{"resolved":true,"match_type":"uuid","source_type":"original_book","book_id":"ea3b7167-0435-4a94-84b9-80a5359326a4","title":"递归与偶然","author":null,"edition":null,"publisher":null,"year":null,"isbn":"9787576004694","requested_by":"printed_page","book":{"source_type":"original_book","book_id":"ea3b7167-0435-4a94-84b9-80a5359326a4","id":"ea3b7167-0435-4a94-84b9-80a5359326a4","uuid":"ea3b7167-0435-4a94-84b9-80a5359326a4","title":"递归与偶然","normalized_title":"递归与偶然","author":null,"edition":null,"publisher":null,"year":null,"isbn":"9787576004694","isbns":["9787576004694"],"parsed":true,"ai_readable":true,"document_type":"book","page_count":401,"navigation_url":"https://workshop.closeclaw.top/ai/book/ea3b7167-0435-4a94-84b9-80a5359326a4","title_url":"https://workshop.closeclaw.top/ai/title/%E9%80%92%E5%BD%92%E4%B8%8E%E5%81%B6%E7%84%B6","fixed_page_url_template":"https://workshop.closeclaw.top/ai/book/ea3b7167-0435-4a94-84b9-80a5359326a4/page/{printed_page}","path_api_url_template":"https://workshop.closeclaw.top/api/ai/book/ea3b7167-0435-4a94-84b9-80a5359326a4/page/{printed_page}","read_url":"https://workshop.closeclaw.top/api/ai/read/ea3b7167-0435-4a94-84b9-80a5359326a4","page_url":"https://workshop.closeclaw.top/api/ai/books/ea3b7167-0435-4a94-84b9-80a5359326a4/page","search_url":"https://workshop.closeclaw.top/api/ai/books/ea3b7167-0435-4a94-84b9-80a5359326a4/search"},"page":{"printed_page":139,"pdf_page":151,"physical_index":151,"physical_index_base":0,"printed_page_status":"recognized","page_sha256":"3a59a19d88e80dc5d515ca6ecc14726b279e9ecc41222954b5dd43a9cc530cb1"},"chapter":"第二章逻辑与偶然","content":"随后，哥德尔在他1931年题为“论数学原理中的形式不可判定命题及相关系统 I”(On Formally Undecidable Proposi-tions of Principia Mathematica and Related Systems I) 的文章中，定义了递归函数，又在随后1934年的论文《论形式数学的不可判定命题》（On Undecidable Propositions of FormalMathematics)中定义了一般递归函数。与符号逻辑的形式化相反，哥德尔1931年的论文用数字把形式系统算术化了(如今被称为哥德尔配数[Gödel numbering])，这样不同公理的关系就可以用数字表示；正如我们在表2.1中见到的：每个符号，包括括号，都被赋予了一个自然数。\n\n这样，哥德尔就可以把证明从逻辑推理变成由递归函数组成的运算。在他1931年的论文中，哥德尔这样定义递归函数：\n\n第二章 逻辑与偶然\n\n139\n\n① 罗德·亚当斯(Rod Adams)，《递归函数和可计算性的早期历史》(An Early History of Recursive Functions and Computability),波上顿：Docent,2011年，第19—32页。","previous_page":138,"next_page":140}