# Bernie Sanders

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. ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ajay — we know we keep emailing you and asking you for$5.

Here’s why: this week, Bernie Sanders’ campaign reported that they raised a staggering $46 million in February. And while that might seem like bad news, we actually have a real chance to compete against this latest haul. Look, given our surge in small-dollar fundraising following our win in South Carolina, we are in a better position than ever before to compete head-to-head against the Sanders campaign’s money. But, we have to keep closing the money gap before polls close tonight, and that’s where you come in, ajay. If you pitch in$5 before the polls close tonight, we’ll be in a terrific position to win tonight in Super Tuesday states and beyond »

# Tell Kamala which one of these issues is most important to you >>

### ᐧ

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>

 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.

# 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.

 Zestimate: $404,958 1909 ‌Payne ‌Ave, ‌Austin, ‌TX 2 bd, 1 ba, 720 sqft Off Market Check out your news feed  6309 Burns St # C-201 listed for rent for$1,299/mo 1mi away • 40m ago
 6309 Burns St # D-307 listed for rent for $1,369/mo 1mi away • 41m ago  2211 W North Loop Blvd APT 223 listed for rent for$999/mo 1mi away • 11h ago
 Median Neighborhood Zestimate +0.8% past 30 days 476,800 Just Listed  3 bd | 2 ba | 1,204 sqft Austin  4 bd | 1 ba | 1,430 sqft Austin  4 bd | 3 ba | 2,300 sqft Austin Sold Nearby  3 bd | 2 ba | 1,102 sqft Austin  4 bd | 3 ba | 1,892 sqft Austin  Zillow, ‌Inc. 1301 ‌Second ‌Avenue, ‌Floor ‌31 Seattle, ‌WA ‌98101 © ‌2006‌-‌2020 Privacy policy | Unsubscribe from this email | Update your preferences # 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 ? 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OH HERE IS JAKARTA ? https://www.google.com/search?q=BHULEKH+Uttar+Pradesh+%E0%A4%96%E0%A4%BE%E0%A4%A4%E0%A4%BE+%E0%A4%B5%E0%A4%BF%E0%A4%B5%E0%A4%B0%E0%A4%A3+(%E0%A4%85%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A4%BE%E0%A4%A3%E0%A4%BF%E0%A4%A4+%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%A4%E0%A4%BF)+%E0%A4%97%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%AE+%E0%A4%95%E0%A4%BE+%E0%A4%A8%E0%A4%BE%E0%A4%AE+:+%E0%A4%AC%E0%A4%95%E0%A5%8D%E0%A4%95%E0%A4%BE%E0%A4%B8+%E0%A4%AA%E0%A4%B0%E0%A4%97%E0%A4%A8%E0%A4%BE+:+(%E0%A4%AE%E0%A5%8B%E0%A4%B9%E0%A4%A8%E0%A4%B2%E0%A4%BE%E0%A4%B2%E0%A4%97%E0%A4%82%E0%A4%9C)+%E0%A4%A4%E0%A4%B9%E0%A4%B8%E0%A5%80%E0%A4%B2+:+%E0%A4%AE%E0%A5%8B%E0%A4%B9%E0%A4%A8%E0%A4%B2%E0%A4%BE%E0%A4%B2%E0%A4%97%E0%A4%82%E0%A4%9C+%E0%A4%9C%E0%A4%A8%E0%A4%AA%E0%A4%A6+:+%E0%A4%B2%E0%A4%96%E0%A4%A8%E0%A4%8A+%E0%A4%AB%E0%A4%B8%E0%A4%B2%E0%A5%80+%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%B7+:+1423-1428+%E0%A4%AD%E0%A4%BE%E0%A4%97+:+1+%E0%A4%96%E0%A4%BE%E0%A4%A4%E0%A4%BE+%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE+:+00074+%E0%A4%96%E0%A4%BE%E0%A4%A4%E0%A5%87%E0%A4%A6%E0%A4%BE%E0%A4%B0+%E0%A4%95%E0%A4%BE+%E0%A4%A8%E0%A4%BE%E0%A4%AE+/+%E0%A4%AA%E0%A4%BF%E0%A4%A4%E0%A4%BE+%E0%A4%AA%E0%A4%A4%E0%A4%BF+%E0%A4%B8%E0%A4%82%E0%A4%B0%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%95+%E0%A4%95%E0%A4%BE+%E0%A4%A8%E0%A4%BE%E0%A4%AE+/+%E0%A4%A8%E0%A4%BF%E0%A4%B5%E0%A4%BE%E0%A4%B8+%E0%A4%B8%E0%A5%8D%E0%A4%A5%E0%A4%BE%E0%A4%A8+%E0%A4%96%E0%A4%B8%E0%A4%B0%E0%A4%BE+%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE+%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AB%E0%A4%B2+(%E0%A4%B9%E0%A5%87.)+%E0%A4%86%E0%A4%A6%E0%A5%87%E0%A4%B6+%E0%A4%9F%E0%A4%BF%E0%A4%AA%E0%A5%8D%E0%A4%AA%E0%A4%A3%E0%A5%80+%E0%A4%B6%E0%A5%8D%E0%A4%B0%E0%A5%87%E0%A4%A3%E0%A5%80+:+1-%E0%A4%95+/+%E0%A4%AD%E0%A5%82%E0%A4%AE%E0%A4%BF+%E0%A4%9C%E0%A5%8B+%E0%A4%B8%E0%A4%82%E0%A4%95%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A4%A3%E0%A5%80%E0%A4%AF+%E0%A4%AD%E0%A5%82%E0%A4%AE%E0%A4%BF%E0%A4%A7%E0%A4%B0%E0%A5%8B%E0%A4%82+%E0%A4%95%E0%A5%87%E0%A4%85%E0%A4%A7%E0%A4%BF%E0%A4%95%E0%A4%BE%E0%A4%B0+%E0%A4%AE%E0%A5%87%E0%A4%82+%E0%A4%B9%E0%A5%8B%E0%A5%A4+%E0%A4%93%E0%A4%AE+%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%95%E0%A4%BE%E0%A4%B6+%E0%A4%B6%E0%A5%8D%E0%A4%B0%E0%A5%80%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5+/+%E0%A4%B8%E0%A5%8D%E0%A4%B5.%E0%A4%8F%E0%A4%AB.%E0%A4%AA%E0%A5%80.%E0%A4%B6%E0%A5%8D%E0%A4%B0%E0%A5%80%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5+/+%E0%A4%AE.%E0%A4%A8.%E0%A4%8F.706+%E0%A4%B8%E0%A5%87.+%E0%A4%B8%E0%A5%80.%E0%A4%AE%E0%A4%B9%E0%A4%BE%E0%A4%A8%E0%A4%97%E0%A4%B0+%E0%A4%B2%E0%A4%96%E0%A4%A8%E0%A4%8A+%E0%A4%B6%E0%A5%8D%E0%A4%B0%E0%A5%80%E0%A4%AE%E0%A4%A4%E0%A5%80+%E0%A4%B0%E0%A5%87%E0%A4%A8%E0%A5%82&source=lnms&tbm=isch&sa=X&ved=2ahUKEwigqd643P7nAhVFT30KHSyvBx4Q_AUoBHoECAYQBg&biw=1536&bih=758 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 � To: ## 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 © 2011-2020 Streak 160 Pine Street, San Francisco, CA 94111 # Someone just viewed: HOTMAN WHAR THIESE? 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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}. ## CLICK TO HEAR THE SOUN DOF PAIN AND OF RAIN CAELD OLGA LEDNICHENKO YONI NETANAYHU PIANO AND PEANO’S ITALIN AXIOMS REFERENCES: The first axiom states that the constant 0 is a natural number: 1. 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] 1. For every natural number xx = x. That is, equality is reflexive. 2. For all natural numbers x and y, if x = y, then y = x. That is, equality is symmetric. 3. For all natural numbers xy and z, if x = y and y = z, then x = z. That is, equality is transitive. 4. 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. 1. For every natural number nS(n) is a natural number. That is, the natural numbers are closed under S. 2. For all natural numbers m and nm = n if and only if S(m) = S(n). That is, S is an injection. 3. For every natural number nS(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. 1. If K is a set such that: • 0 is in K, and • for every natural number nn 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: 1. If φ is a unary predicate such that: • φ(0) is true, and • for every natural number nφ(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}}} 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 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 The usual total order relation ≤ on natural numbers can be defined as follows, assuming 0 is a natural number: For all ab ∈ Na ≤ 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} }, 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 nk ∈ N, if k ≤ n implies that φ(k) is true, then φ(S(n)) is true, then for every n ∈ Nφ(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 ≤ nk ∉ X. Then S(n) ∉ X, for otherwise it would be the least element of X. Thus, by the strong induction principle, for every n ∈ Nn ∉ 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,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(xy1, …, 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}}} 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] 1. {\displaystyle \forall x,y,z\ ((x+y)+z=x+(y+z))}, i.e., addition is associative. 2. {\displaystyle \forall x,y\ (x+y=y+x)}, i.e., addition is commutative. 3. {\displaystyle \forall x,y,z\ ((x\cdot y)\cdot z=x\cdot (y\cdot z))}, i.e., multiplication is associative. 4. {\displaystyle \forall x,y\ (x\cdot y=y\cdot x)}, i.e., multiplication is commutative. 5. {\displaystyle \forall x,y,z\ (x\cdot (y+z)=(x\cdot y)+(x\cdot z))}, i.e., multiplication distributes over addition. 6. {\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]). 7. {\displaystyle \forall x\ (x\cdot 1=x)}, i.e., one is an identity for multiplication. 8. {\displaystyle \forall x,y,z\ (x<y\land y<z\Rightarrow x<z)}, i.e., the ‘<‘ operator is transitive. 9. {\displaystyle \forall x\ (\neg (x<x))}, i.e., the ‘<‘ operator is irreflexive. 10. {\displaystyle \forall x,y\ (x<y\lor x=y\lor y<x)}, i.e., the ordering satisfies trichotomy. 11. {\displaystyle \forall x,y,z\ (x<y\Rightarrow x+z<y+z)}, i.e. the ordering is preserved under addition of the same element. 12. {\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. 13. {\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. 14. {\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. 15. {\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} satisfying this theory has an initial segment (ordered by {\displaystyle \leq }) isomorphic to {\displaystyle \mathbf {N} }. Elements in that segment are called standard elements, while other elements are called nonstandard elements. ## Models model of the Peano axioms is a triple (N, 0, S), where N is a (necessarily infinite) set, 0 ∈ N and SN → N satisfies the axioms above. Dedekind proved in his 1888 book, The Nature and Meaning of Numbers (GermanWas 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, 0ASA) and (NB, 0BSB) 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 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 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, 0XSX) where X is an object of C, and 0X : 1C → X and SX : X → X are C-morphisms. • A morphism φ : (X, 0XSX) → (Y, 0YSY) 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, 0XSX) is any other object, then the unique map u : (N, 0, S) → (X, 0XSX) 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 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 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. Then there is a c in M that is greater than every element of C such that {\displaystyle M\vDash \phi (c,{\bar {a}}).} ## 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}} 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 ## Notes 1. ^ {\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}. # Someone just viewed: FROM PRESIDENT WILLIAM JEFFERSON CLINTON – WHO WON THE MAXIMUM NUMBER OF COUNTIES UP AND DWON THE MISSISPI RIVER WITH OR WITHOT LOISUINA PURCHASE PROGRAMA ND DIXIE CHICSKS OR MASON DIXIES AND FROM CHELSEA AND OLGA CLINTON -> HELLO VERYONE – HOW IS 2 O 0 6 SOUND TO U IN EAR AND IN IN YEAR FORMAT ? AND HILALRY RODHMAN CLINTONREFSUES TO TALK TO LOW LEVEL WHITES NEGROS ABORGINALS OR BROWN OR JEWS EVEN – THANK U FOR YOUR ABIITY AND FOR YOUR COMPREHSNION AND FOR YOUR GEMRORTY AND ALBEGRA AND AIRTHMETCI AND ORTRY ANDFOR YOUR PROSE AND FOR YOUR EXCELLENCE IN LISTEINGA ND FOR YOUR ORATORY AND SPECHS AND FOR ? FOR YOUR DINENR TABLE CONVERSSTIONS – :) Inbox x BIG SHOT FROM CHICAGO x Streak 7:00 PM (21 minutes ago) to me Someone just viewed: “FROM PRESIDENT WILLIAM JEFFERSON CLINTON – WHO WON THE MAXIMUM NUMBER OF COUNTIES UP AND DWON THE MISSISPI RIVER WITH OR WITHOT LOISUINA PURCHASE PROGRAMA ND DIXIE CHICSKS OR MASON DIXIES AND FROM CHELSEA AND OLGA CLINTON -> HELLO VERYONE – HOW IS 2 O 0 6 SOUND TO U IN EAR AND IN IN YEAR FORMAT ? AND HILALRY RODHMAN CLINTONREFSUES TO TALK TO LOW LEVEL WHITES NEGROS ABORGINALS OR BROWN OR JEWS EVEN – THANK U FOR YOUR ABIITY AND FOR YOUR COMPREHSNION AND FOR YOUR GEMRORTY AND ALBEGRA AND AIRTHMETCI AND ORTRY ANDFOR YOUR PROSE AND FOR YOUR EXCELLENCE IN LISTEINGA ND FOR YOUR ORATORY AND SPECHS AND FOR ? FOR YOUR DINENR TABLE CONVERSSTIONS – :)” People on thread: 208 Westhaven Drive 78746 Blog Post By Email Device: PC Location: Chicago, IL # [A] ## https://www.youtube.com/watch?v=-Dx_STgFLLo # [B] # SEA ## SSSSEEEEEEEEE SEA CCCCCCCC SEEEEE # 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] # ]2] # CLICK # 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. ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ajay, There’s a lot on the line today. There are more than one thousand delegates being awarded across the country, and we need to win as many as possible to seal the deal on making Joe the Democratic nominee. Can we count on you for5 today?

If you’ve saved payment information with ActBlue Express, your donation will go through immediately:

Let us break it down, ajay:

IF we can bring in a surge of donations today, we’ll be in great shape to win the votes we need in strategic districts across the country.

IF we can win in these districts, we’ll earn the delegate totals we need to secure the Democratic nomination and send Joe Biden to defeat Donald Trump.

BUT we can only make all of this happen if all our supporters rally around our campaign today and make it happen!

Your donation today can make a huge difference, ajay. Will you chip in \$5 to make sure Joe Biden is our Democratic nominee come July?

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Biden for President

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ajay, thank you so much for supporting Joe Biden’s Presidential campaign.

This isn’t the time to be complacent. If you are ready to fight for the soul of this nation, you can start by donating to elect Joe Biden by clicking the button below.

We know we send a lot of emails, and we are sorry about that. The reason? We are relying on grassroots supporters like you (we’re serious!).

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REFERENCES

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Ереван, Армения

# 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