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Full satisfaction classes, definability, and automorphisms [article]

Bartosz Wcisło
2021 arXiv   pre-print
In particular, the automorphism group of a model expanded with a satisfaction class is never equal to the automorphism group of the original model.  ...  It follows that in every such model, there exists a full satisfaction class which makes every element definable and thus the expanded model is minimal and rigid.  ...  Acknowledgements This research was supported by an NCN MAESTRO grant 2019/34/A/HS1/00399 "Epistemic and Semantic Commitments of Foundational Theories."  ... 
arXiv:2104.09969v1 fatcat:mnoaudy3yzbf7oxd67pnqtunma

Page 4134 of Mathematical Reviews Vol. , Issue 92h [page]

1992 Mathematical Reviews  
Constantin Milici (Timisoara) 92h:03051 03C62 Kossak, Roman (PL-PAN); Schmerl, James H. (1-CT) Minimal satisfaction classes with an application to rigid models of Peano arithmetic. Notre Dame J.  ...  The result is based on a theorem saying that every countable recursively saturated model of PA has minimal inductive satisfac- tion classes. A generalization to x-like models is also given.  ... 

Carnap's Early Semantics

Georg Schiemer
2012 Erkenntnis: An International Journal of Scientific Philosophy  
der Logistik. 123 The main innovation in the 1936 paper is its application to formal models with the aim to classify different limiting types of model structures defined by an AS: 'minimal structures  ...  a categorical (in Carnap's terms "monomorphic") axiomatization. 236 In the case of Peano arithmetic, adding an accessibility axiom or a minimal axiom to the base system enforces model minimality on the  ... 
doi:10.1007/s10670-012-9365-8 fatcat:7jcctf5ajfdhlnshex7iaruija

SET THEORY FROM CANTOR TO COHEN [chapter]

Akihiro Kanamori
2009 Philosophy of Mathematics  
What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been  ...  With the emergence of the cumulative hierarchy picture, set theory can be regarded as becoming a theory of well-foundedness, later to expand to a study of consistency strength.  ...  ACKNOWLEDGEMENTS This is a revision with significant changes of the author's The Mathematical Development of Set Theory from Cantor to Cohen, The Bulletin of Symbolic Logic, volume 2, 1996, pages 1-71,  ... 
doi:10.1016/b978-0-444-51555-1.50014-6 fatcat:vnj2vdlpofdp7pg37tyiemdkiq

Set Theory from Cantor to Cohen [chapter]

Akihiro Kanamori
2012 Handbook of the History of Logic  
What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been  ...  With the emergence of the cumulative hierarchy picture, set theory can be regarded as becoming a theory of well-foundedness, later to expand to a study of consistency strength.  ...  ACKNOWLEDGEMENTS This is a revision with significant changes of the author's The Mathematical Development of Set Theory from Cantor to Cohen, The Bulletin of Symbolic Logic, volume 2, 1996, pages 1-71,  ... 
doi:10.1016/b978-0-444-51621-3.50001-3 fatcat:ogdmpb6mlvgxbdvoh3jqq6umqe

Chapter 1 Pure computable model theory [chapter]

Valentina S. Harizanov
1998 Studies in Logic and the Foundations of Mathematics  
orders, Peano arithmetic, true arithmetic, and the theory of Boolean algebras.  ...  The generalization of the definition of a particular computable algebraic structure to an arbitrary model yields one of the basic concepts of pure computable model theory, an area of logic developed in  ...  I thank Tim McNicholl for proofreading parts of an early draft. I thank Graeme Bailey for technical assistance with word-processing.  ... 
doi:10.1016/s0049-237x(98)80002-5 fatcat:mktikwmmazgale5ejrpnihv6hm

ON THE RELATIONSHIP BETWEEN PLANE AND SOLID GEOMETRY

ANDREW ARANA, PAOLO MANCOSU
2012 The Review of Symbolic Logic  
In this paper our major concern is with methodological issues of purity and thus we treat the connection to other areas of the planimetry/stereometry relation only to the extent necessary to articulate  ...  To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic.  ...  But it was Pieri's fellow countryman Peano who seems to have been the first to find a model of the plane in which Desargues' theorem fails, in an 1894 essay Peano [1894] .  ... 
doi:10.1017/s1755020312000020 fatcat:ycxnc4ue4rgivjm5chzumj3sfu

Algebras, Projective Geometry, Mathematical Logic, and Constructing the World: Intersections in the Philosophy of Mathematics of A. N. Whitehead

I. Grattan-Guinness
2002 Historia Mathematica  
By then his interests had switched to educational issues, and especially to space and time and relativity theory, where his earlier dependence upon logic was extended to an ontology of events and to a  ...  Then in the 1900s he joined Bertrand Russell in an attempt to ground many parts of mathematics in the newly developing mathematical logic.  ...  On model-theoretic aspects of basic notions such as number, zero, and successor in the Peano axioms for arithmetic "might be fitted on to anything. B. Russell doesn't seem to object to this."  ... 
doi:10.1006/hmat.2002.2356 fatcat:af6uawrpibccjgnkqln5ftaism

On the methodology of informal rigour: set theory, semantics, and intuitionism [article]

Walter Dean, Hidenori Kurokawa
2021 arXiv   pre-print
In an appendix, we also offer briefer reconstructions of Kreisel's attempts to apply informal rigour to the discovery of set theoretic axioms, the distinction between standard and nonstandard models of  ...  We conclude by offering a comparison of Kreisel's understanding of informal rigour with Carnap's method of explication.  ...  such as first-order Peano arithmetic [PA] .  ... 
arXiv:2104.14887v1 fatcat:bxlwnrcqizecvfggmwkccchbxu

The Value-Passing Calculus [chapter]

Yuxi Fu
2013 Lecture Notes in Computer Science  
The logical expression ϕ must be evaluated in the model before process (1) fires an action. If ϕ contains free variables, the evaluation is done with respect to an assignment.  ...  A valuepassing calculus consists of a first order theory with boolean completeness and an operational model that makes use of the terms and the boolean expressions of the theory.  ...  Section 7 applies the methodology to the value-passing calculus VPC defined over the Peano arithmetics. The model is shown to be minimal among all the Turing complete value-passing calculi.  ... 
doi:10.1007/978-3-642-39698-4_11 fatcat:ec74pnvkovgelgyuecc34avjm4

Page 81 of Mathematical Reviews Vol. 28, Issue Index [page]

Mathematical Reviews  
recursively saturated models of Peano arithmetic.  ...  A reflection principle and its applications to nonstandard models. 96m:03025 — PA(aa).  ... 

Answer Set Programming's Contributions to Classical Logic [chapter]

Marc Denecker, Joost Vennekens, Hanne Vlaeminck, Johan Wittocx, Maurice Bruynooghe
2011 Lecture Notes in Computer Science  
In this paper, a look at different aspects of ASP, in an effort to identify precisely the limitations of classical logic that they exposed and investigate how the ASP approaches can be transferred back  ...  to the classical setting.  ...  A (very) early example of this is Peano arithmetic, Peano's standard second order theory of the natural numbers where σ = {0, S/1}.  ... 
doi:10.1007/978-3-642-20832-4_2 fatcat:yrydik5mlrgmxd7azmhpuudz7i

POLYMORPHISM AND THE OBSTINATE CIRCULARITY OF SECOND ORDER LOGIC: A VICTIMS' TALE

PAOLO PISTONE
2018 Bulletin of Symbolic Logic  
to Russell's and Poincaré's 1906 "vicious circle" diagnosis.  ...  for the intuitionistic type theory with a type of all types (shown inconsistent by Girard's paradox).The comparison suggests that the question of the circularity of second order logic cannot be reduced  ...  by Gödel, and essentially corresponding to first-order Peano Arithmetics.  ... 
doi:10.1017/bsl.2017.43 fatcat:q3xtgurssnhrfmsgieso3aqhe4

A survey of homogeneous structures

Dugald Macpherson
2011 Discrete Mathematics  
The article discusses connections between these topics, with an emphasis on examples, and on special properties of an amalgamation class which yield important consequences for the automorphism group. language  ...  Recently there has been a focus on connections to topological dynamics, and to constraint satisfaction.  ...  An existential interpretation is one in which all the defining formulas can be taken to be existential; similarly for positive existential interpretation, and p.p. interpretation.  ... 
doi:10.1016/j.disc.2011.01.024 fatcat:fqpfnc25hjdajekkiyaeszw6ye

MODALITY AND AXIOMATIC THEORIES OF TRUTH I: FRIEDMAN-SHEARD

JOHANNES STERN
2014 The Review of Symbolic Logic  
First, we show that Modal Friedman-Sheard preserves theoremhood modulo translation with respect to modal operator logic.  ...  The idea of this strategy is to develop modal theories over axiomatic theories of truth.  ...  F S based on the standard model of arithmetic.  ... 
doi:10.1017/s1755020314000057 fatcat:nmejpurlmjfi7lwmpojw3c6bh4
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