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Full satisfaction classes, definability, and automorphisms
[article]

2021
*
arXiv
*
pre-print

In particular, the automorphism group

arXiv:2104.09969v1
fatcat:mnoaudy3yzbf7oxd67pnqtunma
*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." ...##
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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. ...##
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Carnap's Early Semantics

2012
*
Erkenntnis: An International Journal of Scientific Philosophy
*

der Logistik. 123 The main innovation in the 1936 paper is its

doi:10.1007/s10670-012-9365-8
fatcat:7jcctf5ajfdhlnshex7iaruija
*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 ...##
###
SET THEORY FROM CANTOR TO COHEN
[chapter]

2009
*
Philosophy of Mathematics
*

What follows is

doi:10.1016/b978-0-444-51555-1.50014-6
fatcat:vnj2vdlpofdp7pg37tyiemdkiq
*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, ...##
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Set Theory from Cantor to Cohen
[chapter]

2012
*
Handbook of the History of Logic
*

What follows is

doi:10.1016/b978-0-444-51621-3.50001-3
fatcat:ogdmpb6mlvgxbdvoh3jqq6umqe
*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, ...##
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Chapter 1 Pure computable model theory
[chapter]

1998
*
Studies in Logic and the Foundations of Mathematics
*

orders,

doi:10.1016/s0049-237x(98)80002-5
fatcat:mktikwmmazgale5ejrpnihv6hm
*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. ...##
###
ON THE RELATIONSHIP BETWEEN PLANE AND SOLID GEOMETRY

2012
*
The Review of Symbolic Logic
*

In this paper our major concern is

doi:10.1017/s1755020312000020
fatcat:ycxnc4ue4rgivjm5chzumj3sfu
*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] . ...##
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Algebras, Projective Geometry, Mathematical Logic, and Constructing the World: Intersections in the Philosophy of Mathematics of A. N. Whitehead

2002
*
Historia Mathematica
*

By then his interests had switched

doi:10.1006/hmat.2002.2356
fatcat:af6uawrpibccjgnkqln5ftaism
*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." ...##
###
On the methodology of informal rigour: set theory, semantics, and intuitionism
[article]

2021
*
arXiv
*
pre-print

In

arXiv:2104.14887v1
fatcat:bxlwnrcqizecvfggmwkccchbxu
*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] . ...##
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The Value-Passing Calculus
[chapter]

2013
*
Lecture Notes in Computer Science
*

The logical expression ϕ must be evaluated in the

doi:10.1007/978-3-642-39698-4_11
fatcat:ec74pnvkovgelgyuecc34avjm4
*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. ...##
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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]

2011
*
Lecture Notes in Computer Science
*

In this paper, a look at different aspects

doi:10.1007/978-3-642-20832-4_2
fatcat:yrydik5mlrgmxd7azmhpuudz7i
*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}. ...##
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POLYMORPHISM AND THE OBSTINATE CIRCULARITY OF SECOND ORDER LOGIC: A VICTIMS' TALE

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*. ...

##
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A survey of homogeneous structures

2011
*
Discrete Mathematics
*

The article discusses connections between these topics,

doi:10.1016/j.disc.2011.01.024
fatcat:fqpfnc25hjdajekkiyaeszw6ye
*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. ...##
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MODALITY AND AXIOMATIC THEORIES OF TRUTH I: FRIEDMAN-SHEARD

2014
*
The Review of Symbolic Logic
*

First, we show that Modal Friedman-Sheard preserves theoremhood modulo translation

doi:10.1017/s1755020314000057
fatcat:nmejpurlmjfi7lwmpojw3c6bh4
*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*. ...
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