LEADER 03953nam 22005535 450 001 9910349319103321 005 20200630014251.0 010 $a3-030-28921-4 024 7 $a10.1007/978-3-030-28921-8 035 $a(CKB)4100000009375044 035 $a(DE-He213)978-3-030-28921-8 035 $a(MiAaPQ)EBC5906275 035 $a(PPN)248602268 035 $a(EXLCZ)994100000009375044 100 $a20190926d2019 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 12$aA Model?Theoretic Approach to Proof Theory /$fby Henryk Kotlarski ; edited by Zofia Adamowicz, Teresa Bigorajska, Konrad Zdanowski 205 $a1st ed. 2019. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2019. 215 $a1 online resource (XVIII, 109 p. 53 illus., 1 illus. in color.) 225 1 $aTrends in Logic, Studia Logica Library,$x1572-6126 ;$v51 311 $a3-030-28920-6 327 $aChapter 1. Some combinatorics -- Chapter 2. Some model theory -- Chapter 3. Incompleteness -- Chapter 4. Trans?nite induction -- Chapter 5. Satisfaction classes. 330 $aThis book presents a detailed treatment of ordinal combinatorics of large sets tailored for independence results. It uses model theoretic and combinatorial methods to obtain results in proof theory, such as incompleteness theorems or a description of the provably total functions of a theory. In the first chapter, the authors first discusses ordinal combinatorics of finite sets in the style of Ketonen and Solovay. This provides a background for an analysis of subsystems of Peano Arithmetic as well as for combinatorial independence results. Next, the volume examines a variety of proofs of Gödel's incompleteness theorems. The presented proofs differ strongly in nature. They show various aspects of incompleteness phenomena. In additon, coverage introduces some classical methods like the arithmetized completeness theorem, satisfaction predicates or partial satisfaction classes. It also applies them in many contexts. The fourth chapter defines the method of indicators for obtaining independence results. It shows what amount of transfinite induction we have in fragments of Peano arithmetic. Then, it uses combinatorics of large sets of the first chapter to show independence results. The last chapter considers nonstandard satisfaction classes. It presents some of the classical theorems related to them. In particular, it covers the results by S. Smith on definability in the language with a satisfaction class and on models without a satisfaction class. Overall, the book's content lies on the border between combinatorics, proof theory, and model theory of arithmetic. It offers readers a distinctive approach towards independence results by model-theoretic methods. 410 0$aTrends in Logic, Studia Logica Library,$x1572-6126 ;$v51 606 $aLogic 606 $aLogic, Symbolic and mathematical 606 $aLogic$3https://scigraph.springernature.com/ontologies/product-market-codes/E16000 606 $aMathematical Logic and Foundations$3https://scigraph.springernature.com/ontologies/product-market-codes/M24005 615 0$aLogic. 615 0$aLogic, Symbolic and mathematical. 615 14$aLogic. 615 24$aMathematical Logic and Foundations. 676 $a160 700 $aKotlarski$b Henryk$4aut$4http://id.loc.gov/vocabulary/relators/aut$0780954 702 $aAdamowicz$b Zofia$4edt$4http://id.loc.gov/vocabulary/relators/edt 702 $aBigorajska$b Teresa$4edt$4http://id.loc.gov/vocabulary/relators/edt 702 $aZdanowski$b Konrad$4edt$4http://id.loc.gov/vocabulary/relators/edt 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910349319103321 996 $aModel?Theoretic Approach to Proof Theory$91668083 997 $aUNINA