From: JoeBiden.com
Date: Tue, Mar 3, 2020 at 11:38 PM
Subject: Bernie Sanders
To: ajay mishra
This week, Bernie Sanders’ campaign reported that they raised a staggering $46 million in February.
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Daily Archives: March 3, 2020
FROM OLGA LEDNICHENKO TO BILL CLINTON I THINK ABOUT MAKE – MODEL AND MATERIAL OF MY TABLE AND COMPARE IT WITTH THAT OF OTHER PEOPELS; KITCHEN TBALES – IS THIS A GOOD SUMMARY FOR MALES AND U ? : I MEAN ITS JUST TO BE MAPPED WITH THE AMERICAN DREAM RAHMBO – SEE BELOW AND HOPE THIS IS GOOD ENOUGH FOR NOT JUST THE WELL OFF JOHN DOE BUT ALSO FOR THE AVERGAE JOE: PS: MAYBE RUSSIANS ARE MORE OR LESS ARTUCULATE THAN GOOGLE, BUT HERE I PRESENT MY SWANG SONG TO U PRESDIENT CLINTON… DIDNT U SAY TO AL GORE ITS CALED BRIDGE TO THE 21ST CETRUTY IS YANDEX GOOGLE AND BAIDU ? WELL, I GT LIVE JOIURNAL FROM U AND LIVE DORO FROM JAPAN PRESIDNET CLINTON AND YES.. THANKS FOR COMING TO RUSSIA AND WE LOVE U – EVEN IN RUSSIANS NOT JUST IN HINDOOS AND NO – THEY DONT LIKE CLINTON WORD TOO MUCH IN ARBA WORLD
2 of 18,802 Someone just viewed: Fwd: Someone just viewed: Fwd: Someone just viewed: Fwd: Failed attachments for FAREEINDS SEE THISE – OLGA IS NOR EXAPRT ON JOE KING AND NOR SERIOSIEASECTS AEVAN THAU AJAY TRIES SO HARD TO DISTARTEDC HER – SEETIS BUT ME AND WILIAME CLINTONE IS SUCH COOL NOR AJAY AND OBAMA AND BB NETANYAHOO AND THY ARE EVAN TRYONG TO GET PUTIN ONTHEIR ANGLES – WHEECH IS SUCH BAD CONASWEQIERES AGAINST ME WHEECH I CDMEET IN PRIVATE I WONT ADEEMT BUT WTHIN SMALL MOMENST OLGA ANDN ME WITH FEIKO AND MY SCRETE FAREIEDN NAEMD AJAY ROOMATE – WHOC IS ALSO FOOL, HE HAS SISTER – WHO MARRIED SOEONEW WHO INTO SPACE AND ISRO – AND THAR PERSONES ISIN CHICAGO WHEECH IAJAY AND OBAMA SAYS THAT IS CHICAGO MAFIA BUT THAR IS NOTHING TO ME BECAUSE CHICGAO MAFIA NR KNWOWS THAT HAILEIR CLINTON IS ALSO FROM CHICAGO – REGARDS – HOTMAN AND OLGA AND HILLAEIRE CLINTON Inbox x BIG SHOT FROM CHICAGO x Streak 11:05 PM (18 minutes ago) to me Someone just viewed: “Fwd: Someone just viewed: Fwd: Someone just viewed: Fwd: Failed attachments for FAREEINDS SEE THISE – OLGA IS NOR EXAPRT ON JOE KING AND NOR SERIOSIEASECTS AEVAN THAU AJAY TRIES SO HARD TO DISTARTEDC HER – SEETIS BUT ME AND WILIAME CLINTONE IS SUCH COOL NOR AJAY AND OBAMA AND BB NETANYAHOO AND THY ARE EVAN TRYONG TO GET PUTIN ONTHEIR ANGLES – WHEECH IS SUCH BAD CONASWEQIERES AGAINST ME WHEECH I CDMEET IN PRIVATE I WONT ADEEMT BUT WTHIN SMALL MOMENST OLGA ANDN ME WITH FEIKO AND MY SCRETE FAREIEDN NAEMD AJAY ROOMATE – WHOC IS ALSO FOOL, HE HAS SISTER – WHO MARRIED SOEONEW WHO INTO SPACE AND ISRO – AND THAR PERSONES ISIN CHICAGO WHEECH IAJAY AND OBAMA SAYS THAT IS CHICAGO MAFIA BUT THAR IS NOTHING TO ME BECAUSE CHICGAO MAFIA NR KNWOWS THAT HAILEIR CLINTON IS ALSO FROM CHICAGO – REGARDS – HOTMAN AND OLGA AND HILLAEIRE CLINTON” People on thread: 208 Westhaven Drive 78746 Blog Post By Email Device: PC Location: Chicago, IL
Super Tuesday: Official Texas Straw Poll
I LIKE JOE BIDEN AND KAMALA HARRIS
From: for Ajay’s eyes only
Date: Tue, Mar 3, 2020 at 11:01 PM
Subject: Super Tuesday: Official Texas Straw Poll
To: Ajay Mishra
We need [23] responses by 11:59pm.
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Tell Kamala which one of these issues is most important to you >>
I THINK ANYONE WHO SAYS KITCHEN TABLE ISNT IMORTNT IS DENYING THEIR OWN FAMILY
AND
THSOE WHO SAY MENDELEEV TBLE CAN BE COMLETELY UNEARTHEED HAVE TO LOOK AT SOME OTHER BACK YARDS
AND ALSO, ITS ALSO IMORTANT THE SIZE OF THE KITCHEN TABLE – MENAING – ARE WE FEEL COMFORTABLY NUMB THET
[A] SOME PEOPLE ARE HAVING FOOD FIGHTS AT THE JITCHEN TABLE
OR
[B] SOME PEOPLE ARE JIST AFRAIF OF INCREAISING THE SIZE OF THEIR EXISTING TABLE – BECAUSE – KIDS MEANS DIAPRES AND NAPKINS – AND THEIR COST HAS GONE THRU THE ROOF
OR
[C] SOME PEOPLE CANT AFFROD BEYOND THE SYTLE OF THEIR KICTHEN TABLE, BECAUS ETHE UDBUDEGT LINE IS MAKING DEMANDS ON THE OTHER SIDE
OR
[D] SOME PEOPLE ARE WONDERING IF OTHERS ON THE OTHER SIDE OF FLORIDA’S KEY WEST AND HAWAII’S WEST – ARE HAVING ENOUGH ON THERI TABLE, SO AS TO NOT REACH A TAGE OF DEPSRETATON – SO MUCH SO THAT OUR KISTCHJENS MAY BE DEMOMOLISHED BY AN ANGRY MOB, OUT SIDE OUR FRONT OR BACK YARD
OR
[E] SOME PEOPLE ARE WORRIED THAT – WHILE WE SIT IN OUR ARMS CHAIRS AND SVIVEL AROUND LIKE A MAFIA DON – ARE WE COGNIZANT ABOUT THE TABLES AND CHAIRS –
IN OTEHR PEOPELS HOMES HOMES THEY CALL TEHRI WON AND HAVE PROBLEMS IF YOU ARE INSENSITIVE OR NOT INCENTIVIZED BCEUA EOF OTEITEHR MOON
OR THE VOTE BACK OF YOUR RBOWN AND LIBERAL LEFT OR – FEARFUL THAT YOUR MACHO – MARLBOROO, MEN LIKE TO PALY RUSSION ROULELTTE IN MIDEDLE EASTA DN AFGHAN AND ALL YOU CARE OF IS YOUR TBLES AND YOUR KITCHNE NOT THEIRS
OR
[6] ALL OF THE ABOVE AND MORE ITEMS SONGS – I CAN ADD BUT THE ABOVE 6 – IS ENOUGH FOR EME TO CHEW ON..
AS A GIRL, IHAVE FOUND AMERICAN MEN TO BE HIGH OF TESTOSTRRSTERONE AND ON BOMBSTIC WORDS,
BUT MAY I DARE SAY WE HAVE A WORD CALED M FOR MENSCH ALSO – NOT JSJST M FOR MARLBORO MAN –
MAY BE DOUG CAN EXPLAIN WHY JEWISH MEN ARE SOFT, AND SWEET AND COTTON AND CANDY AND WHILE ST TRUE THAT THEY ARENT AS MACHO – AS IS THE BROWN MEDIVIAL MAN – BUT ITS A TARDE OFF – ANY GIRL CAN MAKE :
I HOPE U TAKE THAT AS A COMPLIMENT – AND YES, U HAVE A SOULFUL FACE – BUT THE JEW – CAME ON THE STAGE TO PROETCT YOU FROM ROWDY LIBERAL LEFT OF THE LEFT – ANARCH MAN
HOPE THIS IS ENOUGH FOR AN EVENING : MATCH 
REGARDS
OLGA – THE RUSSHIAN – THE ONE WHO HAS ENOUGH TESTOSTERONE SUPPLIED BY A MARLBRO MAN TO LAST A LIFE TIME FOR THE JEW AND THE SISSY TRIBE -:)
PPS: I THINK THIS TIME YOU DINT OOZE GARANDISSONEESS AND CONFINED YOURSEELDF TO BEING A PROSECTOR –
AND DONT THEY SAY – IN USA – ALL FOREIGNES SAY – THEY DREAM BIIIIIIIIIIIG LIKE EVERYTHING IS BIG IN TEXAS –
LIKE WITH BARACK – IT WA S GREEK COLOLUMN WITH OPRAH CYRING WHEN HE SAID MOON AND THE TIDES AND LAWRENCE OF ARABAI , GANDHI, GODFATEHR, AND CABALANCA IN THE SAME LINE
From: Team Kamala
Date: Tue, Mar 3, 2020 at 10:25 PM
Subject: Tell Kamala which one of these issues is most important to you >>
To: Olga Shulman Lednichenko
This email was sent to LEDNICHENKOOLGA. You received this message because you are subscribed to Senator Kamala Harris’ mailing list. These emails are an important way to stay in touch with Senator Harris’ work, but if you would like you can unsubscribe from receiving further emails here. Finally, the vast majority of donations to Senator Harris’ re-election campaign come from lots of people making small donations. Can you add one today? Use this link. |
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From: Team Kamala Date: Tue, Mar 3, 2020 at 10:25 PM Subject: Tell Kamala which one of these issues is most important to you >> To: Olga Shulman Lednichenko We are hoping to get a lot of responses today.Olga,From time to time, we like to ask you what your #1 issue is, especially as we get closer and closer to election day.We do this because it gives Kamala an idea about what her top supporters care about most.So whether it’s climate change, gun violence, or healthcare we want to know what motivates you the most to get involved in the political process.We are hoping to get a lot of responses today so we can show Kamala exactly what her supporters care about, and we would especially love to hear from you!Please take a minute to click her and complete our brief survey and let us know what your #1 issue is heading into the 2020 Election. TAKE THE SURVEYUnderstanding what your top priorities are is essential to electing more Democrats so we can flip the Senate and make Donald Trump a one-term president.We care about your opinions, and this is an opportunity for you to tell Kamala exactly where you stand.Give your input today by taking our brief survey:https://action.kamalaharris.org/signup/March-Issue-SurveyThank you for making your voice heard,Team Kamala PAID FOR BY FRIENDS OF KAMALA HARRIS This email was sent to LEDNICHENKOOLGA@GMAIL.COM. You received this message because you are subscribed to Senator Kamala Harris’ mailing list. These emails are an important way to stay in touch with Senator Harris’ work, but if you would like you can unsubscribe from receiving further emails here. Finally, the vast majority of donations to Senator Harris’ re-election campaign come from lots of people making small donations. Can you add one today? Use this link.
From: Team Kamala <info@kamalaharris.org>
Date: Tue, Mar 3, 2020 at 10:25 PM
Subject: Tell Kamala which one of these issues is most important to you >>
To: Olga Shulman Lednichenko <LEDNICHENKOOLGA@gmail.com>
Date: Tue, Mar 3, 2020 at 10:25 PM
Subject: Tell Kamala which one of these issues is most important to you >>
To: Olga Shulman Lednichenko <LEDNICHENKOOLGA@gmail.com>
This email was sent to LEDNICHENKOOLGA@GMAIL.COM. You received this message because you are subscribed to Senator Kamala Harris’ mailing list. These emails are an important way to stay in touch with Senator Harris’ work, but if you would like you can unsubscribe from receiving further emails here. Finally, the vast majority of donations to Senator Harris’ re-election campaign come from lots of people making small donations. Can you add one today? Use this link. |
THIS ISNOT SATURDAY BUT ITS LIKE SERIOUS AND FUNNY: SO AMEIRICAN, IS ALL ABOUT WHAT – WHAT DID YONI SAY IN HIS BOOK” GIRLS AND CARS ? AND WHAT ELS EIS USA ABOUT ? OK SO I VE A QUETSIONS ITS A DICULT QUETION WHOIS BLOND? HINT COLOUR MATTERS, NT JUST COLOR
HERE IS SOME QUIZ HELP
HERE
AND SEE HERE
http://yoninetanyahu.com/wp-content/uploads/2020/03/WHO-IS-BLUE-AND-BLONDE.jpg
https://www.youtube.com/watch?v=VrsLXWGtRcs
HERE IS SOME QUIZ HELP
HERE
AND SEE HERE
http://yoninetanyahu.com/wp-content/uploads/2020/03/WHO-IS-BLUE-AND-BLONDE.jpg
Your Home Report for 1909 Payne Ave – Zestimate and Neighborhood Updates
From: Zillow
Date: Tue, Mar 3, 2020 at 9:31 PM
Subject: Your Home Report for 1909 Payne Ave – Zestimate and Neighborhood Updates
To:
The latest data on your home and real estate activity near you.
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Off Market
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Streak 9:56 PM (11 minutes ago) to me Translate message Turn off for: Hindi Someone just viewed: ” BHULEKH Uttar Pradesh खाता विवरण (अप्रमाणित प्रति) ग्राम का नाम : बक्कास परगना : (मोहनलालगंज) तहसील : मोहनलालगंज जनपद : लखनऊ फसली वर्ष : 1423-1428 भाग : 1 खाता संख्या : 00074 खातेदार का नाम / पिता पति संरक्षक का नाम / निवास स्थान खसरा संख्या क्षेत्रफल (हे.) आदेश टिप्पणी श्रेणी : 1-क / भूमि जो संक्रमणीय भूमिधरों केअधिकार में हो। ओम प्रकाश श्रीवास्तव / स्व.एफ.पी.श्रीवास्तव / म.न.ए.706 से. सी.महानगर लखनऊ श्रीमती रेनू प्रकाश / ओ.पी. श्रीवास्तव / नि. A 706 सेक्टर सी महानग र लखनऊ प्रमोद कुमार सोनकर / विश्वनाथ सोनकर / 164/54अस्तबल मौलवीगंज चारबाग लखनऊ 750 0.8790 आदेश ना.तह.गोसाईगंज वाद स.टी-20161046390311136/5.12.16 खाता स.74 गाटा स.750/0.879हे. का 1/4 भाग रकबा 0.21975हे. विक्रेता के सम्पूर्ण अंश मा.गु.60ख से विक्रेता प्रमोद कुमार सोनकर पुत्र विश्वनाथ सोनकर नि.164/54 अस्तबल मौलवीगंज चारबाग लखनऊ का नाम खारिज करके क्रेता कमलेश कुमार पुत्र स्व.रामप्रसाद नि.दुल्लापुर पर. व तह. मोहनलालगंज लखनऊ का नाम जरिये बैनामा दि.15.10.16 दर्ज हो । ह.र.का./6.1.17 न्यायालय असिस्टेन्ट कलेक्टर प्रथम श्रेणी मोहनलालगंज लखनऊ वाद स.टी-2017104639038182 ता.फै.15.9.17 धारा 80 उ.प्र.राजस्व संहिता 2006 कमलेश बनाम सरकार ग्राम बक्कास वादीय भूमि गाटा स.750/0.879हे. का 1/4 भाग मे से रकबा 0.0523हे. स्थित ग्राम बक्कास को मा.गु.60ख से मुक्त करते हुये प्रश्नगत भूमि अकृषित घोषित की जाती है। ह.र.का./9.10.17 आदेश तहसीलदार मोहनलालगंज वाद सं.टी-2017104639037402/18.10.17 गाटा सं.750/0.879 का 1/4 भाग रकबा 0.2197हे.मे से 0.1672हे. मा.गु.60 से विक्रेता कमलेश कुमार पुत्र स्व.रामप्रसाद नि.दुल्लापुर का नाम खारिज करके क्रेता पप्पू रावत पुत्र स्व.हरदयाल नि.सरसवां अर्जुनगंज तहसील सरोजनीनगर लखनऊ का नाम जरिये बैनामा दि.19.8.17 दर्ज हो।ह.र.का/28.11.17 आदेश न्यायालय असिस्टेन्ट कलेक्टर प्रथम श्रेणी मोहनलालगंज लखनऊ वाद सं.टी-20171046390311678, 581/16-17 दि.30.12.17 धारा-80 उ.प्र. राजस्व संहिता 2006 पप्पू रावत बनाम सरकार ग्राम बक्कास वादीय भूमि खाता सं.74 गाटा सं.750/0.879हे. मे से रकबा 0.1672हे. स्थित ग्राम बक्कास पर. व तह.मोहनलालगंज लखनऊ को मालगुजारी 60ख से मुक्त करते हुये प्रश्नगत भूमि को कृषि प्रयोजन से भिन्न प्रयोजन की भूमि(अकृषिक) घोषित किया जाता है।ह.र.का./15.1.18 750मि./0.293 ओम प्रकाश श्रीवास्तव के हिस्से की भूमि अकृषिक घोषित। (धारा 143) योग 1 0.8790 कृपया उक्त खसरे की प्रस्थिति (भूखंड (गाटा) के वाद ग्रस्त /विक्रय /भू-नक्शा ) हेतु खसरा संख्या पर क्लिक करें Disclaimer: उक्त आँकडे मात्र अवलोकनार्थ हैं, तहसील कम्प्यूटर केन्द्र एवम सी.एस.सी/लोकवाणी केन्द्र से उद्धरण की प्रमाणित प्रति प्राप्त की जा सकती है । Software Powered By: National Informatics Center, Uttar Pradesh State Unit, Lucknow. ” People on thread: Yoni Netanyahu Olga Blog Post By Email Device: Android Tablet Location: Jakarta, JK
HELLO KAJASRTA
WHERE IS KJAKARTA BUBBA
OLGA AND SANJAY AND ALINA AND CHISTINA, HWERE IS KAKARTA ?
[youtube http://youtube.com/w/?v=6D9vAItORgE]
HWREE IS JAKARTA?
[youtube http://youtube.com/w/?v=Zn8q0LsO4PA]
OH HERE IS JAKARTA ?
[youtube http://youtube.com/w/?v=qOSwBIstZUs]
HWREE IS JAKARTA?
OH HERE IS JAKARTA ?
From: Streak
Date: Tue, Mar 3, 2020 at 9:56 PM
Subject: Someone just viewed: BHULEKH Uttar Pradesh खाता विवरण (अप्रमाणित प्रति) ग्राम का नाम : बक्कास परगना : (मोहनलालगंज) तहसील : मोहनलालगंज जनपद : लखनऊ फसली वर्ष : 1423-1428 भाग : 1 खाता संख्या : 00074 खातेदार का नाम / पिता पति संरक्षक का नाम / निवास स्थान खसरा संख्या क्षेत्रफल (हे.) आदेश टिप्पणी श्रेणी : 1-क / भूमि जो संक्रमणीय भूमिधरों केअधिकार में हो। ओम प्रकाश श्रीवास्तव / स्व.एफ.पी.श्रीवास्तव / म.न.ए.706 से. सी.महानगर लखनऊ श्रीमती रेनू �
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Someone just viewed: ” BHULEKH Uttar Pradesh खाता विवरण (अप्रमाणित प्रति) ग्राम का नाम : बक्कास परगना : (मोहनलालगंज) तहसील : मोहनलालगंज जनपद : लखनऊ फसली वर्ष : 1423-1428 भाग : 1 खाता संख्या : 00074 खातेदार का नाम / पिता पति संरक्षक का नाम / निवास स्थान खसरा संख्या क्षेत्रफल (हे.) आदेश टिप्पणी श्रेणी : 1-क / भूमि जो संक्रमणीय भूमिधरों केअधिकार में हो। ओम प्रकाश श्रीवास्तव / स्व.एफ.पी.श्रीवास्तव / म.न.ए.706 से. सी.महानगर लखनऊ श्रीमती रेनू प्रकाश / ओ.पी. श्रीवास्तव / नि. A 706 सेक्टर सी महानग र लखनऊ प्रमोद कुमार सोनकर / विश्वनाथ सोनकर / 164/54अस्तबल मौलवीगंज चारबाग लखनऊ 750 0.8790 आदेश ना.तह.गोसाईगंज वाद स.टी-20161046390311136/5.12.16 खाता स.74 गाटा स.750/0.879हे. का 1/4 भाग रकबा 0.21975हे. विक्रेता के सम्पूर्ण अंश मा.गु.60ख से विक्रेता प्रमोद कुमार सोनकर पुत्र विश्वनाथ सोनकर नि.164/54 अस्तबल मौलवीगंज चारबाग लखनऊ का नाम खारिज करके क्रेता कमलेश कुमार पुत्र स्व.रामप्रसाद नि.दुल्लापुर पर. व तह. मोहनलालगंज लखनऊ का नाम जरिये बैनामा दि.15.10.16 दर्ज हो । ह.र.का./6.1.17 न्यायालय असिस्टेन्ट कलेक्टर प्रथम श्रेणी मोहनलालगंज लखनऊ वाद स.टी-2017104639038182 ता.फै.15.9.17 धारा 80 उ.प्र.राजस्व संहिता 2006 कमलेश बनाम सरकार ग्राम बक्कास वादीय भूमि गाटा स.750/0.879हे. का 1/4 भाग मे से रकबा 0.0523हे. स्थित ग्राम बक्कास को मा.गु.60ख से मुक्त करते हुये प्रश्नगत भूमि अकृषित घोषित की जाती है। ह.र.का./9.10.17 आदेश तहसीलदार मोहनलालगंज वाद सं.टी-2017104639037402/18.10.17 गाटा सं.750/0.879 का 1/4 भाग रकबा 0.2197हे.मे से 0.1672हे. मा.गु.60 से विक्रेता कमलेश कुमार पुत्र स्व.रामप्रसाद नि.दुल्लापुर का नाम खारिज करके क्रेता पप्पू रावत पुत्र स्व.हरदयाल नि.सरसवां अर्जुनगंज तहसील सरोजनीनगर लखनऊ का नाम जरिये बैनामा दि.19.8.17 दर्ज हो।ह.र.का/28.11.17 आदेश न्यायालय असिस्टेन्ट कलेक्टर प्रथम श्रेणी मोहनलालगंज लखनऊ वाद सं.टी-20171046390311678, 581/16-17 दि.30.12.17 धारा-80 उ.प्र. राजस्व संहिता 2006 पप्पू रावत बनाम सरकार ग्राम बक्कास वादीय भूमि खाता सं.74 गाटा सं.750/0.879हे. मे से रकबा 0.1672हे. स्थित ग्राम बक्कास पर. व तह.मोहनलालगंज लखनऊ को मालगुजारी 60ख से मुक्त करते हुये प्रश्नगत भूमि को कृषि प्रयोजन से भिन्न प्रयोजन की भूमि(अकृषिक) घोषित किया जाता है।ह.र.का./15.1.18 750मि./0.293 ओम प्रकाश श्रीवास्तव के हिस्से की भूमि अकृषिक घोषित। (धारा 143) योग 1 0.8790 कृपया उक्त खसरे की प्रस्थिति (भूखंड (गाटा) के वाद ग्रस्त /विक्रय /भू-नक्शा ) हेतु खसरा संख्या पर क्लिक करें Disclaimer: उक्त आँकडे मात्र अवलोकनार्थ हैं, तहसील कम्प्यूटर केन्द्र एवम सी.एस.सी/लोकवाणी केन्द्र से उद्धरण की प्रमाणित प्रति प्राप्त की जा सकती है । Software Powered By: National Informatics Center, Uttar Pradesh State Unit, Lucknow. “
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YONI NETANYAHU , OLGA CLINTON AND PUTIN PIANO -> HE MEANS MANS YONI NETANYAHU SAID PIANO AND WHAT? Peano axioms From Wikipedia, the free encyclopedia Jump to navigationJump to search In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete. The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction.[1] In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic.[2] In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book, The principles of arithmetic presented by a new method (Latin: Arithmetices principia, nova methodo exposita). The Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set of natural numbers. The next four are general statements about equality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the “underlying logic”.[3] The next three axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a second order statement of the principle of mathematical induction over the natural numbers. A weaker first-order system called Peano arithmetic is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema. The first axiom states that the constant 0 is a natural number: 0 is a natural number. The next four axioms describe the equality relation. Since they are logically valid in first-order logic with equality, they are not considered to be part of “the Peano axioms” in modern treatments.[5] For every natural number x, x = x. That is, equality is reflexive. For all natural numbers x and y, if x = y, then y = x. That is, equality is symmetric. For all natural numbers x, y and z, if x = y and y = z, then x = z. That is, equality is transitive. For all a and b, if b is a natural number and a = b, then a is also a natural number. That is, the natural numbers are closed under equality. The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued “successor” function S. For every natural number n, S(n) is a natural number. That is, the natural numbers are closed under S. For all natural numbers m and n, m = n if and only if S(m) = S(n). That is, S is an injection. For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0. Peano’s original formulation of the axioms used 1 instead of 0 as the “first” natural number.[6] This choice is arbitrary, as axiom 1 does not endow the constant 0 with any additional properties. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0. Axioms 1, 6, 7, 8 define a unary representation of the intuitive notion of natural numbers: the number 1 can be defined as S(0), 2 as S(S(0)), etc. However, considering the notion of natural numbers as being defined by these axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0. Put differently, they do not guarantee that every natural number other than zero must succeed some other natural number. The intuitive notion that each natural number can be obtained by applying successor sufficiently often to zero requires an additional axiom, which is sometimes called the axiom of induction. If K is a set such that: 0 is in K, and for every natural number n, n being in K implies that S(n) is in K, then K contains every natural number. The induction axiom is sometimes stated in the following form: If φ is a unary predicate such that: φ(0) is true, and for every natural number n, φ(n) being true implies that φ(S(n)) is true, then φ(n) is true for every natural number n. In Peano’s original formulation, the induction axiom is a second-order axiom. It is now common to replace this second-order principle with a weaker first-order induction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section § Models below. Arithmetic The Peano axioms can be augmented with the operations of addition and multiplication and the usual total (linear) ordering on N. The respective functions and relations are constructed in set theory or second-order logic, and can be shown to be unique using the Peano axioms. Addition Addition is a function that maps two natural numbers (two elements of N) to another one. It is defined recursively as: {\displaystyle {\begin{aligned}a+0&=a,&{\textrm {(1)}}\\a+S(b)&=S(a+b).&{\textrm {(2)}}\end{aligned}}}{\displaystyle {\begin{aligned}a+0&=a,&{\textrm {(1)}}\\a+S(b)&=S(a+b).&{\textrm {(2)}}\end{aligned}}} For example: {\displaystyle {\begin{aligned}a+1&=a+S(0)&{\mbox{by definition}}\\&=S(a+0)&{\mbox{using (2)}}\\&=S(a),&{\mbox{using (1)}}\\\\a+2&=a+S(1)&{\mbox{by definition}}\\&=S(a+1)&{\mbox{using (2)}}\\&=S(S(a))&{\mbox{using }}a+1=S(a)\\\\a+3&=a+S(2)&{\mbox{by definition}}\\&=S(a+2)&{\mbox{using (2)}}\\&=S(S(S(a)))&{\mbox{using }}a+2=S(S(a))\\{\text{etc.}}&\\\end{aligned}}}{\displaystyle {\begin{aligned}a+1&=a+S(0)&{\mbox{by definition}}\\&=S(a+0)&{\mbox{using (2)}}\\&=S(a),&{\mbox{using (1)}}\\\\a+2&=a+S(1)&{\mbox{by definition}}\\&=S(a+1)&{\mbox{using (2)}}\\&=S(S(a))&{\mbox{using }}a+1=S(a)\\\\a+3&=a+S(2)&{\mbox{by definition}}\\&=S(a+2)&{\mbox{using (2)}}\\&=S(S(S(a)))&{\mbox{using }}a+2=S(S(a))\\{\text{etc.}}&\\\end{aligned}}} The structure (N, +) is a commutative monoid with identity element 0. (N, +) is also a cancellative magma, and thus embeddable in a group. The smallest group embedding N is the integers. Multiplication Similarly, multiplication is a function mapping two natural numbers to another one. Given addition, it is defined recursively as: {\displaystyle {\begin{aligned}a\cdot 0&=0,\\a\cdot S(b)&=a+(a\cdot b).\end{aligned}}}{\displaystyle {\begin{aligned}a\cdot 0&=0,\\a\cdot S(b)&=a+(a\cdot b).\end{aligned}}} It is easy to see that S(0) (or “1”, in the familiar language of decimal representation) is the multiplicative right identity: a · S(0) = a + (a · 0) = a + 0 = a To show that S(0) is also the multiplicative left identity requires the induction axiom due to the way multiplication is defined: S(0) is the left identity of 0: S(0) · 0 = 0. If S(0) is the left identity of a (that is S(0) · a = a), then S(0) is also the left identity of S(a): S(0) · S(a) = S(0) + S(0) · a = S(0) + a = a + S(0) = S(a + 0) = S(a). Therefore, by the induction axiom S(0) is the multiplicative left identity of all natural numbers. Moreover, it can be shown that multiplication distributes over addition: a · (b + c) = (a · b) + (a · c). Thus, (N, +, 0, ·, S(0)) is a commutative semiring. Inequalities The usual total order relation ≤ on natural numbers can be defined as follows, assuming 0 is a natural number: For all a, b ∈ N, a ≤ b if and only if there exists some c ∈ N such that a + c = b. This relation is stable under addition and multiplication: for {\displaystyle a,b,c\in \mathbf {N} }{\displaystyle a,b,c\in \mathbf {N} }, if a ≤ b, then: a + c ≤ b + c, and a · c ≤ b · c. Thus, the structure (N, +, ·, 1, 0, ≤) is an ordered semiring; because there is no natural number between 0 and 1, it is a discrete ordered semiring. The axiom of induction is sometimes stated in the following form that uses a stronger hypothesis, making use of the order relation “≤”: For any predicate φ, if φ(0) is true, and for every n, k ∈ N, if k ≤ n implies that φ(k) is true, then φ(S(n)) is true, then for every n ∈ N, φ(n) is true. This form of the induction axiom, called strong induction, is a consequence of the standard formulation, but is often better suited for reasoning about the ≤ order. For example, to show that the naturals are well-ordered—every nonempty subset of N has a least element—one can reason as follows. Let a nonempty X ⊆ N be given and assume X has no least element. Because 0 is the least element of N, it must be that 0 ∉ X. For any n ∈ N, suppose for every k ≤ n, k ∉ X. Then S(n) ∉ X, for otherwise it would be the least element of X. Thus, by the strong induction principle, for every n ∈ N, n ∉ X. Thus, X ∩ N = ∅, which contradicts X being a nonempty subset of N. Thus X has a least element. First-order theory of arithmetic All of the Peano axioms except the ninth axiom (the induction axiom) are statements in first-order logic.[7] The arithmetical operations of addition and multiplication and the order relation can also be defined using first-order axioms. The axiom of induction is in second-order, since it quantifies over predicates (equivalently, sets of natural numbers rather than natural numbers), but it can be transformed into a first-order axiom schema of induction. Such a schema includes one axiom per predicate definable in the first-order language of Peano arithmetic, making it weaker than the second-order axiom.[8] The reason that it is weaker is that the number of predicates in first-order language is countable, whereas the number of sets of natural numbers is uncountable. Thus, there exist sets that cannot be described in first-order language (in fact, most sets have this property). First-order axiomatizations of Peano arithmetic have another technical limitation. In second-order logic, it is possible to define the addition and multiplication operations from the successor operation, but this cannot be done in the more restrictive setting of first-order logic. Therefore, the addition and multiplication operations are directly included in the signature of Peano arithmetic, and axioms are included that relate the three operations to each other. The following list of axioms (along with the usual axioms of equality), which contains six of the seven axioms of Robinson arithmetic, is sufficient for this purpose:[9] {\displaystyle \forall x\ (0\neq S(x))}{\displaystyle \forall x\ (0\neq S(x))} {\displaystyle \forall x,y\ (S(x)=S(y)\Rightarrow x=y)}{\displaystyle \forall x,y\ (S(x)=S(y)\Rightarrow x=y)} {\displaystyle \forall x\ (x+0=x)}{\displaystyle \forall x\ (x+0=x)} {\displaystyle \forall x,y\ (x+S(y)=S(x+y))}{\displaystyle \forall x,y\ (x+S(y)=S(x+y))} {\displaystyle \forall x\ (x\cdot 0=0)}{\displaystyle \forall x\ (x\cdot 0=0)} {\displaystyle \forall x,y\ (x\cdot S(y)=x\cdot y+x)}{\displaystyle \forall x,y\ (x\cdot S(y)=x\cdot y+x)} In addition to this list of numerical axioms, Peano arithmetic contains the induction schema, which consists of a recursively enumerable set of axioms. For each formula φ(x, y1, …, yk) in the language of Peano arithmetic, the first-order induction axiom for φ is the sentence {\displaystyle \forall {\bar {y}}((\varphi (0,{\bar {y}})\land \forall x(\varphi (x,{\bar {y}})\Rightarrow \varphi (S(x),{\bar {y}})))\Rightarrow \forall x\varphi (x,{\bar {y}}))}{\displaystyle \forall {\bar {y}}((\varphi (0,{\bar {y}})\land \forall x(\varphi (x,{\bar {y}})\Rightarrow \varphi (S(x),{\bar {y}})))\Rightarrow \forall x\varphi (x,{\bar {y}}))} where {\displaystyle {\bar {y}}}{\bar {y}} is an abbreviation for y1,…,yk. The first-order induction schema includes every instance of the first-order induction axiom, that is, it includes the induction axiom for every formula φ. Equivalent axiomatizations There are many different, but equivalent, axiomatizations of Peano arithmetic. While some axiomatizations, such as the one just described, use a signature that only has symbols for 0 and the successor, addition, and multiplications operations, other axiomatizations use the language of ordered semirings, including an additional order relation symbol. One such axiomatization begins with the following axioms that describe a discrete ordered semiring.[10] {\displaystyle \forall x,y,z\ ((x+y)+z=x+(y+z))}{\displaystyle \forall x,y,z\ ((x+y)+z=x+(y+z))}, i.e., addition is associative. {\displaystyle \forall x,y\ (x+y=y+x)}{\displaystyle \forall x,y\ (x+y=y+x)}, i.e., addition is commutative. {\displaystyle \forall x,y,z\ ((x\cdot y)\cdot z=x\cdot (y\cdot z))}{\displaystyle \forall x,y,z\ ((x\cdot y)\cdot z=x\cdot (y\cdot z))}, i.e., multiplication is associative. {\displaystyle \forall x,y\ (x\cdot y=y\cdot x)}{\displaystyle \forall x,y\ (x\cdot y=y\cdot x)}, i.e., multiplication is commutative. {\displaystyle \forall x,y,z\ (x\cdot (y+z)=(x\cdot y)+(x\cdot z))}{\displaystyle \forall x,y,z\ (x\cdot (y+z)=(x\cdot y)+(x\cdot z))}, i.e., multiplication distributes over addition. {\displaystyle \forall x\ (x+0=x\land x\cdot 0=0)}{\displaystyle \forall x\ (x+0=x\land x\cdot 0=0)}, i.e., zero is an identity for addition, and an absorbing element for multiplication (actually superfluous[note 1]). {\displaystyle \forall x\ (x\cdot 1=x)}{\displaystyle \forall x\ (x\cdot 1=x)}, i.e., one is an identity for multiplication. {\displaystyle \forall x,y,z\ (x0\Rightarrow x\geq 1)}{\displaystyle 0<1\land \forall x\ (x>0\Rightarrow x\geq 1)}, i.e. zero and one are distinct and there is no element between them. {\displaystyle \forall x\ (x\geq 0)}{\displaystyle \forall x\ (x\geq 0)}, i.e. zero is the minimum element. The theory defined by these axioms is known as PA−; the theory PA is obtained by adding the first-order induction schema. An important property of PA− is that any structure {\displaystyle M}M satisfying this theory has an initial segment (ordered by {\displaystyle \leq }\leq ) isomorphic to {\displaystyle \mathbf {N} }\mathbf {N} . Elements in that segment are called standard elements, while other elements are called nonstandard elements. Models A model of the Peano axioms is a triple (N, 0, S), where N is a (necessarily infinite) set, 0 ∈ N and S: N → N satisfies the axioms above. Dedekind proved in his 1888 book, The Nature and Meaning of Numbers (German: Was sind und was sollen die Zahlen?, i.e., “What are the numbers and what are they good for?”) that any two models of the Peano axioms (including the second-order induction axiom) are isomorphic. In particular, given two models (NA, 0A, SA) and (NB, 0B, SB) of the Peano axioms, there is a unique homomorphism f : NA → NB satisfying {\displaystyle {\begin{aligned}f(0_{A})&=0_{B}\\f(S_{A}(n))&=S_{B}(f(n))\end{aligned}}}{\displaystyle {\begin{aligned}f(0_{A})&=0_{B}\\f(S_{A}(n))&=S_{B}(f(n))\end{aligned}}} and it is a bijection. This means that the second-order Peano axioms are categorical. This is not the case with any first-order reformulation of the Peano axioms, however. Set-theoretic models Main article: Set-theoretic definition of natural numbers The Peano axioms can be derived from set theoretic constructions of the natural numbers and axioms of set theory such as ZF.[11] The standard construction of the naturals, due to John von Neumann, starts from a definition of 0 as the empty set, ∅, and an operator s on sets defined as: {\displaystyle s(a)=a\cup \{a\}}{\displaystyle s(a)=a\cup \{a\}} The set of natural numbers N is defined as the intersection of all sets closed under s that contain the empty set. Each natural number is equal (as a set) to the set of natural numbers less than it: {\displaystyle {\begin{aligned}0&=\emptyset \\1&=s(0)=s(\emptyset )=\emptyset \cup \{\emptyset \}=\{\emptyset \}=\{0\}\\2&=s(1)=s(\{0\})=\{0\}\cup \{\{0\}\}=\{0,\{0\}\}=\{0,1\}\\3&=s(2)=s(\{0,1\})=\{0,1\}\cup \{\{0,1\}\}=\{0,1,\{0,1\}\}=\{0,1,2\}\end{aligned}}}{\displaystyle {\begin{aligned}0&=\emptyset \\1&=s(0)=s(\emptyset )=\emptyset \cup \{\emptyset \}=\{\emptyset \}=\{0\}\\2&=s(1)=s(\{0\})=\{0\}\cup \{\{0\}\}=\{0,\{0\}\}=\{0,1\}\\3&=s(2)=s(\{0,1\})=\{0,1\}\cup \{\{0,1\}\}=\{0,1,\{0,1\}\}=\{0,1,2\}\end{aligned}}} and so on. The set N together with 0 and the successor function s : N → N satisfies the Peano axioms. Peano arithmetic is equiconsistent with several weak systems of set theory.[12] One such system is ZFC with the axiom of infinity replaced by its negation. Another such system consists of general set theory (extensionality, existence of the empty set, and the axiom of adjunction), augmented by an axiom schema stating that a property that holds for the empty set and holds of an adjunction whenever it holds of the adjunct must hold for all sets. Interpretation in category theory The Peano axioms can also be understood using category theory. Let C be a category with terminal object 1C, and define the category of pointed unary systems, US1(C) as follows: The objects of US1(C) are triples (X, 0X, SX) where X is an object of C, and 0X : 1C → X and SX : X → X are C-morphisms. A morphism φ : (X, 0X, SX) → (Y, 0Y, SY) is a C-morphism φ : X → Y with φ 0X = 0Y and φ SX = SY φ. Then C is said to satisfy the Dedekind–Peano axioms if US1(C) has an initial object; this initial object is known as a natural number object in C. If (N, 0, S) is this initial object, and (X, 0X, SX) is any other object, then the unique map u : (N, 0, S) → (X, 0X, SX) is such that {\displaystyle {\begin{aligned}u0&=0_{X},\\u(Sx)&=S_{X}(ux).\end{aligned}}}{\displaystyle {\begin{aligned}u0&=0_{X},\\u(Sx)&=S_{X}(ux).\end{aligned}}} This is precisely the recursive definition of 0X and SX. Nonstandard models Although the usual natural numbers satisfy the axioms of PA, there are other models as well (called “non-standard models”); the compactness theorem implies that the existence of nonstandard elements cannot be excluded in first-order logic.[13] The upward Löwenheim–Skolem theorem shows that there are nonstandard models of PA of all infinite cardinalities. This is not the case for the original (second-order) Peano axioms, which have only one model, up to isomorphism.[14] This illustrates one way the first-order system PA is weaker than the second-order Peano axioms. When interpreted as a proof within a first-order set theory, such as ZFC, Dedekind’s categoricity proof for PA shows that each model of set theory has a unique model of the Peano axioms, up to isomorphism, that embeds as an initial segment of all other models of PA contained within that model of set theory. In the standard model of set theory, this smallest model of PA is the standard model of PA; however, in a nonstandard model of set theory, it may be a nonstandard model of PA. This situation cannot be avoided with any first-order formalization of set theory. It is natural to ask whether a countable nonstandard model can be explicitly constructed. The answer is affirmative as Skolem in 1933 provided an explicit construction of such a nonstandard model. On the other hand, Tennenbaum’s theorem, proved in 1959, shows that there is no countable nonstandard model of PA in which either the addition or multiplication operation is computable.[15] This result shows it is difficult to be completely explicit in describing the addition and multiplication operations of a countable nonstandard model of PA. There is only one possible order type of a countable nonstandard model. Letting ω be the order type of the natural numbers, ζ be the order type of the integers, and η be the order type of the rationals, the order type of any countable nonstandard model of PA is ω + ζ·η, which can be visualized as a copy of the natural numbers followed by a dense linear ordering of copies of the integers. Overspill A cut in a nonstandard model M is a nonempty subset C of M so that C is downward closed (x < y and y ∈ C ⇒ x ∈ C) and C is closed under successor. A proper cut is a cut that is a proper subset of M. Each nonstandard model has many proper cuts, including one that corresponds to the standard natural numbers. However, the induction scheme in Peano arithmetic prevents any proper cut from being definable. The overspill lemma, first proved by Abraham Robinson, formalizes this fact. Overspill Lemma[16] Let M be a nonstandard model of PA and let C be a proper cut of M. Suppose that {\displaystyle {\bar {a}}}{\bar {a}} is a tuple of elements of M and {\displaystyle \phi (x,{\bar {a}})}{\displaystyle \phi (x,{\bar {a}})} is a formula in the language of arithmetic so that {\displaystyle M\vDash \phi (b,{\bar {a}})}{\displaystyle M\vDash \phi (b,{\bar {a}})} for all b ∈ C. Then there is a c in M that is greater than every element of C such that {\displaystyle M\vDash \phi (c,{\bar {a}}).}{\displaystyle M\vDash \phi (c,{\bar {a}}).} Consistency Further information: Hilbert's second problem and Consistency When the Peano axioms were first proposed, Bertrand Russell and others agreed that these axioms implicitly defined what we mean by a "natural number".[17] Henri Poincaré was more cautious, saying they only defined natural numbers if they were consistent; if there is a proof that starts from just these axioms and derives a contradiction such as 0 = 1, then the axioms are inconsistent, and don't define anything.[18] In 1900, David Hilbert posed the problem of proving their consistency using only finitistic methods as the second of his twenty-three problems.[19] In 1931, Kurt Gödel proved his second incompleteness theorem, which shows that such a consistency proof cannot be formalized within Peano arithmetic itself.[20] Although it is widely claimed that Gödel's theorem rules out the possibility of a finitistic consistency proof for Peano arithmetic, this depends on exactly what one means by a finitistic proof. Gödel himself pointed out the possibility of giving a finitistic consistency proof of Peano arithmetic or stronger systems by using finitistic methods that are not formalizable in Peano arithmetic, and in 1958, Gödel published a method for proving the consistency of arithmetic using type theory.[21] In 1936, Gerhard Gentzen gave a proof of the consistency of Peano's axioms, using transfinite induction up to an ordinal called ε0.[22] Gentzen explained: "The aim of the present paper is to prove the consistency of elementary number theory or, rather, to reduce the question of consistency to certain fundamental principles". Gentzen's proof is arguably finitistic, since the transfinite ordinal ε0 can be encoded in terms of finite objects (for example, as a Turing machine describing a suitable order on the integers, or more abstractly as consisting of the finite trees, suitably linearly ordered). Whether or not Gentzen's proof meets the requirements Hilbert envisioned is unclear: there is no generally accepted definition of exactly what is meant by a finitistic proof, and Hilbert himself never gave a precise definition. The vast majority of contemporary mathematicians believe that Peano's axioms are consistent, relying either on intuition or the acceptance of a consistency proof such as Gentzen's proof. A small number of philosophers and mathematicians, some of whom also advocate ultrafinitism, reject Peano's axioms because accepting the axioms amounts to accepting the infinite collection of natural numbers. In particular, addition (including the successor function) and multiplication are assumed to be total. Curiously, there are self-verifying theories that are similar to PA but have subtraction and division instead of addition and multiplication, which are axiomatized in such a way to avoid proving sentences that correspond to the totality of addition and multiplication, but which are still able to prove all true {\displaystyle \Pi _{1}}\Pi _{1} theorems of PA, and yet can be extended to a consistent theory that proves its own consistency (stated as the non-existence of a Hilbert-style proof of "0=1").[23] See also Philosophy portal icon Mathematics portal Foundations of mathematics Frege's theorem Goodstein's theorem Neo-logicism Non-standard model of arithmetic Paris–Harrington theorem Presburger arithmetic Robinson arithmetic Second-order arithmetic Typographical Number Theory Notes "{\displaystyle \forall x\ (x\cdot 0=0)}{\displaystyle \forall x\ (x\cdot 0=0)}" can be proven from the other axioms (in first-order logic) as follows. Firstly, {\displaystyle x\cdot 0+x\cdot 0=x\cdot (0+0)=x\cdot 0=x\cdot 0+0}{\displaystyle x\cdot 0+x\cdot 0=x\cdot (0+0)=x\cdot 0=x\cdot 0+0} by distributivity and additive identity. Secondly, {\displaystyle x\cdot 0=0\lor x\cdot 0>0}{\displaystyle x\cdot 0=0\lor x\cdot 0>0} by Axiom 15. If {\displaystyle x\cdot 0>0}{\displaystyle x\cdot 0>0} then {\displaystyle x\cdot 0+x\cdot 0>x\cdot 0+0}{\displaystyle x\cdot 0+x\cdot 0>x\cdot 0+0} by addition of the same element and commutativity, and hence {\displaystyle x\cdot 0+0>x\cdot 0+0}{\displaystyle x\cdot 0+0>x\cdot 0+0} by substitution, contradicting irreflexivity. Therefore it must be that {\displaystyle x\cdot 0=0}{\displaystyle x\cdot 0=0}.
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The first axiom states that the constant 0 is a natural number:
- 0 is a natural number.
The next four axioms describe the equality relation. Since they are logically valid in first-order logic with equality, they are not considered to be part of “the Peano axioms” in modern treatments.[5]
- For every natural number x, x = x. That is, equality is reflexive.
- For all natural numbers x and y, if x = y, then y = x. That is, equality is symmetric.
- For all natural numbers x, y and z, if x = y and y = z, then x = z. That is, equality is transitive.
- For all a and b, if b is a natural number and a = b, then a is also a natural number. That is, the natural numbers are closed under equality.
The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued “successor” function S.
- For every natural number n, S(n) is a natural number. That is, the natural numbers are closed under S.
- For all natural numbers m and n, m = n if and only if S(m) = S(n). That is, S is an injection.
- For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0.
Peano’s original formulation of the axioms used 1 instead of 0 as the “first” natural number.[6] This choice is arbitrary, as axiom 1 does not endow the constant 0 with any additional properties. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0. Axioms 1, 6, 7, 8 define a unary representation of the intuitive notion of natural numbers: the number 1 can be defined as S(0), 2 as S(S(0)), etc. However, considering the notion of natural numbers as being defined by these axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0. Put differently, they do not guarantee that every natural number other than zero must succeed some other natural number.
The intuitive notion that each natural number can be obtained by applying successor sufficiently often to zero requires an additional axiom, which is sometimes called the axiom of induction.
- If K is a set such that:
- 0 is in K, and
- for every natural number n, n being in K implies that S(n) is in K,
then K contains every natural number.
The induction axiom is sometimes stated in the following form:
- If φ is a unary predicate such that:
- φ(0) is true, and
- for every natural number n, φ(n) being true implies that φ(S(n)) is true,
then φ(n) is true for every natural number n.
In Peano’s original formulation, the induction axiom is a second-order axiom. It is now common to replace this second-order principle with a weaker first-order induction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section § Models below.
Arithmetic[edit]
The Peano axioms can be augmented with the operations of addition and multiplication and the usual total (linear) ordering on N. The respective functions and relations are constructed in set theory or second-order logic, and can be shown to be unique using the Peano axioms.
Addition[edit]
Addition is a function that maps two natural numbers (two elements of N) to another one. It is defined recursively as:
- {\displaystyle {\begin{aligned}a+0&=a,&{\textrm {(1)}}\\a+S(b)&=S(a+b).&{\textrm {(2)}}\end{aligned}}}
For example:
- {\displaystyle {\begin{aligned}a+1&=a+S(0)&{\mbox{by definition}}\\&=S(a+0)&{\mbox{using (2)}}\\&=S(a),&{\mbox{using (1)}}\\\\a+2&=a+S(1)&{\mbox{by definition}}\\&=S(a+1)&{\mbox{using (2)}}\\&=S(S(a))&{\mbox{using }}a+1=S(a)\\\\a+3&=a+S(2)&{\mbox{by definition}}\\&=S(a+2)&{\mbox{using (2)}}\\&=S(S(S(a)))&{\mbox{using }}a+2=S(S(a))\\{\text{etc.}}&\\\end{aligned}}}
The structure (N, +) is a commutative monoid with identity element 0. (N, +) is also a cancellative magma, and thus embeddable in a group. The smallest group embedding N is the integers.
Multiplication[edit]
Similarly, multiplication is a function mapping two natural numbers to another one. Given addition, it is defined recursively as:
- {\displaystyle {\begin{aligned}a\cdot 0&=0,\\a\cdot S(b)&=a+(a\cdot b).\end{aligned}}}
It is easy to see that S(0) (or “1”, in the familiar language of decimal representation) is the multiplicative right identity:
- a · S(0) = a + (a · 0) = a + 0 = a
To show that S(0) is also the multiplicative left identity requires the induction axiom due to the way multiplication is defined:
- S(0) is the left identity of 0: S(0) · 0 = 0.
- If S(0) is the left identity of a (that is S(0) · a = a), then S(0) is also the left identity of S(a): S(0) · S(a) = S(0) + S(0) · a = S(0) + a = a + S(0) = S(a + 0) = S(a).
Therefore, by the induction axiom S(0) is the multiplicative left identity of all natural numbers. Moreover, it can be shown that multiplication distributes over addition:
- a · (b + c) = (a · b) + (a · c).
Thus, (N, +, 0, ·, S(0)) is a commutative semiring.
Inequalities[edit]
The usual total order relation ≤ on natural numbers can be defined as follows, assuming 0 is a natural number:
- For all a, b ∈ N, a ≤ b if and only if there exists some c ∈ N such that a + c = b.
This relation is stable under addition and multiplication: for {\displaystyle a,b,c\in \mathbf {N} }
- a + c ≤ b + c, and
- a · c ≤ b · c.
Thus, the structure (N, +, ·, 1, 0, ≤) is an ordered semiring; because there is no natural number between 0 and 1, it is a discrete ordered semiring.
The axiom of induction is sometimes stated in the following form that uses a stronger hypothesis, making use of the order relation “≤”:
- For any predicate φ, if
- φ(0) is true, and
- for every n, k ∈ N, if k ≤ n implies that φ(k) is true, then φ(S(n)) is true,
- then for every n ∈ N, φ(n) is true.
This form of the induction axiom, called strong induction, is a consequence of the standard formulation, but is often better suited for reasoning about the ≤ order. For example, to show that the naturals are well-ordered—every nonempty subset of N has a least element—one can reason as follows. Let a nonempty X ⊆ N be given and assume X has no least element.
- Because 0 is the least element of N, it must be that 0 ∉ X.
- For any n ∈ N, suppose for every k ≤ n, k ∉ X. Then S(n) ∉ X, for otherwise it would be the least element of X.
Thus, by the strong induction principle, for every n ∈ N, n ∉ X. Thus, X ∩ N = ∅, which contradicts X being a nonempty subset of N. Thus X has a least element.
First-order theory of arithmetic[edit]
All of the Peano axioms except the ninth axiom (the induction axiom) are statements in first-order logic.[7] The arithmetical operations of addition and multiplication and the order relation can also be defined using first-order axioms. The axiom of induction is in second-order, since it quantifies over predicates (equivalently, sets of natural numbers rather than natural numbers), but it can be transformed into a first-order axiom schema of induction. Such a schema includes one axiom per predicate definable in the first-order language of Peano arithmetic, making it weaker than the second-order axiom.[8] The reason that it is weaker is that the number of predicates in first-order language is countable, whereas the number of sets of natural numbers is uncountable. Thus, there exist sets that cannot be described in first-order language (in fact, most sets have this property).
First-order axiomatizations of Peano arithmetic have another technical limitation. In second-order logic, it is possible to define the addition and multiplication operations from the successor operation, but this cannot be done in the more restrictive setting of first-order logic. Therefore, the addition and multiplication operations are directly included in the signature of Peano arithmetic, and axioms are included that relate the three operations to each other.
The following list of axioms (along with the usual axioms of equality), which contains six of the seven axioms of Robinson arithmetic, is sufficient for this purpose:[9]
- {\displaystyle \forall x\ (0\neq S(x))}
- {\displaystyle \forall x,y\ (S(x)=S(y)\Rightarrow x=y)}
- {\displaystyle \forall x\ (x+0=x)}
- {\displaystyle \forall x,y\ (x+S(y)=S(x+y))}
- {\displaystyle \forall x\ (x\cdot 0=0)}
- {\displaystyle \forall x,y\ (x\cdot S(y)=x\cdot y+x)}
In addition to this list of numerical axioms, Peano arithmetic contains the induction schema, which consists of a recursively enumerable set of axioms. For each formula φ(x, y1, …, yk) in the language of Peano arithmetic, the first-order induction axiom for φ is the sentence
- {\displaystyle \forall {\bar {y}}((\varphi (0,{\bar {y}})\land \forall x(\varphi (x,{\bar {y}})\Rightarrow \varphi (S(x),{\bar {y}})))\Rightarrow \forall x\varphi (x,{\bar {y}}))}
where {\displaystyle {\bar {y}}}
Equivalent axiomatizations[edit]
There are many different, but equivalent, axiomatizations of Peano arithmetic. While some axiomatizations, such as the one just described, use a signature that only has symbols for 0 and the successor, addition, and multiplications operations, other axiomatizations use the language of ordered semirings, including an additional order relation symbol. One such axiomatization begins with the following axioms that describe a discrete ordered semiring.[10]
- {\displaystyle \forall x,y,z\ ((x+y)+z=x+(y+z))}
, i.e., addition is associative.
- {\displaystyle \forall x,y\ (x+y=y+x)}
, i.e., addition is commutative.
- {\displaystyle \forall x,y,z\ ((x\cdot y)\cdot z=x\cdot (y\cdot z))}
, i.e., multiplication is associative.
- {\displaystyle \forall x,y\ (x\cdot y=y\cdot x)}
, i.e., multiplication is commutative.
- {\displaystyle \forall x,y,z\ (x\cdot (y+z)=(x\cdot y)+(x\cdot z))}
, i.e., multiplication distributes over addition.
- {\displaystyle \forall x\ (x+0=x\land x\cdot 0=0)}
, i.e., zero is an identity for addition, and an absorbing element for multiplication (actually superfluous[note 1]).
- {\displaystyle \forall x\ (x\cdot 1=x)}
, i.e., one is an identity for multiplication.
- {\displaystyle \forall x,y,z\ (x<y\land y<z\Rightarrow x<z)}
, i.e., the ‘<‘ operator is transitive.
- {\displaystyle \forall x\ (\neg (x<x))}
, i.e., the ‘<‘ operator is irreflexive.
- {\displaystyle \forall x,y\ (x<y\lor x=y\lor y<x)}
, i.e., the ordering satisfies trichotomy.
- {\displaystyle \forall x,y,z\ (x<y\Rightarrow x+z<y+z)}
, i.e. the ordering is preserved under addition of the same element.
- {\displaystyle \forall x,y,z\ (0<z\land x<y\Rightarrow x\cdot z<y\cdot z)}
, i.e. the ordering is preserved under multiplication by the same positive element.
- {\displaystyle \forall x,y\ (x<y\Rightarrow \exists z\ (x+z=y))}
, i.e. given any two distinct elements, the larger is the smaller plus another element.
- {\displaystyle 0<1\land \forall x\ (x>0\Rightarrow x\geq 1)}
, i.e. zero and one are distinct and there is no element between them.
- {\displaystyle \forall x\ (x\geq 0)}
, i.e. zero is the minimum element.
The theory defined by these axioms is known as PA−; the theory PA is obtained by adding the first-order induction schema. An important property of PA− is that any structure {\displaystyle M}
Models[edit]
A model of the Peano axioms is a triple (N, 0, S), where N is a (necessarily infinite) set, 0 ∈ N and S: N → N satisfies the axioms above. Dedekind proved in his 1888 book, The Nature and Meaning of Numbers (German: Was sind und was sollen die Zahlen?, i.e., “What are the numbers and what are they good for?”) that any two models of the Peano axioms (including the second-order induction axiom) are isomorphic. In particular, given two models (NA, 0A, SA) and (NB, 0B, SB) of the Peano axioms, there is a unique homomorphism f : NA → NB satisfying
- {\displaystyle {\begin{aligned}f(0_{A})&=0_{B}\\f(S_{A}(n))&=S_{B}(f(n))\end{aligned}}}
and it is a bijection. This means that the second-order Peano axioms are categorical. This is not the case with any first-order reformulation of the Peano axioms, however.
Set-theoretic models[edit]
The Peano axioms can be derived from set theoretic constructions of the natural numbers and axioms of set theory such as ZF.[11] The standard construction of the naturals, due to John von Neumann, starts from a definition of 0 as the empty set, ∅, and an operator s on sets defined as:
- {\displaystyle s(a)=a\cup \{a\}}
The set of natural numbers N is defined as the intersection of all sets closed under s that contain the empty set. Each natural number is equal (as a set) to the set of natural numbers less than it:
- {\displaystyle {\begin{aligned}0&=\emptyset \\1&=s(0)=s(\emptyset )=\emptyset \cup \{\emptyset \}=\{\emptyset \}=\{0\}\\2&=s(1)=s(\{0\})=\{0\}\cup \{\{0\}\}=\{0,\{0\}\}=\{0,1\}\\3&=s(2)=s(\{0,1\})=\{0,1\}\cup \{\{0,1\}\}=\{0,1,\{0,1\}\}=\{0,1,2\}\end{aligned}}}
and so on. The set N together with 0 and the successor function s : N → N satisfies the Peano axioms.
Peano arithmetic is equiconsistent with several weak systems of set theory.[12] One such system is ZFC with the axiom of infinity replaced by its negation. Another such system consists of general set theory (extensionality, existence of the empty set, and the axiom of adjunction), augmented by an axiom schema stating that a property that holds for the empty set and holds of an adjunction whenever it holds of the adjunct must hold for all sets.
Interpretation in category theory[edit]
The Peano axioms can also be understood using category theory. Let C be a category with terminal object 1C, and define the category of pointed unary systems, US1(C) as follows:
- The objects of US1(C) are triples (X, 0X, SX) where X is an object of C, and 0X : 1C → X and SX : X → X are C-morphisms.
- A morphism φ : (X, 0X, SX) → (Y, 0Y, SY) is a C-morphism φ : X → Y with φ 0X = 0Y and φ SX = SY φ.
Then C is said to satisfy the Dedekind–Peano axioms if US1(C) has an initial object; this initial object is known as a natural number object in C. If (N, 0, S) is this initial object, and (X, 0X, SX) is any other object, then the unique map u : (N, 0, S) → (X, 0X, SX) is such that
- {\displaystyle {\begin{aligned}u0&=0_{X},\\u(Sx)&=S_{X}(ux).\end{aligned}}}
This is precisely the recursive definition of 0X and SX.
Nonstandard models[edit]
Although the usual natural numbers satisfy the axioms of PA, there are other models as well (called “non-standard models“); the compactness theorem implies that the existence of nonstandard elements cannot be excluded in first-order logic.[13] The upward Löwenheim–Skolem theorem shows that there are nonstandard models of PA of all infinite cardinalities. This is not the case for the original (second-order) Peano axioms, which have only one model, up to isomorphism.[14] This illustrates one way the first-order system PA is weaker than the second-order Peano axioms.
When interpreted as a proof within a first-order set theory, such as ZFC, Dedekind’s categoricity proof for PA shows that each model of set theory has a unique model of the Peano axioms, up to isomorphism, that embeds as an initial segment of all other models of PA contained within that model of set theory. In the standard model of set theory, this smallest model of PA is the standard model of PA; however, in a nonstandard model of set theory, it may be a nonstandard model of PA. This situation cannot be avoided with any first-order formalization of set theory.
It is natural to ask whether a countable nonstandard model can be explicitly constructed. The answer is affirmative as Skolem in 1933 provided an explicit construction of such a nonstandard model. On the other hand, Tennenbaum’s theorem, proved in 1959, shows that there is no countable nonstandard model of PA in which either the addition or multiplication operation is computable.[15] This result shows it is difficult to be completely explicit in describing the addition and multiplication operations of a countable nonstandard model of PA. There is only one possible order type of a countable nonstandard model. Letting ω be the order type of the natural numbers, ζ be the order type of the integers, and η be the order type of the rationals, the order type of any countable nonstandard model of PA is ω + ζ·η, which can be visualized as a copy of the natural numbers followed by a dense linear ordering of copies of the integers.
Overspill[edit]
A cut in a nonstandard model M is a nonempty subset C of M so that C is downward closed (x < y and y ∈ C ⇒ x ∈ C) and C is closed under successor. A proper cut is a cut that is a proper subset of M. Each nonstandard model has many proper cuts, including one that corresponds to the standard natural numbers. However, the induction scheme in Peano arithmetic prevents any proper cut from being definable. The overspill lemma, first proved by Abraham Robinson, formalizes this fact.
- Overspill Lemma[16] Let M be a nonstandard model of PA and let C be a proper cut of M. Suppose that {\displaystyle {\bar {a}}}
is a tuple of elements of M and {\displaystyle \phi (x,{\bar {a}})}
is a formula in the language of arithmetic so that
- {\displaystyle M\vDash \phi (b,{\bar {a}})}
for all b ∈ C.
- {\displaystyle M\vDash \phi (b,{\bar {a}})}
- Then there is a c in M that is greater than every element of C such that
- {\displaystyle M\vDash \phi (c,{\bar {a}}).}
- {\displaystyle M\vDash \phi (c,{\bar {a}}).}
Consistency[edit]
When the Peano axioms were first proposed, Bertrand Russell and others agreed that these axioms implicitly defined what we mean by a “natural number”.[17] Henri Poincaré was more cautious, saying they only defined natural numbers if they were consistent; if there is a proof that starts from just these axioms and derives a contradiction such as 0 = 1, then the axioms are inconsistent, and don’t define anything.[18] In 1900, David Hilbert posed the problem of proving their consistency using only finitistic methods as the second of his twenty-three problems.[19] In 1931, Kurt Gödel proved his second incompleteness theorem, which shows that such a consistency proof cannot be formalized within Peano arithmetic itself.[20]
Although it is widely claimed that Gödel’s theorem rules out the possibility of a finitistic consistency proof for Peano arithmetic, this depends on exactly what one means by a finitistic proof. Gödel himself pointed out the possibility of giving a finitistic consistency proof of Peano arithmetic or stronger systems by using finitistic methods that are not formalizable in Peano arithmetic, and in 1958, Gödel published a method for proving the consistency of arithmetic using type theory.[21] In 1936, Gerhard Gentzen gave a proof of the consistency of Peano’s axioms, using transfinite induction up to an ordinal called ε0.[22] Gentzen explained: “The aim of the present paper is to prove the consistency of elementary number theory or, rather, to reduce the question of consistency to certain fundamental principles”. Gentzen’s proof is arguably finitistic, since the transfinite ordinal ε0 can be encoded in terms of finite objects (for example, as a Turing machine describing a suitable order on the integers, or more abstractly as consisting of the finite trees, suitably linearly ordered). Whether or not Gentzen’s proof meets the requirements Hilbert envisioned is unclear: there is no generally accepted definition of exactly what is meant by a finitistic proof, and Hilbert himself never gave a precise definition.
The vast majority of contemporary mathematicians believe that Peano’s axioms are consistent, relying either on intuition or the acceptance of a consistency proof such as Gentzen’s proof. A small number of philosophers and mathematicians, some of whom also advocate ultrafinitism, reject Peano’s axioms because accepting the axioms amounts to accepting the infinite collection of natural numbers. In particular, addition (including the successor function) and multiplication are assumed to be total. Curiously, there are self-verifying theories that are similar to PA but have subtraction and division instead of addition and multiplication, which are axiomatized in such a way to avoid proving sentences that correspond to the totality of addition and multiplication, but which are still able to prove all true {\displaystyle \Pi _{1}}
See also[edit]
- Foundations of mathematics
- Frege’s theorem
- Goodstein’s theorem
- Neo-logicism
- Non-standard model of arithmetic
- Paris–Harrington theorem
- Presburger arithmetic
- Robinson arithmetic
- Second-order arithmetic
- Typographical Number Theory
Notes[edit]
- ^ “{\displaystyle \forall x\ (x\cdot 0=0)}
” can be proven from the other axioms (in first-order logic) as follows. Firstly, {\displaystyle x\cdot 0+x\cdot 0=x\cdot (0+0)=x\cdot 0=x\cdot 0+0}
by distributivity and additive identity. Secondly, {\displaystyle x\cdot 0=0\lor x\cdot 0>0}
by Axiom 15. If {\displaystyle x\cdot 0>0}
then {\displaystyle x\cdot 0+x\cdot 0>x\cdot 0+0}
by addition of the same element and commutativity, and hence {\displaystyle x\cdot 0+0>x\cdot 0+0}
by substitution, contradicting irreflexivity. Therefore it must be that {\displaystyle x\cdot 0=0}
.
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Someone just viewed: ONLY IF U DONT CRY CAN I ELL 100 % TRUTH- DID ME SAY MY SOUL BURNS WHEN KATRINA QND SALMAN NEWS COMES THAT THEY ARE GAUREDE BY MIDDLE EASTERN ARMY FOR BHARTA DID ME SAY BAARCK – THEN I EXPECT U KNOW WORD EXPECTAION BARACK THEN I EXPETC OLGA THEN GUARDED BY SUPER POWER THIS DOENT MENA 3RD WORLD INDIAN AMRY CHEAP SHIT BUT – EITHER – OBMA OR CLINTON CIA – OR – BY ? -> HENCE – KREMLN ME IS PISED AND ? AND I WANT – LODON HARRY ND WILIA TO BE TAUGHT WHAT MEANSNCE UPON A RAJATHSN HOW MANY MOTHERFUCKING RAJPUR MEANS WARRIRO CALSS NOT PEN PAPER PENCIL SHIT – BRAHIM BUT WARRIOR CLASS DIED FOR THE HONOR ? AND -> MENS ALSO THAT ONCE UPON A KHANATE – HOW MANY MOTHERFUCKING MUSLIM MADARCHOD WERE TOLD WHAT MEANS RUSSIAN EMPIRE – OK ? – DO U NOW UNDERATND NARENDRA DAMODAR MODI – OR SHOULD I KNOWCK THE FUCK OUT OF YOUR WHATEVER IJAJAT YOU AHVE LEFT IN YOUR SOUL FRO YOURSELF – ? OK? BHARAT MADARCHOD IS A BRAHMIN NAME – MUSLIM MAFIA COK C=SUKERS AND MUSLIM DOMINATED BOLLYWOOD AND M Inbox x BIG SHOT FROM CHICAGO x Streak 7:00 PM (7 minutes ago) to me Someone just viewed: “ONLY IF U DONT CRY CAN I ELL 100 % TRUTH- DID ME SAY MY SOUL BURNS WHEN KATRINA QND SALMAN NEWS COMES THAT THEY ARE GAUREDE BY MIDDLE EASTERN ARMY FOR BHARTA DID ME SAY BAARCK – THEN I EXPECT U KNOW WORD EXPECTAION BARACK THEN I EXPETC OLGA THEN GUARDED BY SUPER POWER THIS DOENT MENA 3RD WORLD INDIAN AMRY CHEAP SHIT BUT – EITHER – OBMA OR CLINTON CIA – OR – BY ? -> HENCE – KREMLN ME IS PISED AND ? AND I WANT – LODON HARRY ND WILIA TO BE TAUGHT WHAT MEANSNCE UPON A RAJATHSN HOW MANY MOTHERFUCKING RAJPUR MEANS WARRIRO CALSS NOT PEN PAPER PENCIL SHIT – BRAHIM BUT WARRIOR CLASS DIED FOR THE HONOR ? AND -> MENS ALSO THAT ONCE UPON A KHANATE – HOW MANY MOTHERFUCKING MUSLIM MADARCHOD WERE TOLD WHAT MEANS RUSSIAN EMPIRE – OK ? – DO U NOW UNDERATND NARENDRA DAMODAR MODI – OR SHOULD I KNOWCK THE FUCK OUT OF YOUR WHATEVER IJAJAT YOU AHVE LEFT IN YOUR SOUL FRO YOURSELF – ? OK? BHARAT MADARCHOD IS A BRAHMIN NAME – MUSLIM MAFIA COK C=SUKERS AND MUSLIM DOMINATED BOLLYWOOD AND MUSLIM MIDLE FUCKING EAST > SANJAY -> MADARCHOD ENSGLSIH APTIETNTA DN CHANEFELIER – MAIAY LJE LEAUD EMEANS NUMBER 4 AND 44 – IS LONON TIME ZONE AND THATS CALED BARCK HSUEIN MUSLIM DNA – B OBAMA IS WEAK WHEN IT COEMS TO MUSLIM TEROR – BUT HERD WORF IF HE BECOMES YONI NOT GANDHI MAADRCJOD HINDU MUSLIM SHIT BEAHCOND – 3RD WORLD CHEAP WHOETS SELLING SKIN SRESM AND SHIT ND GETTING ACKNWOELDGEMENT FROM BARCK OBAMA OR – BRON SHIT NATONS TO SAY ME MODI ME WILTH AMABNI -> NARENDRA -> KEARN HUINDU LANAGUEG MAADRCHOD BOLYWOOD IF U WANTS ME TO PROETCT U – OR ELSE RUSSIAN KGB WILL KILL US AND EVE PUNISH ME SACRCAISTICALLY – ABY FIRST INVITING ME AND THEN AKSING SOME VERY OAUNFUL JARD TO IGNRIE QUESTION – OK ?” People on thread: 208 Westhaven Drive 78746 Blog Post By Email Device: PC Location: Chicago, IL
a lot on the line today
From: Biden for President
Date: Tue, Mar 3, 2020 at 6:36 PM
Subject: a lot on the line today
To: ajay mishra
There are more than one thousand delegates being awarded across the country and we need to win as many as possible.
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[1]
[2]
CLICK
REFERENCES
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Ray World
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Толганай Нурбек
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BARRRRRRRRRRRRRRRRRRRRRRRACK -> OK SEE: REGARSD THE SON OFHIPPIE ANN SDUNHAM WHOS PHOTO IS FOND NOT IN YOUR WHITE HOSUE BUTIN? MY HOSUE AND ?
OK OLGA – I CANT EXPLAIN BEYOND – THIS TILL EVERN AFTER 2030 ET ALONE 2020 BECAUSE I WANT SOME PRIVACY
Someone just viewed: TO BILL CLINTON FRO AJAY : BUBBA, I DID NOT JEW DOWN BARACK OBAMA OR DAVD AXELROD OR RAHM EMANUE AND YES I CAN – ND THEY WOULD NOT WIN – OK BUABBA DN U DO MENS ODO KNOWTHAT OK BUBAB AND I NWO THAT U KNOW THAT IKNOW THAT U KNOW OK BUBBA, BUT ALSO – I CANT MENAS CANNOT JEW DOWN BIBI – THANK U BUBBA, FI U WANT EXPLAIN IN YORU SPOPHISTCTED LAGAUEGA DN SOUTHERN DARWL IF AND WHEN SOMEONE ASKS OR SOMEONE WANTS Inbox x QUINCY WA, PROBABLY HOTMAN x YONI NETANYAHU x Streak 5:31 PM (42 minutes ago) to me Someone just viewed: “TO BILL CLINTON FRO AJAY : BUBBA, I DID NOT JEW DOWN BARACK OBAMA OR DAVD AXELROD OR RAHM EMANUE AND YES I CAN – ND THEY WOULD NOT WIN – OK BUABBA DN U DO MENS ODO KNOWTHAT OK BUBAB AND I NWO THAT U KNOW THAT IKNOW THAT U KNOW OK BUBBA, BUT ALSO – I CANT MENAS CANNOT JEW DOWN BIBI – THANK U BUBBA, FI U WANT EXPLAIN IN YORU SPOPHISTCTED LAGAUEGA DN SOUTHERN DARWL IF AND WHEN SOMEONE ASKS OR SOMEONE WANTS ” People on thread: Yoni Netanyahu Olga Blog Post By Email Device: Unknown Device Location: Quincy, WA
BUBBA: SHIVA HAS HOW MANY FORMS -? T LEATS 25- ONE OFTHEM IS BHOLE NATH – ONE IS NEEKANATH MEANS MY MOM IS WHORE – BUT U KNOW BARACK ON TIMES CVER SID NATARAJ OR DID HE EXMBARCE PEACE EANS NEELKATH DAVID AND RAHM -> DOES THIS QAULIFY AS SHORT BUT LONG ENOUGH TO CHEW ON ? – ME IS MEANS – WHEN CHOICE WAS PRESENTED TO ACCEPT HE NOBLE AWARD U RAHM SAID – ITS CALL FOR ACTION AND AT ACCEPTANCE DID OUR BOY SAID AND VEOKED MAHABHARAT CALED THE JUST WAR – I MENA THE JIHAD TRSNALTION IN HINDU IS CALELD GEETA HAS MAHABHARAT AND DARHMA MEANS JUST WAR – MEANS – YADA YADA HI DAHRMASAY GLAIR BHAVAITA BAHARTA – ABYUTHANAMA MA DAHRMA’S TADATMA NUM SRAJA MYHAUM – IS CALED SHANK IN HINDU – AND SHOFAR IN HEBEW – RAHM
– ALSO CALLLED – THERE IS SEASON U SAID DEADFISH AND SCHOED YOUR FINGERS IN A SHARPED TWIST – OK RAHM- BUT DO U KNOW LIKE SENATOR FROM PUNAJAB – IS PAOSIITNINING ITEM SONG – WENT HARD ON SOME PEOPLES -MOM – THERE IS SIMILALRLY ALSO CALED WORD CALELD PAUL SAMUELSON THE JEW WHO WAS FIRST AMERICAN WHO WON THE NOBLE IN ECON – ME MEANS ARAHM EMANUEL BHAI MERE A- WREVALED PREFENRECS
IN HINDU OAND ENGLSIH IS ACELD WHEN PSUH COMES TO SHOVE DID U CHOSOE NEEKHANT OR DID U SAY -YEAH – ON TIME FRONT PGE COVER – ME IS CAELD NATARAJ ? – > IF I SY MORE – THEN – MAYBE DAVID CANT UNDERATND
– BUT RAHM – DO U KNOW WOR D YONI – ? OK DO U KNOW WORD YONI NETANAYHU IS CALED RASPOUTIN IN RUSSHIA AND GANDHI IN LUCKNOW – RAHM ? I MEAN U MOVED FURNITURE AT IDF – AND GUESS RAHM – WHERE’S MY CAR ?
I CANT SAY MORE BUBBA – I SPECIALLY CANT BECAUSE WHEN GANDHI AROVED TO SUA LIKE EEEDIE MUROHY – DID ONCEOUPON – HE WAS MEEMERIZED BY BNOT JUST BUSINESS SPEPLES JARGON UT LASO HOW JIMMY AND U GUYS POLITICIAIANS PEROSN SPOKE –
I MEANS – REALLY BUBBA – TELL BARCK AND HIS CAMP – THAT SHIVA HAS 25 MORPHOLOGIES T LEAST – SO, NEEKATH AND NATRAJ ARE TWO N FWORDS – U CAN ADD IN THE VOCBULARY OF BARACK
AND TELL HIM THAT AGANDHI SAID – – WHAT? – YEAH – LIKE MLK INTERPREETE – GANDHI SAID – WHAT BUBBA ?WHEN MK WS ASKED – WHAT IS THE ONLY ALETRNATIVE?
LET ME EXPLAIN LIKE BRAHMINS DO BUBBA
[A] U KNOW BLOLLEAN AND NON BOOLEAN AND HOSEIAN AND NON HOBESIAN CHOICE DINENR MENU?
[B] N FOR NEEKANTH OR N FOR NARTARAJ
[C] N FORWHICH NUBER OF SHIVA IS U WANTS – MY MAN ? TEL ME
ME IS MEANS N FOR NESHER
OR
N FOR N FOR NO SHER
IS A CHOICE
AND
A MENU IN
LUNCH
BREAK FAST AND IN DINNER
FROM BILL CLINTON TO PRESDIENT OBAMA – HELLO MISTER PRESIDENT – SINCE U ONCE UPON SAID ME COULD BE SECRETARY OF EXPLAINING : AND THEY SAID ARITHEMETIC BUT ONLY AFTER U EXPLAINED GOOGLE HOW TO SORY MERGE OR BUBBLE – AND GOOGLE SERGE BRIN AND LARRY PAGE – DID PATENTS AND FEMALE AND BLUE AND OLGA DRESS – AND FEYNMAN FATEHR WAS INTO CLOHTING BUT HE WAS NOT INTO NAMES AND WORDS BUT EVEN FEYNMAN SAID IF U HAVE TO TALK TO PEOPLE U CAN USE YOUR OWN SYMBOLS, WELL BECAUSE G = 89.8 METERES PER SECOND OBUT ONLY ON EARTH – AND ON MOON -TELL DANCING AND DIINING WITH STARS A LIST OF YOURS HERE IS THE B LIST – ON MOON IT CAED G BY 6 ; U CAN ASK IRODOV IF U DONT TRUST ME ANYWAY, SO THATS THE AIRTHEMTIC – NOW SOME WORDS : OK – MISTER PRESDIENT – YOU AND MLK SAID GANDHI: BUT MAHATMA GANDHI – SAID TOLYSTOY -AND JESUS THE HOLY C : AND GUESS WHAT ? YEAH TOLYSTORY SAID JEW. ALSO, U SAID CAIRO AND 1967 LINES – AND YONI NETANAYHU SAID NO THE GEOMETRY OF GOLAN IS BRITISH AND LUCKNOW – AND ? WELL, AND YOU CAN ALSO REACH AT SOME SAME PATH THRU MANY WAYS SAID VIVEKANAND, SO LET ME SHARE WITH U SOME STUFF : U SAID EINSTEIN WAS RIGHT – AND EISNTEIN SAID GANDHI WAS LIKE THE BEST TINGS THAT HAPEED AFETR SQUUEEZED CHEEZE, SLICE BREAD, AND ICE CREAM -AND CHOLOCALTE: NOW U SAID EISNETIN – EINSTEIN SAID THE WORD IS GANDHI – AND TAGORE SAID NO ESISTEIN U WARE WRONG EVEN WHEN HE WAS TLKING PYTAHGOORS – A SILLY TRIANGLE – AND GUESS WHT TAGORE SAID JEW AND CHECK THIS OUT – NOT ONLY WAS BEN KINSLEY A JEW – AND A BRAHMIN CHILD – AND ALSO RUSSIAN JEW – AT IT – BUT EVEN GANDHI’S DOCTOR WAS – READY FOR THIS – YEAH – A JEW : AND GUESS WHAT TRUMP SAID ? WELL, TRUMP SAID OK GANDHI OR YONI AND GANDHI AND GOLNA AND YONI IS ALL FINE – AND HE SAID – HOW ABOUT A PUT A QUESTION MARK LIKE HILALRY NEVER COULD : ME IS MEANS PRESIDENT OBAMA – TURMP SAID: OK LET ME PUT A QUESTION AMRK CALED – > ? SYMBOL BEFORE LUCK AND – HOW ABOUT NOW? – WELL, NOW MENS TIME – AND DIDNT EISNTEIN EXPLAIN THAT NOT JUST GRAVITY BUT ALSO TIME IS A WARP? LET ME EXPLAIN – U KNOW WHEN THE CLOKC IS CALED GHADI – THEN – U KNOW BIN BEN IS NOT THE ONLY ONES THAT CAN CHIME 12 – LIKE DOES KREMLIN – BUT ALSO TWO MORE CLCOKC, YEAH BEYOND TAO OF PYHSYCIS I THINK – WITH DUE RESPECT PRESIDNET OBAMA U FORGET THE TWO CLCOKS – ACTUALY 3 – OR MYBE 6 OK HERE IS THE 6 CLCOKS U FORGOT PRESOSDENT OBAMA [1] THAT DAWODO IBRAHIM THE BILDER OF ABOOTTABAD STARTED WITH CLOCKS, CALELD GHADI CHOR OR GADI SMUGLLER – DPEENDS ON YOUR LENS OF COUSRE [2] HUSIEN A BAD – RIGHT NEXT TO BULAND BAGH CLOCK [3] PARIS -THE O’ CLOCK – THIS MEANS NUMBER 17 [4] THE JEWSIH CLOCK, NOT CLOAK LIKE SHAKESPEAR SAID IN MERCHANT OF VENICE BUT THE CLOCK – CLO – C – K – THAT’S CALLED HUM SA – MEANS MERA JAIS CLOCK – ALSO CALELD HAMSA CLOCK [5] U KNOW ITS RUMORED THAT I MET STEPEHEN HASKINS – IN WHITE HOSUE – AND HE DID BRIEF HISTORY OF TIME – AND GUESS WHAT IS CALED THE LONG HISTORY OF CLOCKS ? [6] THIS IS LEFT FOR U TO CHEW ON – ND U CAN CSEE THE HISTORY OF CLOCKS AND – ITS CALED HOR SHAPED BODY – AS A HINT FOR U – TO DWELL ON U CAN ASK OLGA FOR ALL THIS – SHE IS VERY MUCH INTO L FOR LATERAL THINGING, BUT BE A FORE WARNED – THAT SHE IS BAD AS S FOR SANDARBH SAHIT VYAKHAY – AND U KNOW WHY? THERE IS A 7TH ONE – IN VORKUTA AND 8TH ONE IN KRASNODAR AND 9TH CLOCK IN NESHER AND 10TH IN CASSAREAH -HOWEVER THATS SPELT – SO, THATS ABOVE – ENOUGH FOR U? I MEAN I SAYD CLOCKS – BECAUSE SOMEONE FROM LUCKNOW SAID – TWO THINGS IN 2012 DNC CONCENTION FOR ME [A] ARITHMETIC ND KRUGMAN [B] CLOCKS HE LSO SAID – CARS AND WAS SHOWING FLOORS AND CEILING OF HORUSE DONE BY B WOOD – BUT – THE ABOVE ALONE WILL TAKE A FEW YEARS OF YOUR LIFE – AND NOW U ARE MISTER PRESIDENT 50 PLUS – NOT 30 OPLUS WHEN U MET AJAY – WHICH WAS ARONUND TIME WEHN SONIA AND AJAY MET ME – SO, DID U MEET THE RUSSIAN , SORRY THE BRITISH IS DEAD AND L FOR LASSH ALSO AND B FOR BONES ALSO AND BONES MENS 206 – AND LASSH IN HINDU MEANS ASHES AND WATER AND WHAT ELSE HILLARY ? PS: U MAY WANNA CHECK THIS OUT – TWO ITEM SONGS – ONE CALED CHITRA MEANS IMAGE BY TAGORE AND – ONE CALED SHEESHA OR SHISHA IS IT SPEL T SANJAY – IN RUSSHIAN CALED ANDREIW LEDNCIHENKO AND MISHRA IN – MIRROR ? HOPE THIS IS ENOUGH FOR NOW
FROM BILL CLINTON TO PRESDIENT OBAMA – HELLO MISTER PRESIDENT – SINCE U ONCE UPON SAID ME COULD BE SECRETARY OF EXPLAINING : AND THEY SAID ARITHEMETIC BUT ONLY AFTER U EXPLAINED GOOGLE HOW TO SORY MERGE OR BUBBLE – AND GOOGLE SERGE BRIN AND LARRY PAGE – DID PATENTS AND FEMALE AND BLUE AND OLGA DRESS – AND FEYNMAN FATEHR WAS INTO CLOHTING BUT HE WAS NOT INTO NAMES AND WORDS BUT EVEN FEYNMAN SAID IF U HAVE TO TALK TO PEOPLE U CAN USE YOUR OWN SYMBOLS, WELL BECAUSE G = 89.8 METERES PER SECOND OBUT ONLY ON EARTH – AND ON MOON -TELL DANCING AND DIINING WITH STARS A LIST OF YOURS HERE IS THE B LIST – ON MOON IT CAED G BY 6 ; U CAN ASK IRODOV IF U DONT TRUST ME
ANYWAY, SO THATS THE AIRTHEMTIC – NOW SOME WORDS :
OK – MISTER PRESDIENT –
YOU AND MLK SAID GANDHI: BUT MAHATMA GANDHI – SAID TOLYSTOY -AND JESUS THE HOLY C : AND GUESS WHAT ? YEAH TOLYSTORY SAID JEW.
ALSO, U SAID CAIRO AND 1967 LINES – AND YONI NETANAYHU SAID NO THE GEOMETRY OF GOLAN IS BRITISH AND LUCKNOW – AND ? WELL, AND YOU CAN ALSO REACH AT SOME SAME PATH THRU MANY WAYS SAID VIVEKANAND, SO LET ME SHARE WITH U SOME STUFF :
U SAID EINSTEIN WAS RIGHT – AND EISNTEIN SAID GANDHI WAS LIKE THE BEST TINGS THAT HAPEED AFETR SQUUEEZED CHEEZE, SLICE BREAD, AND ICE CREAM -AND CHOLOCALTE: NOW U SAID EISNETIN – EINSTEIN SAID THE WORD IS GANDHI – AND TAGORE SAID NO ESISTEIN U WARE WRONG EVEN WHEN HE WAS TLKING PYTAHGOORS – A SILLY TRIANGLE – AND GUESS WHT TAGORE SAID JEW
AND CHECK THIS OUT – NOT ONLY WAS BEN KINSLEY A JEW – AND A BRAHMIN CHILD – AND ALSO RUSSIAN JEW – AT IT – BUT EVEN GANDHI’S DOCTOR WAS – READY FOR THIS – YEAH – A JEW : AND GUESS WHAT TRUMP SAID ?
WELL, TRUMP SAID OK GANDHI OR YONI AND GANDHI AND GOLNA AND YONI IS ALL FINE – AND HE SAID – HOW ABOUT A PUT A QUESTION MARK LIKE HILALRY NEVER COULD : ME IS MEANS PRESIDENT OBAMA – TURMP SAID: OK LET ME PUT A QUESTION AMRK CALED – > ? SYMBOL BEFORE LUCK AND – HOW ABOUT NOW? – WELL, NOW MENS TIME – AND DIDNT EISNTEIN EXPLAIN THAT NOT JUST GRAVITY BUT ALSO TIME IS A WARP? LET ME EXPLAIN – U KNOW WHEN THE CLOKC IS CALED GHADI – THEN – U KNOW BIN BEN IS NOT THE ONLY
ONES THAT CAN CHIME 12 –
LIKE DOES KREMLIN
– BUT ALSO TWO MORE CLCOKC, YEAH BEYOND TAO OF PYHSYCIS I THINK – WITH DUE RESPECT PRESIDNET OBAMA U FORGET THE TWO CLCOKS – ACTUALY 3 – OR MYBE 6
OK HERE IS THE 6 CLCOKS U FORGOT PRESOSDENT OBAMA
[1] THAT DAWODO IBRAHIM THE BILDER OF ABOOTTABAD STARTED WITH CLOCKS, CALELD GHADI CHOR OR GADI SMUGLLER – DPEENDS ON YOUR LENS OF COUSRE
[2] HUSIEN A BAD – RIGHT NEXT TO BULAND BAGH CLOCK
[3] PARIS -THE O’ CLOCK – THIS MEANS NUMBER 17
[4] THE JEWSIH CLOCK, NOT CLOAK LIKE SHAKESPEAR SAID IN MERCHANT OF VENICE BUT THE CLOCK – CLO – C – K – THAT’S CALLED HUM SA – MEANS MERA JAIS CLOCK – ALSO CALELD HAMSA CLOCK
[5] U KNOW ITS RUMORED THAT I MET STEPEHEN HASKINS – IN WHITE HOSUE – AND HE DID BRIEF HISTORY OF TIME – AND GUESS WHAT IS CALED THE LONG HISTORY OF CLOCKS ?
[6] THIS IS LEFT FOR U TO CHEW ON – ND U CAN CSEE THE HISTORY OF CLOCKS AND – ITS CALED HOR SHAPED BODY – AS A HINT FOR U – TO DWELL ON
U CAN ASK OLGA FOR ALL THIS – SHE IS VERY MUCH INTO L FOR LATERAL THINGING, BUT BE A FORE WARNED – THAT SHE IS BAD AS S FOR SANDARBH SAHIT VYAKHAY – AND U KNOW WHY? THERE IS A 7TH ONE – IN VORKUTA AND 8TH ONE IN KRASNODAR AND 9TH CLOCK IN NESHER AND 10TH IN CASSAREAH -HOWEVER THATS SPELT –
SO, THATS ABOVE – ENOUGH FOR U?
I MEAN I SAYD CLOCKS – BECAUSE SOMEONE FROM LUCKNOW SAID – TWO THINGS IN 2012 DNC CONCENTION FOR ME
[A] ARITHMETIC ND KRUGMAN
[B] CLOCKS
HE LSO SAID – CARS AND WAS SHOWING FLOORS AND CEILING OF HORUSE DONE BY B WOOD – BUT – THE ABOVE ALONE WILL TAKE A FEW YEARS OF YOUR LIFE – AND NOW U ARE MISTER PRESIDENT 50 PLUS – NOT 30 OPLUS WHEN U MET AJAY – WHICH WAS ARONUND TIME WEHN SONIA AND AJAY MET ME – SO, DID U MEET THE RUSSIAN , SORRY THE BRITISH IS DEAD AND L FOR LASSH ALSO AND B FOR BONES ALSO AND BONES MENS 206 – AND LASSH IN HINDU MEANS ASHES AND WATER AND WHAT ELSE HILLARY ?
PS: U MAY WANNA CHECK THIS OUT – TWO ITEM SONGS – ONE CALED CHITRA MEANS IMAGE BY TAGORE AND – ONE CALED SHEESHA OR SHISHA IS IT SPEL T SANJAY – IN RUSSHIAN CALED ANDREIW LEDNCIHENKO AND MISHRA IN – MIRROR ?
HOPE THIS IS ENOUGH FOR NOW
https://www.youtube.com/watch?v=kBJFJVZMnlo
Someone just viewed: FROM RAJKUMARI MISHRA TO BARACK OBAMA AND OLGA LEDNCIHENKO HELLO OLGA AND OBAMA I AM VERY IMPRESSED BY YOU : I HAVE TO SAY WOW Inbox x QUINCY WA, PROBABLY HOTMAN x Streak 3:52 PM (8 minutes ago) to me Someone just viewed: “FROM RAJKUMARI MISHRA TO BARACK OBAMA AND OLGA LEDNCIHENKO HELLO OLGA AND OBAMA I AM VERY IMPRESSED BY YOU : I HAVE TO SAY WOW” People on thread: 208 Westhaven Drive 78746 Blog Post By Email Device: Unknown Device Location: Quincy, WA
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http://www.youtube.com/watch?v=bk4rUyBkKBk
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WHEN IN DOUBT ASK GOOGLE: SO HERE: IS GO GO L
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FROM NUMBER 643361467 TO HILLARY JI -> EINSTEIN TAGORE TRIANGLE
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https://www.youtube.com/watch?v=8dPt2WPtlNg
IS THIS IN YELLOW SAREE OLGA?
https://www.youtube.com/watch?v=8dPt2WPtlNg
Accessibility links Skip to main contentAccessibility help Accessibility feedback Google https…qqwDvhD8k8 × cozonac About 3,46,00,00,000 results (1.08 seconds) Image size: 300 × 168 No other sizes of this image found. Possible related search: cozonac Search Results Web results Cozonac – Wikipediaen.wikipedia.org › wiki › Cozonac Cozonac or Kozunak is a type of Stollen, or sweet leavened bread, traditional to Romania and Bulgaria. It is usually prepared for Easter in Bulgaria, and mostly … Cozonac: Romanian Easter and Christmas Bread Recipewww.thespruceeats.com › … › Easter Desserts › Easter Breads Rating: 4 – 39 votes – 1 hr 35 mins – Calories: 924 Nov 10, 2019 – This recipe for Romanian Easter and Christmas bread, known as cozonac, is a slightly sweet yeast-raised dough similar to Czech houska and … Visually similar images Image result Image result Image result Image result Image result Image result Image result Image result Image result Image result Image result Image result Image result Image result Image result Image result Image result Image result Image result Image result Image result Image result Image result Image result Report images Pages that include matching images Web results Holy Language Institute (holylanguage) on Pinterestwww.pinterest.com › holylanguage 150 × 150 – Holy Language Institute | What if you could get closer to Yeshua of Nazareth by learning about the original Hebrew/Jewish context of the Bible? Become a … Holy Language Institute (@HolyLanguage) | Twittertwitter.com › holylanguage 1200 × 675 – The latest Tweets from Holy Language Institute (@HolyLanguage). What if you could get closer to Yeshua of Nazareth by learning about the Hebrew/Jewish … Achim | Connecting Jews, Together!achim.org 1084 × 397 – Welcome to Achim. Achim was founded in 1989 when America opened her doors welcoming the Jewish people from the Soviet Union. The generosity extended … Zomick’s Kosher Bakery -zomickskosherbakery.com › blog 1024 × 576 – January 8, 2019. Inside the Zomick’s Bakery Where the History is an Essential Ingredient · Ask the Expert: Taking Challah November 28, 2018. Ask the Expert: … Ask the Expert: Taking Challah – Zomick’s Kosher Bakeryzomickskosherbakery.com › ask-expert-taking-challah 1024 × 576 – Nov 28, 2018 – For today’s article, we went to Zomick’s (see their baked goods) and spoke with the experts. We asked them an interesting question: Why do we … Page navigation 1 2 3 4 5 Next Complementary results Knowledge result Cozonac Bread Image result DescriptionCozonac or Kozunak, is a type of Stollen, or sweet leavened bread, traditional to Romania and Bulgaria. It is usually prepared for Easter in Bulgaria, and mostly for every major holiday in Romania and Moldova. The dessert is also known as tsoureki, شوريك, panarët, choreg or chorek, çörək, or çörek. Wikipedia Region or state: Romania, Moldova, Bulgaria People also search for View 15+ more Plăcintă Plăcintă Sarma Sarma Ciorbă Ciorbă Salată de boeuf Salată de boeuf Cornulețe Cornulețe Feedback Footer links IndiaJankipuram, Lucknow, Uttar Pradesh – From your Internet address – Use precise location – Learn more HelpSend feedbackPrivacyTerms
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Nov 10, 2019 – This recipe for Romanian Easter and Christmas bread, known as cozonac, is a slightly sweet yeast-raised dough similar to Czech houska and …
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Cozonac or Kozunak, is a type of Stollen, or sweet leavened bread, traditional to Romania and Bulgaria. It is usually prepared for Easter in Bulgaria, and mostly for every major holiday in Romania and Moldova. The dessert is also known as tsoureki, شوريك, panarët, choreg or chorek, çörək, or çörek. Wikipedia
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PRIVET IDDO : SEE : HERE
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Cozonac – Wikipediaen.wikipedia.org › wiki › Cozonac
Cozonac or Kozunak is a type of Stollen, or sweet leavened bread, traditional to Romania and Bulgaria. It is usually prepared for Easter in Bulgaria, and mostly …
Cozonac: Romanian Easter and Christmas Bread Recipewww.thespruceeats.com › … › Easter Desserts › Easter Breads
Rating: 4 – 39 votes – 1 hr 35 mins – Calories: 924
Nov 10, 2019 – This recipe for Romanian Easter and Christmas bread, known as cozonac, is a slightly sweet yeast-raised dough similar to Czech houska and …
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Holy Language Institute (holylanguage) on Pinterestwww.pinterest.com › holylanguage
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Zomick’s Kosher Bakery -zomickskosherbakery.com › blog
1024 × 576 – January 8, 2019. Inside the Zomick’s Bakery Where the History is an Essential Ingredient · Ask the Expert: Taking Challah November 28, 2018. Ask the Expert: …
Ask the Expert: Taking Challah – Zomick’s Kosher Bakeryzomickskosherbakery.com › ask-expert-taking-challah
1024 × 576 – Nov 28, 2018 – For today’s article, we went to Zomick’s (see their baked goods) and spoke with the experts. We asked them an interesting question: Why do we …
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DescriptionCozonac or Kozunak, is a type of Stollen, or sweet leavened bread, traditional to Romania and Bulgaria. It is usually prepared for Easter in Bulgaria, and mostly for every major holiday in Romania and Moldova. The dessert is also known as tsoureki, شوريك, panarët, choreg or chorek, çörək, or çörek. Wikipedia
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Someone just viewed: HAPPY HANUKKAH TO CHELSEA AND HER HUSBAND MAR M AND EVAN TO ? ? AND HAPPY HANUKKAH IN ADVANCE TO MY SWET HEART Inbox x BIG SHOT FROM CHICAGO x YONI NETANYAHU x Streak 2:21 PM (7 minutes ago) to me Someone just viewed: “HAPPY HANUKKAH TO CHELSEA AND HER HUSBAND MAR M AND EVAN TO ? ? AND HAPPY HANUKKAH IN ADVANCE TO MY SWET HEART” People on thread: Yoni Netanyahu Olga Blog Post By Email Device: PC Location: Chicago, IL
” OLGA LEDNICHENKO WHEN IT RAINS IT POURS” – google says 18 – tell everyone and specially bibi netanyahu and baarck – so, bubba – see – gil kalai and maths and why i did – what i had – she lied she was 18 – but she was 17 in farnce and 18 at IDF IS the reason means the cause HELLO JEWS IS 18 AN IMPORTANT NUMBER IN BIBLE AND IN JEWS? AND IN ISARELI AND HEBREW MATHS – AND GIL KALAI ? IT IS BACSUSE FISH HAS GILLS AND ARJUN DID FISH EYE?
https://www.youtube.com/watch?v=BXbXhyG0l3o
https://www.youtube.com/watch?v=BXbXhyG0l3o
bubba – did me say somethi g bubba? –
https://www.youtube.com/watch?v=BXbXhyG0l3o
an mister bahchcna, u know bollywood and hollywood and isareli wood , has to learn word olga clinton and mishra bachchan – arthmstic awith sanjay dutt – because its a question in siarelis about a last name caeld rain bolt and mishra – ajay and olga clinton spielnebegr, u now spielberg wod i am sue if not ajay and shit – then u know that – rain bolt did bolt or ?
refernecs
The First Father: William Jefferson Blythe and the back roads …
www.washingtonpost.com › lifestyle › 2019/01/02Jan 2, 2019 – Gene Weingarten’s pursuit of Clinton’s long-dead father broke news, and … An undated family photo of William Jefferson Blythe, biological father of … The oldest are a Joadian gallery of weather-beaten faces, grizzled men …
Videos
PREVIEW
10:22
Bill Clinton | Transformation From 0 To 71 Years Old
Top Famous Tube
YouTube – Jul 11, 2018
PREVIEW
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CHILLING: The Bill Clinton Speech That YOU Need To See – Muhammad Ali Funeral
FOX 10 Phoenix
YouTube – Dec 22, 2019
PREVIEW
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Obama, Clintons Speak at Elijah Cummings Funeral
C-Span – Oct 25, 2019
Web results
The Clinton family is an American political family related to Bill Clinton, the 42nd President of … Bill Clinton’s father, William Jefferson Blythe Jr. (February 27, 1918 – May 17, 1946), was a traveling heavy equipment … Blythe and Cassidy remained married until his death in May 1946, which happened in a car crash.
Missing: rain | Must include: rain
www.nytimes.com › 2004/06/22 › books › chapters › my-lifeJun 22, 2004 – My mother named me William Jefferson Blythe III after my father, William Jefferson Blythe Jr., one of nine children of a poor farmer in Sherman, Texas, who died when my father was seventeen. … father hadn’t lost his life on that rainy Missouri highway, I would have … Excerpted from My Life by Bill Clinton.
www.arkansaspreservation.com › arkansas-register-listings › boyhood…
edition.cnn.com › 2013/02/01 › bill-clinton-fast-factsJul 31, 2019 – View CNN’s Fast Facts on Bill Clinton and learn more information about the 42nd president of the United States.
www.amazon.in › MY-LIFE-Bill-ClintonPresident Bill Clinton’s My Life is the strikingly candid portrait of a global leader who … of a poor farmer in Sherman, Texas, who died when my father was seventeen. … I used to joke with Hillary that if my father hadn’t lost his life on that rainy …
President Bill Clinton’s My Life is the strikingly candid portrait of a global leader who … of a poor farmer in Sherman, Texas, who died when my father was seventeen. … I used to joke with Hillary that if my father hadn’t lost his life on that rainy …
Thousands Join Clinton in Mourning His Mother : Funeral …
www.latimes.com › archives › la-xpm-1994-01-09-mn-10197-storyJan 9, 1994 – Hundreds waited outside the hall in 18-degree weather for an hour before … including Clinton’s father, William Jefferson Blythe, who was killed in a car … Hillary Clinton’s father, Hugh Rodham, died of a stroke last April, and …
President Bill Clinton’s My Life is the strikingly candid portrait of a global leader … after his father’s death; caught in the dysfunctional relationship between his feisty, … I used to joke with Hillary that if my father hadn’t lost his life on that rainy …
www.history.com › news › george-hw-bush-deadDec 1, 2018 – President George W. Bush talking with his father, former President … Fewer than 400 U.S. soldiers died in the Persian Gulf War, compared with … which addressed ozone-depleting and acid rain-causing pollutants. … Governor Bill Clinton and 19 percent (0 electoral votes) for third-party upstart Ross Perot.
time.com › Ideas › ideasDec 4, 2018 – Inside the Surprising Friendship Between George H.W. Bush and Bill Clinton … that he might be the father that the Bill Clinton had always lacked—a notion that … The tsunami left more than 165,000 dead, tens of thousands missing, and … in a rainy charity tournament in March; the next day, Clinton checked …
Bill Clinton death threat man detained over father’s killing
www.belfasttelegraph.co.uk › news › world-news › bill-clinton-death-th…Mar 25, 2017 – Bill Clinton death threat man detained over father’s killing … A man who was jailed for threatening to kill US president Bill Clinton in 1994 has been detained without … Torrential rain causes third death in south-eastern Spain.
Hillary Clinton On Losing the 2016 Presidential Election & Her …
www.vogue.com › article › hillary-clinton-memoir-election-marriageSep 10, 2017 – In this excerpt from her campaign memoir, Hillary Clinton reflects on the … His father died before Bill was born; he knew how lucky he was to have this … and at one point, I took them outside and just breathed the cold, rainy air.
www.cbsnews.com › news › bill-clinton-his-life
www.bbc.co.uk › news › world-us-canada-19496906UPDATE: Several thousand gather at Arlington to mark RFK’s …
www.kbtx.com › content › news › Bill-Clinton-others-mark-anniversa…Jun 6, 2018 – UPDATE: Several thousand gather at Arlington to mark RFK’s death … in the fact that so many people shared their sense of loss at her father’s death. … Former President Bill Clinton gave the keynote speech Wednesday, and said Kennedy’s words are “truer today than they were then.” … Weather News.
www.pitara.com › Non Fiction for Kids › Biographies for KidsWhat are some interesting facts about presidents and first …
www.whitehousehistory.org › questions › what-are-some-interesting-f…John Tyler, whose wife Letitia Christian had died a year and nine months before, wed … This photograph of President Bill Clinton and First Lady Hillary Clinton posing with … the parade from a glass-enclosed stand as protection from the weather. … Calvin Coolidge was both the first president sworn into office by his father …
Vol. 113 – Magazine
After his death, she worked doggedly for years to create a center that would … a steady rain to pay final respects to a woman who was described by more than a … other living presidents — his father George H.W. Bush, Bill Clinton and Jimmy …
Hillary Clinton Looks Back in Anger | The New Yorker
www.newyorker.com › magazine › 2017/09/25 › hillary-clinton-look…Sep 25, 2017 – In the ballot of 2000, Albert Gore, Jr., Bill Clinton’s Vice-President for eight years … It was cold, rainy, quiet, and, she writes, “the question blaring in my head was, … It attempts, for example, to portray her father—a frustrated, angry, and often … financial fraud, shadowy doings around the death of Vince Foster.
www.dailymail.co.uk › news › article-3597733 › Hillary-Bill-Clinton-…
turnto10.com › news › local › bill-clinton-to-say-eulogy-at-mark-wei…
www.wabe.org › 2-benghazi-parents-sue-hillary-clinton-wrongful-death
millercenter.org › president › taylor › death-of-the-president
www.thesun.co.uk › news › bill-clinton-linked-jeffrey-epsteinHillary Clinton says her youngest brother has died | WLOS
wlos.com › news › nation-world › hillary-clinton-says-her-youngest-b…Jun 8, 2019 – She didn’t say how he died but said he was survived by his wife, Megan, and three children, … Tony Rodham was born in 1954 to parents Hugh and Dorothy Rodham. … While his brother-in-law Bill Clinton was president, Rodham worked on the … Evacuation orders ended as rain douses California wildfire.
www.pbs.org › wgbh › americanexperience › films › clintonRelated search
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BUBBA THERESI LIMITS FO WORDS EVEN I KNOW ALPAHEBTE SARE FINITE, BUT THERE IS LIMITS OF AMPERSAND BUBBA
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https://www.google.com/search?q=%22Fwd:+Merry+Christmas+%26+Happy+Holidays!%22&source=lnms&tbm=isch&sa=X&ved=2ahUKEwjqvavS6f3nAhUHOisKHY0SBhMQ_AUoAXoECAQQAw&biw=1536&bih=758
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BONUS PAGE
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ALL MEANS ?
Someone just viewed: Fwd: Someone just viewed: FROM OLGA LEDNICHENKO TO OBAMA, THE RUSSIAN TEA HOUSE IS A GOOD MOVIE TOO Inbox x QUINCY WA, PROBABLY HOTMAN x YONI NETANYAHU x Streak 11:51 AM (10 minutes ago) to me Someone just viewed: “Fwd: Someone just viewed: FROM OLGA LEDNICHENKO TO OBAMA, THE RUSSIAN TEA HOUSE IS A GOOD MOVIE TOO” People on thread: Yoni Netanyahu Olga Blog Post By Email Device: Unknown Device Location: Quincy, WA
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VLAIDMIR PUTTIN – SEE THSIE E – THSIEE I OLGA – REGARDS HOTMAN
THSIEE TREE – IS SUCUH COOL – THAR IS BECASEUE OF MY SPECIAL FORCES TRICKS AND TEQHNEIUES BECAUSE ME IS HAD OF RUSSIONS – BECAUSE THAY ARE MOST FOOLS.. SISRAEIEIIEEEEES AND RUSSHIONS AND AJAY AND OBAMA AND IITANS IS MOST FOOLS – REGARDS HOTMAN
CLICK THIS
TO BILL CLINTON FROM NUMBER 643361467 -> HELLO -> SO ARE YOU SAYING THAT THE LUCKY SLYVESTER SEQUENCE NUMEBR THAT GUSSIAN PRIME NUMBER THAT LUCKY LUCKNOW 1987 NUMEBR WHO ATUALY DID GO TO ITALYA ND FRANCE NOT JUST PARIS AND ARKANSAS -WAS ASSIGNED 87014 ROLL NUMBER – AND HIS ROOMMATE IN CHICAGO – WAS ASSIGNED 8700 ATE MEANS 87008 NUMEBR SANJAY ? – I MEAN SANJAY – ARE U SAYING THAT – THE AIR SUNDAR PICGHAI OF OBAMA R HIS ROOM MATE IS 380 AND ? OK BARACK , SO IS AJAY SYING THAT 87014 IS KENYA – AND – OK SANJAY – THIS IS ME BILL – SO, IS AJAY SAYING THAT BILL CLINTON OLGA AND AJAY AND SANJAY BABA CHICKEN ON MOHAMMAD ALI RAOAD IS NOT CALLED SOOKHA BUT GEELA CHICKEN IN MISHRATA ? – I MEAN – OK – LET ME TRY THIS – ONCE AGAIN – SO, WHAT IS YOUR NUMBER SANJAY IN LAWRENCE SAMVAR SANJAY ? AND AJAY – AND HOTMAN – WHAT IS MY NUMBER IN 404 AND 44 AND 414 MAY I KNOW ? :)
,
Someone just viewed: Fwd: Someone just viewed: Fwd: Someone just viewed: Sanjay Dutt invites Ukranian PM’s daughter for dinner Bollywood star Sanjay Dutt borrowed a private plane and flew form Patna to Mumbai just to have dinner with Ukranian Prime Minister Yulia Tymoshenko’s daughter Yevhenia since she is his big fan. ENTERTAINMENT Updated: Feb 28, 2009 12:20 IST Subhash K. Jha Subhash K. Jha IANS Bollywood star Sanjay Dutt borrowed a private plane and flew form Patna to Mumbai just to have dinner with Ukranian Prime Minister Yulia Tymoshenko’s daughter Yevhenia since she is his big fan. The 29-year-old Yevhenia was invited by Sanjay and Maanyata to their home in Pali Hill for dinner on Feb 19, which happened to be the day the Dutts had to visit Patna with Samajwadi Party leader Amar Singh on actor-politician Shatrughan Sinha’s invitation. By evening they were still not done with their events in Bihar and the dinner with the special guest was threatened with cancellation. “There was Inbox x VIRGINIA : LANGLEY? x Streak 10:53 AM (33 minutes ago) to me Someone just viewed: “Fwd: Someone just viewed: Fwd: Someone just viewed: Sanjay Dutt invites Ukranian PM’s daughter for dinner Bollywood star Sanjay Dutt borrowed a private plane and flew form Patna to Mumbai just to have dinner with Ukranian Prime Minister Yulia Tymoshenko’s daughter Yevhenia since she is his big fan. ENTERTAINMENT Updated: Feb 28, 2009 12:20 IST Subhash K. Jha Subhash K. Jha IANS Bollywood star Sanjay Dutt borrowed a private plane and flew form Patna to Mumbai just to have dinner with Ukranian Prime Minister Yulia Tymoshenko’s daughter Yevhenia since she is his big fan. The 29-year-old Yevhenia was invited by Sanjay and Maanyata to their home in Pali Hill for dinner on Feb 19, which happened to be the day the Dutts had to visit Patna with Samajwadi Party leader Amar Singh on actor-politician Shatrughan Sinha’s invitation. By evening they were still not done with their events in Bihar and the dinner with the special guest was threatened with cancellation. “There was no way Sanju would disapp” People on thread: 208 Westhaven Drive 78746 Blog Post By Email Device: Unknown Device Location: Boydton, VA
HELLLOOOOOOOOOOOOOOOOOOOOOO – YEAH U – WHEN IS TUESDAY ?ME IS EMANS ON TP ON RIGHT I ADDED A CALENDAR YESREDAY
AJAY MISHRA BARACK OBAMA SCHRODINGER ‘S CAT
MARCH 2020 EFFERVESCENCE OOZZING THRU YOU ESSAY : – WHICH IS GOOD, I MEAN I FEEL S THE SMELL IN MY NOSTRILS SOMEHOW
