何亭算术的解读——用集合符号的语言分析

Martin Stein
{"title":"何亭算术的解读——用集合符号的语言分析","authors":"Martin Stein","doi":"10.1016/0003-4843(80)90018-2","DOIUrl":null,"url":null,"abstract":"<div><p>Well-known interpretations of Heyting's arithmetic of all finite types are the Diller-Nahm λ-interpretation [1] and Kreisel's modified realizability, subsequently called mr-interpretation [4]. For both interpretations one can define hybrids λ q resp. mq.</p><p>In Section 4 a chain of interpretations—called <strong>M</strong>-interpretations—is defined (it was introduced in [6], filling the “gap” between λ-interpretation and mr-interpretation.</p><p>In this paper it is shwon that it is possible to prove <em>in one stroke</em> the soundness resp. characterization theorems for <em>all</em> interpretations of HA<sub>ω 〈〉</sub> (Heyting's arithmetic of all finite types with functionals for coding finite sequences). This is done by means of interpretations of systems which contain set-symbols. For these so called <em>M</em>-interpretations, soundness-resp. characterization theorems can be proved simultaneously (Theorem 2.51. Special translations of set symbols and of the formula (<em>λωϵW</em>)<em>A</em> — this means, special decisions about the size of the set <em>W</em>; see Sections 3 and 4 — yield the corresponding results for all interpretations of HA<sub>ω〈〉</sub> mentioned.</p><p>The terminology of set theoretical language — we consider an extension of HA<sub>ω〈〉</sub> by an extensively weak fragment only, which leads to a conservative extension of HA<sub>ω〈〉</sub> — is of good use for studying realizing terms of different interpretations: if HA<sub><em>ω</em></sub>&lt;&gt;⊢<em>A</em>, <em>A</em><sup><em>M</em></sup>∃<em>υ</em> ∀<em>w</em> <em>A</em><sub><em>M</em></sub>, and ⊢<em>A</em><sub><em>M</em></sub>[<em>t</em><sub><em>M</em></sub>, <em>w</em>] by soundness theorem for <em>M</em>-interpretations, there exists a simple operation which maps <span><math><mtext>v</mtext><mtext>̄</mtext><mtext> </mtext><mtext>to</mtext><mtext> </mtext><mtext>t</mtext><mtext>̄</mtext><msub><mi></mi><mn><mtext>mr</mtext></mn></msub></math></span>, the realizing term for modified realizability. For interpretations of Heyting's arithmetic — λ-interpretation. <strong>M</strong>-interpretations and mr-interpretation — this leads to the following stability result for existence theorems: if <span><math><mtext>∃λ A </mtext><mtext>and</mtext><mtext> t</mtext><msub><mi></mi><mn>^</mn></msub><mtext> </mtext><mtext>resp.</mtext><mtext> t</mtext><msub><mi></mi><mn><mtext>M</mtext><mtext>M</mtext></mn></msub></math></span> is the term computed by λ-interpretation. resp. <strong>M</strong>-interpretation, with <span><math><mtext>∃A[t</mtext><msub><mi></mi><mn><mtext>M</mtext></mn></msub><mtext>]</mtext></math></span>, then — using extensional equality and ω-rule for equations — we can prove that <span><math><mtext>t</mtext><msub><mi></mi><mn>λ</mn></msub><mtext> = t</mtext><msub><mi></mi><mn><mtext>M</mtext></mn></msub><mtext> = t</mtext><msub><mi></mi><mn><mtext>mr</mtext></mn></msub></math></span> (Section 5).</p></div>","PeriodicalId":100093,"journal":{"name":"Annals of Mathematical Logic","volume":"19 1","pages":"Pages 1-31"},"PeriodicalIF":0.0000,"publicationDate":"1980-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/0003-4843(80)90018-2","citationCount":"11","resultStr":"{\"title\":\"Interpretations of Heyting's arithmetic—An analysis by means of a language with set symbols\",\"authors\":\"Martin Stein\",\"doi\":\"10.1016/0003-4843(80)90018-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Well-known interpretations of Heyting's arithmetic of all finite types are the Diller-Nahm λ-interpretation [1] and Kreisel's modified realizability, subsequently called mr-interpretation [4]. For both interpretations one can define hybrids λ q resp. mq.</p><p>In Section 4 a chain of interpretations—called <strong>M</strong>-interpretations—is defined (it was introduced in [6], filling the “gap” between λ-interpretation and mr-interpretation.</p><p>In this paper it is shwon that it is possible to prove <em>in one stroke</em> the soundness resp. characterization theorems for <em>all</em> interpretations of HA<sub>ω 〈〉</sub> (Heyting's arithmetic of all finite types with functionals for coding finite sequences). This is done by means of interpretations of systems which contain set-symbols. For these so called <em>M</em>-interpretations, soundness-resp. characterization theorems can be proved simultaneously (Theorem 2.51. Special translations of set symbols and of the formula (<em>λωϵW</em>)<em>A</em> — this means, special decisions about the size of the set <em>W</em>; see Sections 3 and 4 — yield the corresponding results for all interpretations of HA<sub>ω〈〉</sub> mentioned.</p><p>The terminology of set theoretical language — we consider an extension of HA<sub>ω〈〉</sub> by an extensively weak fragment only, which leads to a conservative extension of HA<sub>ω〈〉</sub> — is of good use for studying realizing terms of different interpretations: if HA<sub><em>ω</em></sub>&lt;&gt;⊢<em>A</em>, <em>A</em><sup><em>M</em></sup>∃<em>υ</em> ∀<em>w</em> <em>A</em><sub><em>M</em></sub>, and ⊢<em>A</em><sub><em>M</em></sub>[<em>t</em><sub><em>M</em></sub>, <em>w</em>] by soundness theorem for <em>M</em>-interpretations, there exists a simple operation which maps <span><math><mtext>v</mtext><mtext>̄</mtext><mtext> </mtext><mtext>to</mtext><mtext> </mtext><mtext>t</mtext><mtext>̄</mtext><msub><mi></mi><mn><mtext>mr</mtext></mn></msub></math></span>, the realizing term for modified realizability. For interpretations of Heyting's arithmetic — λ-interpretation. <strong>M</strong>-interpretations and mr-interpretation — this leads to the following stability result for existence theorems: if <span><math><mtext>∃λ A </mtext><mtext>and</mtext><mtext> t</mtext><msub><mi></mi><mn>^</mn></msub><mtext> </mtext><mtext>resp.</mtext><mtext> t</mtext><msub><mi></mi><mn><mtext>M</mtext><mtext>M</mtext></mn></msub></math></span> is the term computed by λ-interpretation. resp. <strong>M</strong>-interpretation, with <span><math><mtext>∃A[t</mtext><msub><mi></mi><mn><mtext>M</mtext></mn></msub><mtext>]</mtext></math></span>, then — using extensional equality and ω-rule for equations — we can prove that <span><math><mtext>t</mtext><msub><mi></mi><mn>λ</mn></msub><mtext> = t</mtext><msub><mi></mi><mn><mtext>M</mtext></mn></msub><mtext> = t</mtext><msub><mi></mi><mn><mtext>mr</mtext></mn></msub></math></span> (Section 5).</p></div>\",\"PeriodicalId\":100093,\"journal\":{\"name\":\"Annals of Mathematical Logic\",\"volume\":\"19 1\",\"pages\":\"Pages 1-31\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1980-11-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/0003-4843(80)90018-2\",\"citationCount\":\"11\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annals of Mathematical Logic\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/0003484380900182\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Mathematical Logic","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/0003484380900182","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 11

摘要

所有有限类型的Heyting算法的著名解释是Diller-Nahm λ-解释[1]和Kreisel的修正可实现性,后来被称为mr-解释[4]。对于这两种解释,我们都可以定义混合λ q resp。mq。在第4节中,定义了一个解释链——称为m -解释(于2010年引入),填补了λ-解释和mr-解释之间的“空白”。在本文中,证明了可以一次性地证明其完备性。HAω < >的所有解释的表征定理(编码有限序列的泛函的所有有限类型的Heyting算法)。这是通过对包含集合符号的系统的解释来实现的。对于这些所谓的“m -解释”,即“健全-回应”。表征定理可以同时被证明(定理2.51)。集合符号和公式(λωϵW)A的特殊转换-这意味着,关于集合W大小的特殊决定;参见第3节和第4节,对上述所有的HAω < >的解释都得到相应的结果。集合理论语言的术语——我们只考虑广义弱片段对HAω < >的扩展,它会导致HAω < >的保守扩展——对于研究不同解释的实现项很有用:如果HAω<> & a, AM,并且通过m -解释的稳健性定理,存在一个简单的操作,可以将v´映射到t´mr,即修改可实现性的实现项。对于何亭的算术解释——λ解释。m -解释和mr-解释-这导致了存在定理的如下稳定性结果:如果∃λ A和t^ resp。tMM是λ解释计算的项。分别地。使用∃A[tM]进行m解释,然后-使用方程的扩展等式和ω规则-我们可以证明tλ = tM = tmr(第5节)。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Interpretations of Heyting's arithmetic—An analysis by means of a language with set symbols

Well-known interpretations of Heyting's arithmetic of all finite types are the Diller-Nahm λ-interpretation [1] and Kreisel's modified realizability, subsequently called mr-interpretation [4]. For both interpretations one can define hybrids λ q resp. mq.

In Section 4 a chain of interpretations—called M-interpretations—is defined (it was introduced in [6], filling the “gap” between λ-interpretation and mr-interpretation.

In this paper it is shwon that it is possible to prove in one stroke the soundness resp. characterization theorems for all interpretations of HAω 〈〉 (Heyting's arithmetic of all finite types with functionals for coding finite sequences). This is done by means of interpretations of systems which contain set-symbols. For these so called M-interpretations, soundness-resp. characterization theorems can be proved simultaneously (Theorem 2.51. Special translations of set symbols and of the formula (λωϵW)A — this means, special decisions about the size of the set W; see Sections 3 and 4 — yield the corresponding results for all interpretations of HAω〈〉 mentioned.

The terminology of set theoretical language — we consider an extension of HAω〈〉 by an extensively weak fragment only, which leads to a conservative extension of HAω〈〉 — is of good use for studying realizing terms of different interpretations: if HAω<>⊢A, AM∃υw AM, and ⊢AM[tM, w] by soundness theorem for M-interpretations, there exists a simple operation which maps v̄ to t̄mr, the realizing term for modified realizability. For interpretations of Heyting's arithmetic — λ-interpretation. M-interpretations and mr-interpretation — this leads to the following stability result for existence theorems: if ∃λ A and t^ resp. tMM is the term computed by λ-interpretation. resp. M-interpretation, with ∃A[tM], then — using extensional equality and ω-rule for equations — we can prove that tλ = tM = tmr (Section 5).

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信