E EXI/4 ,NO =MULA1 NO >R[ NOTA;N1 NO ,,TREE"<"> NO?+ PROGRAMMABLE C KEEP UP4 ,Z ( #BJBD1 ,,BB"<#E"> 0 F9ALLY PROV$ BY A COLLABORATIVE PRO( EF=T "<#DG1AGF1HGJ /EPS">2 ,,BB"<#F"> REMA9S UNCOMPUT$ & IS "KN TO EXCE$ #AJ#AE ALR 2Y PHYSICAL DESCRIBABIL;Y4 #C4 ,! P>ADOX AT ! TOP :Y 8BI7E/0 EATS XF ,! ,B]RY P>ADOX3 3SID] 8! SMALLE/ POSITIVE 9TEG] N DEF9ABLE 9 "U SIXTY LRS40 ,T S5T;E DEF9ES X 9 "U SIXTY LRS4 ,3TRADIC;N4 ,? IS :Y ! BI7E/-NUMB] GAME C'T J SAY 8! BI7E/ DEF9ABLE NUMB]0 T'S SELF-EAT+4 ,! R1L SPORT IS DEF9ABIL;Y )9 FIX$ RULES3 PICK A LANGUAGE1 C.T ! SYMBOLS1 "N ! L>GE/ NUMB] T LANGUAGE C EXPRESS4 ,: IS EXACTLY :AT ! F9AL 5TRY W1PONIZ$4 #B4 ,! ,BIG ,NUMB] ,DUEL ,,MIT1 #BJJG ,ON #BF ,JANU>Y #BJJG1 PHILOSOPH]S ,AGU/;N ,RAYO & ,ADAM ,ELGA FAC$ (F AT ,,MIT 9 ! 8,BIG ,NUMB] ,DUEL0 WRITE ! BI7E/ F9ITE NUMB] Y C DEF9E1 & ! RULES =BID 8,ELGA'S NUMB] PLUS "O40 ,ELGA OP5$ /R;G4 ,RAYO ANSW]$ ) ! MO/ AUDACI\S DEF9I;N 9 ! HI/ORY ( ! SPORT3 8! SMALLE/ NUMB] BI7] ?AN ANY F9ITE NUMB] "ND BY AN EXPRES.N 9 ! LANGUAGE ( F/-ORD] SET !ORY ) A GOOGOL SYMBOLS OR LESS40 ,CALL X ,RAYO'S NUMB] ,RAYO(#10^100) 4 ,E NUMB] 9 ? >TICLE ,GRAHAM'S1 ,,TREE"<#C">1 ,,SSCG"<#C">1 ,BUSY ,B1V] VALUES UP TO ANY DESCRIBABLE B.D C 2 DEF9$ 9 F> LESS ?AN A GOOGOL SYMBOLS ( SET !ORY4 ,RAYO'S NUMB] IS BI7] ?AN ALL ( !M BY 3/RUC;N4 ,X WON ! DUEL4 #A4 ,"! IS NO W9N] & T'S ! PUN*L9E ,! H"O/ CR[N 2L;GS TO A FACT1 N A NUMB]3 "! IS NO BI7E/ NUMB] ADD "O & Y'RE BI7]4 ,:AT ? C.TD[N ACTUALLY RANKS IS HUMAN;Y'S ABIL;Y TO DESCRIBE L>GE;S4 ,! LADD] RUNS3 NOTA;N " IT]AT$ NOTA;N "<,GRAHAM'S >R[S"> PROVABLE T]M9A;N "<,,TREE"> ! UNCOMPUTABLE "<,BUSY ,B1V]"> DEF9ABIL;Y XF "<,RAYO">4 ,EA* RUNG EXI/S 2C MA!MATICS KEPT NE$+ NUMB]S BI7] ?AN XS NOTA;N CD HOLD & SOLV$ ! PROBLEM BY 9V5T+ M NOTA;N4 ,! BI7E/ NUMB]S >5'T RECORDS4 ,!Y'RE A HI/ORY ( ! HUMAN IMAG9A;N 4COV]+ X _H1 ONCE AG1 RUN \ ( ROOM4 ,! L$G] _? ,NUMB] ,WRITT5 Z ,BI7] ?AN ,FAM\S = #H ,GOOGOLPLEX #AJ@5"<#AJPP"> E ATOM 9 ! UNIV]SE A #I-YE>-OLD "ND X #G ,SKEWES' E@5""> A GOOGOLPLEX1 MO/LY BI7E/ 9 A R1L PRO( "<#AICC"> #F ,GRAHAM'S ;G "<>R[ T[]"> ANY WRITABLE EXPON5T T[] ! ,GU9;S RECORD HOLD] #E ,,TREE"<#C"> F9ITE1 PROVABLY ,GRAHAM'S ;,H #J GR[S 2Y COMPREH5.N #D ,BUSY ,B1V] ,,BB"<;N"> E COMPUTABLE FUNC;N UNCOMPUTABLE BY !OREM #B ,RAYO'S ,RAYO"<#AJPP"> "EY?+ DEF9ABLE 9 #AJPP SYMBOLS WON ! ,,MIT DUEL #A " ! R1L ANSW]3 NO BI7E/ EXI/S ,FREQU5TLY ASK$ ,:AT IS ! BI7E/ NUMB] 9 ! _W8 ,"! ISN'T "O ADD "O4 ,! 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