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Where is the Gödel-point hiding: Gentzen’s Consistency Proof of 1936 and His Representation of Constructive Ordinals 豆 0.0分
资源最后更新于 2020-08-19 15:50:48
作者：Anna Horská
出版社：Springer
出版日期：0000-01
ISBN：9783319021713
文件格式： pdf
标签： 逻辑学
简介· · · · · ·
This book explains the first published consistency proof of PA. It contains the original Gentzen's proof, but it uses modern terminology and examples to illustrate the essential notions. The author comments on Gentzen's steps which are supplemented with exact calculations and parts of formal derivations. A notable aspect of the proof is the representation of ordinal numbers tha...
目录
1 Introduction ........................................ 1
References....................................... ... 8
2 Preliminaries..................................... ... 11
2.1 Sequents..................................... ... 11
2.2 InitialSequents................................ ... 12
2.3 RulesofInference.............................. ... 12
2.4 OverviewoftheProof........................... ... 14
2.5 ReductionStepsforSequents...................... ... 19
2.6 ReductionofInitialSequentstoEndform.............. ... 21
2.7 DefinitionofaNewDerivation..................... ... 24
2.8 ChainRule................................... ... 24
References....................................... ... 28
3 OrdinalNumbers.................................. ... 29
3.1 Definition.................................... ... 29
3.2 AbouttheOrdering............................. ... 31
3.3 The Relationship Between Gentzen’s Notation and Standard Notation of Ordinal Numbers...................... ... 32
3.4 An Algorithm for Translating Gentzen’s Notation of Ordinal Numbers to Cantor Normal Form ...................... 35
References.......................................... 40
4 Consistency Proof .................................... 41
4.1 How to Assign Ordinal Numbers to Derivations............ 41
4.2 Lowering the Ordinal Numbers After Reduction Steps for Derivations................................... 45
Appendix
A: Removal of Logical Operations ∨, ⊃ and ∃ from a Derivation................... .......... 71
References....................................... ... 8
2 Preliminaries..................................... ... 11
2.1 Sequents..................................... ... 11
2.2 InitialSequents................................ ... 12
2.3 RulesofInference.............................. ... 12
2.4 OverviewoftheProof........................... ... 14
2.5 ReductionStepsforSequents...................... ... 19
2.6 ReductionofInitialSequentstoEndform.............. ... 21
2.7 DefinitionofaNewDerivation..................... ... 24
2.8 ChainRule................................... ... 24
References....................................... ... 28
3 OrdinalNumbers.................................. ... 29
3.1 Definition.................................... ... 29
3.2 AbouttheOrdering............................. ... 31
3.3 The Relationship Between Gentzen’s Notation and Standard Notation of Ordinal Numbers...................... ... 32
3.4 An Algorithm for Translating Gentzen’s Notation of Ordinal Numbers to Cantor Normal Form ...................... 35
References.......................................... 40
4 Consistency Proof .................................... 41
4.1 How to Assign Ordinal Numbers to Derivations............ 41
4.2 Lowering the Ordinal Numbers After Reduction Steps for Derivations................................... 45
Appendix
A: Removal of Logical Operations ∨, ⊃ and ∃ from a Derivation................... .......... 71
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