新弗雷格算术的一致性和解释性
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摘要
如果真要探究弗雷格算术系统中的非一致性根源,那么,不仅需要找到算术片段一致性的模型,而且也要找出算术片段的可解释性。因此,本文的主线索有两条：一条是证明弗雷格算术一阶片段和二阶片段的一致性；另一条是证明诸片段的可解释性。但是,两者并不是泾渭分明的,而是经常交叉在一起的。关于PA2,HP2,BLC2子系统和一致性和解释性,我们取得的主要成果有：Frege本人实际上阐明PA2(?)HP2；Heck和Linnebo阐明Π11-CA0(?)Π11-HP0；Boolos阐明Π11-HP0(?)Π11-CA0；同时得到Π11-CA0≡Π11-CA0；Heck｜阐述ABL0(?)Q；Ganea和Visser｜阐述Q(?)ABL0；同时得到ABL0≡Q；Burgess｜阐述AHP0(?)Q；Ferreira和Wehmeier阐述△11-BL0是一致的；对此两人证明的微小改进表明∑11-LB0是一致的；对整个证明的观察会表明∑11-LB0(?)Π11-CA0。Walsh用超算术理论证明∑11-LB0+(?)Σ11-AC0；根据递归饱和域,最终证明ACA(?)∑11-PH0.在对弗雷格算术片段的一致性证明过程中,Burgess使用了有穷论的证明论方法；而Heck等人采用了无穷论的模型论方法。而在采用模型论证明的过程中,Heck采用了变元-约束项-形成算子,而Wehmeier采用了△11-概括公式。由此,近30年来,弗雷格数学哲学研究主要采用了如下三种方法：超算术理论,计算模型理论和逆数学理论。弗雷格算术是由二阶逻辑和休谟原则构成的：而弗雷格定理阐述的是,二阶皮亚诺算术的所有公理都是可以从弗雷格算术和FD中推导出来的。引出弗雷格算术的意义在于,近一个世纪以来,非形式算术几乎无一例外地都被赋以某种Peano-Dedekind式公理化形式。这些公理化形式把自然数认作有穷序数,通过它们在ω-序列中所处的位置而得以个体化。然后,弗雷格定理表明,一个可替代的和概念上完全不同的算术公理化形式也是可能的,而基本的思路就是自然数是有穷基数,通过对概念取数的方式即概念数的基数性而得以个体化。
If we want to probe the reason why there occurs the inconsistency in Frege's Grundge-setze, then we should not only find out the model in which we can avoid this phenomenon of consistency, but also the interpretability relating to this model of consistent arithmetic. Out of this train of thought, My work can be divided into these two parts:The first one is to prove the consistency in the first-order portion and second-order fragment of arithmetic, The other one is to prove the interpretability strength of these fragment of arithmetic. The major previous results on the interpretability strength of the subsystems of PA2, HP2, BLV2 can be described as follows. Frege once showed that PA2(?)HP2; Heck and Linnebo noted that Frege's proofs in fact show thatΠ11-CA0(?)Π11-HP0; Further, Boolos showed that the converse holds, so that one hasΠ11-CA0≡Π11-CA0; Heck then showed that ABL0(?)Q, Ganea and Visser independently showed that the converse holds, so that ABL0≡Q.Burgess showed that AHP0(?)Q; Ferreira and Wehmeier showed thatΔ11-BL0 is consistent. Recently, Walsh showed thatΣ11-LB0+(?)Σ11-AC0 and that ACA0(?)Σ11-PH0. Over the past 30 years, philosophers have studied the systems closely related to subsystems of second-order arithmetic. These constructions use tools from computability theory, including:hyperarithmetic theory, computable model theory and reverse mathematics. The reason why we introduced Frege Arithmetic and Frege's The-orem is that, more than one century from now, informal arithmetic has almost without exception been given some Peano style axiomatization. These axiomatizations regard the natural numbers as finite ordinals, individuated by their position in anω-sequence. Frege's Theorem shows that an alternative and conceptually completely different ax-iomatization of arithmetic is possible, based on the idea that the natural numbers are finite cardinals, individuated by the cardinalities of the concepts whose numbers they are.
If we want to probe the reason why there occurs the inconsistency in Frege's Grundge-setze, then we should not only find out the model in which we can avoid this phenomenon of consistency, but also the interpretability relating to this model of consistent arithmetic. Out of this train of thought, My work can be divided into these two parts:The first one is to prove the consistency in the first-order portion and second-order fragment of arithmetic, The other one is to prove the interpretability strength of these fragment of arithmetic. The major previous results on the interpretability strength of the subsystems of PA2, HP2, BLV2 can be described as follows. Frege once showed that PA2(?)HP2; Heck and Linnebo noted that Frege's proofs in fact show thatΠ11-CA0(?)Π11-HP0; Further, Boolos showed that the converse holds, so that one hasΠ11-CA0≡Π11-CA0; Heck then showed that ABL0(?)Q, Ganea and Visser independently showed that the converse holds, so that ABL0≡Q.Burgess showed that AHP0(?)Q; Ferreira and Wehmeier showed thatΔ11-BL0 is consistent. Recently, Walsh showed thatΣ11-LB0+(?)Σ11-AC0 and that ACA0(?)Σ11-PH0. Over the past 30 years, philosophers have studied the systems closely related to subsystems of second-order arithmetic. These constructions use tools from computability theory, including:hyperarithmetic theory, computable model theory and reverse mathematics. The reason why we introduced Frege Arithmetic and Frege's The-orem is that, more than one century from now, informal arithmetic has almost without exception been given some Peano style axiomatization. These axiomatizations regard the natural numbers as finite ordinals, individuated by their position in anω-sequence. Frege's Theorem shows that an alternative and conceptually completely different ax-iomatization of arithmetic is possible, based on the idea that the natural numbers are finite cardinals, individuated by the cardinalities of the concepts whose numbers they are.
引文
1具体请参考[Slug80]第4章：寻找逻辑对象；第五章：意义分析。Sluga在这两章中分别对《算术基础》和《算术基本定律》进行了详细地探讨。他从弗雷格的文本出发,对Dummett的相关核心观点进行了批评。上述语境原则地位转换的议题就是Sluga所重点批判的。
2 经过12天的激烈讨论,在2006年8月24日于捷克首都布拉格举行的第26届国际天文学联合会大会上位居太阳系九大行星末席70多年的冥王星,最终以237票赞成,157票反对,17票弃权的表决结果,被逐出太阳系九大行星之列。至此,传统意义上的太阳系九大行星变为水星、金星、地球、火星、木星、土星、天王星和海王星八大行星。在《弗雷格：数学哲学》出版的时候,这一决议尚未发生。
3 普特南曾经用置换论证来反对形而上学实在论,而戴维森用它来展现不同于真理概念,指称概念和意义理论是纯内在的。
4 根据Dummett, Bartlett这篇没有被足够重视的文献是关于弗雷格的本体论和语义学的,这是Bartlett本人的博士论文,可参考：James M. Bartlett, Funktion und Gegenstand, Munich,1961.
5 g(ξ)是如下公式的缩写：对每个(?),如果ξ是(?)的值域,那么不能有(?)(ξ)；
6 Dummett提出另外一种建议,替代有穷子集和补有穷子集,我们做出如下表述：包括进外延中的函数,它的值区别于另一个仅对有穷多个主目的函数；但是,由于一个集合被当作一个函数的值域,该函数仅有真值作为它的值,且所有弗雷格的异于抽象和描述算子的原始函数符号指称这些被包括进外延的函数,那么从集合中就可以构造出整个模型。
7 正是基于上述Dummett和Boolos两人的工作,[Heck96]最终证明了简单直谓二阶逻辑和分叉直谓二阶逻辑是一致的,且罗宾逊算术Q在简单直谓二阶逻辑中是可解释的。
8 Dummett:Truth and other Enigmas [1973],和 The interpretation of Frege-'s philosophy [1981]等。
9 具体请参考Boolos:Saving Frege from Contradiction中对弗雷格修正方案的讨论。
10 具体请参考本文第2章第3,4,5,6,7节中的内容。其中第3,4,5节证明了弗雷格算术系统一阶片段的一致性；第6,7节证明了弗雷格算术系统二阶子系统的一致性。
11 具体请参考下述文章中的新弗雷格主义宣言：Wright & Hale:Logicism in the Twenty-first century,发表在[Scha05]。
12 请参考[WCBH01]中第2部分：对诸批判的回应；第3部分：休谟原则。其中第2部分的四篇论文基本上是对Dummett诸质疑的答复；第3部分的四篇论文基本上是对Boolos & Heck的答复。
13 否则,就会出现一种恶性循环,会出现一种非直谓性的情况。为了避免该情况的出现,Wright等人消除第五基本定律,而从休谟原则出发,在二阶逻辑内对标准皮亚诺算术进行推导。这正是[WCBH01]所做的工作,尤其是第3章和第4章；对该工作的深入讨论,还要等到[Burg05]第3章对非直谓性以及全概括公理的分析。
14 Wright将基数算子命名为N=,Boolos将之记做x.Nx,后者是一种已被广为接受的记法。
15 关于抽象原则可接受性的详尽探讨,请参考[Fine02]第3章。
16 该称呼出现在[Burg05]第3章第2节中。该学派是由Wright领导的,主要的合作伙伴为Hale,所在机构为苏格兰圣安德鲁斯大学本原研究所。现在Wright在纽约大学哲学系工作,同行有Hartry Field和Kit Fine。
17 关于一阶和二阶逻辑,请参考[Ende01]; Ebbinghaus:Mathematical logic等经典数理逻辑教材。
18 具体请参考[Simp09]中导论部分,对下述定义的具体证明请参考该书中相关章节。
19 具体请参考[Burg05]第1章第5节中的相关细节。
20 关于罗宾逊算术,请参考[BGBJ07]第16章第4节。
21 不存在Q的一致的,完全的,可公理化的扩展。具体请参考[BGBJ07]定理17.7。
22 令T是理论P的一个一致的,可公理化的扩展。那么T中一致性句子在T中是不可证的。具体请参考[BGBJ07]定理18.1。
23 关于上述提到的原始递归函数和Ackermann-算术等理论,请参考[BGBJ07]第6-8章。
24 关于递归可枚举集和分离公理等理论,请参考Computability and Logic第8章；Jech(2005)第1章。
25 关于一阶算术,请参考[HPPP98]；关于直谓性,请参考[Scha05]第19章；关于类型论,请参考[Russ08]。
26 关于置换论证,最早出现在Dummett:The Interpretatation of Frege's Phi-losophy第403页：Dummett对置换论证做出模型论解释,而且把该论证当作可恒等性论题的一个正确的基础；关于该问题的新近的一篇较为权威的文献,请参考Wehmeier & Schroeder:Frege's Permutation Argument Rivisied。
27 可参考：[Freg93.03]第48页。
28 到目前为止的弗雷格研究也表明,彼特对此是第一个提出质疑的。[ParT87]和Wehmeier & Schroeder[2005]给出了某些积极的结果。
29 对于之前一直使用的秩概念,我们在这里做一个简短的解释。von Neumann-宇宙V是一个良基集合类。一个良基集合的秩被归纳地定义为最小序数,这个序数是大于集合中所有元素的秩的。特别地,空集的秩是零,每个序数都有一个同自身相等的秩。 具体细节请参考：[Jech02]定理2.27。
30 令Γ是一个(?)-句子集。那么Γ是可满足的当且仅当Γ句子替代例示的每个有穷集是真值函项地可满足的。可参考[BGBJ07]定理19.11。
31 如果A蕴涵C,那么存在一个句子B,A蕴涵B,C蕴涵B,且不包含除了比如既在A中又在C中的非逻辑符号。可参考[BGBJ07]定理20.3。
32 对每个集合X,｜X｜<｜P(X)｜。具体可参考[Jech02]定理3.1。
33 具体思路请参考弗雷格：[Freg84]第65-83节；具体证明请参考弗雷格：[Freg93/03]定理32和定理49。
34 请参考弗雷格：《算术基本定律》,第一条对应着定理71和89；第二条对应着定理108；第三条对应着定理107。
35 第一篇论文是Burgess & Hazen:Arithmetic and Predicative Logic；另一篇论文是Burgess:on a Consistent Subsystem of Frege's Grundgesetze.
36 关于递归饱和模型的一阶可定义集,具体请参考论文[BanS75]；关于初等皮亚诺算术的递归饱和模型,请参考[Kaye94]；关于递归模型理论,还可参考[CK1990]等模型论经典著作。
37 Boolos[1987]和[1990]给出了两个证明：首先,弗雷格算术和二阶皮亚诺算术是等一致的；其次,从弗雷格算术和弗雷格关于零、前趋和自然数三个定义中,可以推导出二阶皮亚诺算术。Heck和Linnebo继承了Boolos的处理手法,尤其是积极地使用弗雷格本人的定义,由此,基本上可以把他们称为纯粹弗雷格主义者。具体的评论可参考本章第5节。
38 可参考[Gold01]。
39 即使当二位恒等逻辑谓词是被允许的,一元谓词逻辑是可判定的。具体可参考Computability and Logic第21章；亦可参考[Behm22]和[Lowe15]。
40 根据Burgess,我们在此简单地介绍一下其他人的工作。[Wrig83]第一次将两个重要的论题结合在一起：首先,如果去除第五基本定律,而只从休谟原则出发,我们就不会得出罗素类r；其次,从休谟原则出发,在二阶逻辑中可以推导出经典二阶皮亚诺算术。但在Wright之前,这两个论题是分离的,它们分别是由两个人得到的：其中第二个结果就是下文中的Geach模型；而第一个论题是由Charles Parsons在1965年的论文《弗雷格关于数的理论》中给出的。Harold Hodes[1984]重述了Wright提出的论题；Burgess[1984]和Hazen[1985]对Wright的书评指出了借助于弗雷格算术中的一个简单模型就可以给出算术的一致性；而该模型也正是Geach模型；Neil Tennant[1987]第一次给出了从弗雷格算术中导出皮亚诺公设的严格论证。
41 关于本节中Burgess所引出的术语,请参考第4章1-4节以及[Burg05]中表格1-10；下同。
42 我们给出Burgess[2005]中戴德金定理[1]和[2]：戴德金定理[1]：仅有的满足加法、乘法、指数及其他递归等式的自然数运算存在性,从皮亚诺公式中在二元二阶逻辑下是可推导的；戴德金定理[2]：二阶皮亚诺算术在带有二元二阶逻辑的UUST系统中是可解释的；在带有一元二阶逻辑的UST系统中也是可解释的；具体可参考第29页和第119页。43 关于非弗雷格的演绎方式,具体请参考：[Burg05]第3章第2节到第6节。44 关于变元-约束项-形成算子,具体请参考本文第2章第6节。45 关于一阶片段的模型论证明,具体请参考本文第2章第3节。46 关于Shoenfield定理,具体请参考[BGBJ07]第25章第3节；Burgess(2005)第1章注释3247 关于这个并非成功的尝试,请参考Wehmeier(1999)；但随后,FW(2001)给出了正确的证明。48 关于两个合作的成果,请参考上一个注释以及本文第2章第7节。49 Lowenheim-Skolem定理：如果一个句子集有一个模型,那么它有一个可枚举模型。具体请参考：Computability and Logic定理12.14。50 关于微模型论,请参考本章第6节以及Visser(2009).51 所以这样命名,这是因为Ganea最初的证明思路并不是使用简单集的定义,而是有限时序机代码(codes for finite automata)这个概念,即{0,1}
的正则子集(regular subsets).因而,原初的思路就是这样的：使用概念变元去包摄有限时序机代码这样一种方法对I△0+Ω1>PV进行解释。但是Samuel Buss建议Ganea使用自然数简单集这种用法,大大地简化了证明步骤的复杂性。Ganea原来的证明有10页纸的篇幅,经简化后只有6页的篇幅。52 Lob定理：如果F(?)xP(x,n=φn),那么F(?)φn。其中x是对带有哥德尔数n公式证明的哥德尔数,n是公式φn的哥德尔数数字。请参考：Hinman(2005)；或者Boolos(1995).53 我们来简单地介绍一下Henkin-Feferman构造的历史。1939年,希尔伯特和博尼斯曾经给出了形式化后的哥德尔完全性定理,他们把哥德尔本人的构造形式化；1952年,王浩做出了一个扩展；Visser在此把到目前为止的所有理论称为Godel-(Hilbert +Bernays)-Wang构造。1960年,Slomon Feferman进一步改善了上述结果,使用一种建立在Henkin构造基础上的算术构造。这样就完成了整个Henkin-Feferman构造。Feferman的结果使用了△(?)0-归纳,而使用缩短可定义切割的方法就可以继续改善Feferman的结果。这项工作是由Solovay在他的一篇未公开发表的论文中完成的。 1976年,Solovay发现了缩短可定义切割方法。1990年,Visser详细地探讨了弱理论下的Henkin-Feferman构造；1991年,Visser对他在1990年取得的成果又做出了改善。因而,我们把到目前为止的方法称之为Solovay-Visser构造,以替代Visser在他的论文中称之为Henkin-Feferman构造的这种记法。
54 关于Lob定理,请参考注释52。
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2 经过12天的激烈讨论,在2006年8月24日于捷克首都布拉格举行的第26届国际天文学联合会大会上位居太阳系九大行星末席70多年的冥王星,最终以237票赞成,157票反对,17票弃权的表决结果,被逐出太阳系九大行星之列。至此,传统意义上的太阳系九大行星变为水星、金星、地球、火星、木星、土星、天王星和海王星八大行星。在《弗雷格：数学哲学》出版的时候,这一决议尚未发生。
3 普特南曾经用置换论证来反对形而上学实在论,而戴维森用它来展现不同于真理概念,指称概念和意义理论是纯内在的。
4 根据Dummett, Bartlett这篇没有被足够重视的文献是关于弗雷格的本体论和语义学的,这是Bartlett本人的博士论文,可参考：James M. Bartlett, Funktion und Gegenstand, Munich,1961.
5 g(ξ)是如下公式的缩写：对每个(?),如果ξ是(?)的值域,那么不能有(?)(ξ)；
6 Dummett提出另外一种建议,替代有穷子集和补有穷子集,我们做出如下表述：包括进外延中的函数,它的值区别于另一个仅对有穷多个主目的函数；但是,由于一个集合被当作一个函数的值域,该函数仅有真值作为它的值,且所有弗雷格的异于抽象和描述算子的原始函数符号指称这些被包括进外延的函数,那么从集合中就可以构造出整个模型。
7 正是基于上述Dummett和Boolos两人的工作,[Heck96]最终证明了简单直谓二阶逻辑和分叉直谓二阶逻辑是一致的,且罗宾逊算术Q在简单直谓二阶逻辑中是可解释的。
8 Dummett:Truth and other Enigmas [1973],和 The interpretation of Frege-'s philosophy [1981]等。
9 具体请参考Boolos:Saving Frege from Contradiction中对弗雷格修正方案的讨论。
10 具体请参考本文第2章第3,4,5,6,7节中的内容。其中第3,4,5节证明了弗雷格算术系统一阶片段的一致性；第6,7节证明了弗雷格算术系统二阶子系统的一致性。
11 具体请参考下述文章中的新弗雷格主义宣言：Wright & Hale:Logicism in the Twenty-first century,发表在[Scha05]。
12 请参考[WCBH01]中第2部分：对诸批判的回应；第3部分：休谟原则。其中第2部分的四篇论文基本上是对Dummett诸质疑的答复；第3部分的四篇论文基本上是对Boolos & Heck的答复。
13 否则,就会出现一种恶性循环,会出现一种非直谓性的情况。为了避免该情况的出现,Wright等人消除第五基本定律,而从休谟原则出发,在二阶逻辑内对标准皮亚诺算术进行推导。这正是[WCBH01]所做的工作,尤其是第3章和第4章；对该工作的深入讨论,还要等到[Burg05]第3章对非直谓性以及全概括公理的分析。
14 Wright将基数算子命名为N=,Boolos将之记做x.Nx,后者是一种已被广为接受的记法。
15 关于抽象原则可接受性的详尽探讨,请参考[Fine02]第3章。
16 该称呼出现在[Burg05]第3章第2节中。该学派是由Wright领导的,主要的合作伙伴为Hale,所在机构为苏格兰圣安德鲁斯大学本原研究所。现在Wright在纽约大学哲学系工作,同行有Hartry Field和Kit Fine。
17 关于一阶和二阶逻辑,请参考[Ende01]; Ebbinghaus:Mathematical logic等经典数理逻辑教材。
18 具体请参考[Simp09]中导论部分,对下述定义的具体证明请参考该书中相关章节。
19 具体请参考[Burg05]第1章第5节中的相关细节。
20 关于罗宾逊算术,请参考[BGBJ07]第16章第4节。
21 不存在Q的一致的,完全的,可公理化的扩展。具体请参考[BGBJ07]定理17.7。
22 令T是理论P的一个一致的,可公理化的扩展。那么T中一致性句子在T中是不可证的。具体请参考[BGBJ07]定理18.1。
23 关于上述提到的原始递归函数和Ackermann-算术等理论,请参考[BGBJ07]第6-8章。
24 关于递归可枚举集和分离公理等理论,请参考Computability and Logic第8章；Jech(2005)第1章。
25 关于一阶算术,请参考[HPPP98]；关于直谓性,请参考[Scha05]第19章；关于类型论,请参考[Russ08]。
26 关于置换论证,最早出现在Dummett:The Interpretatation of Frege's Phi-losophy第403页：Dummett对置换论证做出模型论解释,而且把该论证当作可恒等性论题的一个正确的基础；关于该问题的新近的一篇较为权威的文献,请参考Wehmeier & Schroeder:Frege's Permutation Argument Rivisied。
27 可参考：[Freg93.03]第48页。
28 到目前为止的弗雷格研究也表明,彼特对此是第一个提出质疑的。[ParT87]和Wehmeier & Schroeder[2005]给出了某些积极的结果。
29 对于之前一直使用的秩概念,我们在这里做一个简短的解释。von Neumann-宇宙V是一个良基集合类。一个良基集合的秩被归纳地定义为最小序数,这个序数是大于集合中所有元素的秩的。特别地,空集的秩是零,每个序数都有一个同自身相等的秩。 具体细节请参考：[Jech02]定理2.27。
30 令Γ是一个(?)-句子集。那么Γ是可满足的当且仅当Γ句子替代例示的每个有穷集是真值函项地可满足的。可参考[BGBJ07]定理19.11。
31 如果A蕴涵C,那么存在一个句子B,A蕴涵B,C蕴涵B,且不包含除了比如既在A中又在C中的非逻辑符号。可参考[BGBJ07]定理20.3。
32 对每个集合X,｜X｜<｜P(X)｜。具体可参考[Jech02]定理3.1。
33 具体思路请参考弗雷格：[Freg84]第65-83节；具体证明请参考弗雷格：[Freg93/03]定理32和定理49。
34 请参考弗雷格：《算术基本定律》,第一条对应着定理71和89；第二条对应着定理108；第三条对应着定理107。
35 第一篇论文是Burgess & Hazen:Arithmetic and Predicative Logic；另一篇论文是Burgess:on a Consistent Subsystem of Frege's Grundgesetze.
36 关于递归饱和模型的一阶可定义集,具体请参考论文[BanS75]；关于初等皮亚诺算术的递归饱和模型,请参考[Kaye94]；关于递归模型理论,还可参考[CK1990]等模型论经典著作。
37 Boolos[1987]和[1990]给出了两个证明：首先,弗雷格算术和二阶皮亚诺算术是等一致的；其次,从弗雷格算术和弗雷格关于零、前趋和自然数三个定义中,可以推导出二阶皮亚诺算术。Heck和Linnebo继承了Boolos的处理手法,尤其是积极地使用弗雷格本人的定义,由此,基本上可以把他们称为纯粹弗雷格主义者。具体的评论可参考本章第5节。
38 可参考[Gold01]。
39 即使当二位恒等逻辑谓词是被允许的,一元谓词逻辑是可判定的。具体可参考Computability and Logic第21章；亦可参考[Behm22]和[Lowe15]。
40 根据Burgess,我们在此简单地介绍一下其他人的工作。[Wrig83]第一次将两个重要的论题结合在一起：首先,如果去除第五基本定律,而只从休谟原则出发,我们就不会得出罗素类r；其次,从休谟原则出发,在二阶逻辑中可以推导出经典二阶皮亚诺算术。但在Wright之前,这两个论题是分离的,它们分别是由两个人得到的：其中第二个结果就是下文中的Geach模型；而第一个论题是由Charles Parsons在1965年的论文《弗雷格关于数的理论》中给出的。Harold Hodes[1984]重述了Wright提出的论题；Burgess[1984]和Hazen[1985]对Wright的书评指出了借助于弗雷格算术中的一个简单模型就可以给出算术的一致性；而该模型也正是Geach模型；Neil Tennant[1987]第一次给出了从弗雷格算术中导出皮亚诺公设的严格论证。
41 关于本节中Burgess所引出的术语,请参考第4章1-4节以及[Burg05]中表格1-10；下同。
42 我们给出Burgess[2005]中戴德金定理[1]和[2]：戴德金定理[1]：仅有的满足加法、乘法、指数及其他递归等式的自然数运算存在性,从皮亚诺公式中在二元二阶逻辑下是可推导的；戴德金定理[2]：二阶皮亚诺算术在带有二元二阶逻辑的UUST系统中是可解释的；在带有一元二阶逻辑的UST系统中也是可解释的；具体可参考第29页和第119页。43 关于非弗雷格的演绎方式,具体请参考：[Burg05]第3章第2节到第6节。44 关于变元-约束项-形成算子,具体请参考本文第2章第6节。45 关于一阶片段的模型论证明,具体请参考本文第2章第3节。46 关于Shoenfield定理,具体请参考[BGBJ07]第25章第3节；Burgess(2005)第1章注释3247 关于这个并非成功的尝试,请参考Wehmeier(1999)；但随后,FW(2001)给出了正确的证明。48 关于两个合作的成果,请参考上一个注释以及本文第2章第7节。49 Lowenheim-Skolem定理：如果一个句子集有一个模型,那么它有一个可枚举模型。具体请参考：Computability and Logic定理12.14。50 关于微模型论,请参考本章第6节以及Visser(2009).51 所以这样命名,这是因为Ganea最初的证明思路并不是使用简单集的定义,而是有限时序机代码(codes for finite automata)这个概念,即{0,1}
的正则子集(regular subsets).因而,原初的思路就是这样的：使用概念变元去包摄有限时序机代码这样一种方法对I△0+Ω1>PV进行解释。但是Samuel Buss建议Ganea使用自然数简单集这种用法,大大地简化了证明步骤的复杂性。Ganea原来的证明有10页纸的篇幅,经简化后只有6页的篇幅。52 Lob定理：如果F(?)xP(x,n=φn),那么F(?)φn。其中x是对带有哥德尔数n公式证明的哥德尔数,n是公式φn的哥德尔数数字。请参考：Hinman(2005)；或者Boolos(1995).53 我们来简单地介绍一下Henkin-Feferman构造的历史。1939年,希尔伯特和博尼斯曾经给出了形式化后的哥德尔完全性定理,他们把哥德尔本人的构造形式化；1952年,王浩做出了一个扩展；Visser在此把到目前为止的所有理论称为Godel-(Hilbert +Bernays)-Wang构造。1960年,Slomon Feferman进一步改善了上述结果,使用一种建立在Henkin构造基础上的算术构造。这样就完成了整个Henkin-Feferman构造。Feferman的结果使用了△(?)0-归纳,而使用缩短可定义切割的方法就可以继续改善Feferman的结果。这项工作是由Solovay在他的一篇未公开发表的论文中完成的。 1976年,Solovay发现了缩短可定义切割方法。1990年,Visser详细地探讨了弱理论下的Henkin-Feferman构造；1991年,Visser对他在1990年取得的成果又做出了改善。因而,我们把到目前为止的方法称之为Solovay-Visser构造,以替代Visser在他的论文中称之为Henkin-Feferman构造的这种记法。
54 关于Lob定理,请参考注释52。
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