Florio, Salvatore

Linnebo, Øystein

Walsh, Sean

Welch, Philip

University of Birmingham Research Portal

  University of BirminghamIntroductionFlorio, Salvatore; Linnebo, Øystein; Walsh, Sean; Welch, PhilipLicense:None: All rights reservedDocument VersionPeer reviewed versionCitation for published version (Harvard):Florio, S, Linnebo, Ø, Walsh, S & Welch, P 2015, 'Introduction: Set Theory and Higher-Order Logic: FundationalIssues and Mathematical Development', Notre Dame Journal of Formal Logic, vol. 56, no. 1.Link to publication on Research at Birmingham portalGeneral rightsUnless a licence is specified above, all rights (including copyright and moral rights) in this document are retained by the authors and/or thecopyright holders. The express permission of the copyright holder must be obtained for any use of this material other than for purposespermitted by law.•	Users may freely distribute the URL that is used to identify this publication.•	Users may download and/or print one copy of the publication from the University of Birmingham research portal for the purpose of privatestudy or non-commercial research.•	User may use extracts from the document in line with the concept of ‘fair dealing’ under the Copyright, Designs and Patents Act 1988 (?)•	Users may not further distribute the material nor use it for the purposes of commercial gain.Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document.When citing, please reference the published version.Take down policyWhile the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has beenuploaded in error or has been deemed to be commercially or otherwise sensitive.If you believe that this is the case for this document, please contact UBIRA@lists.bham.ac.uk providing details and we will remove access tothe work immediately and investigate.Download date: 01. Mar. 2020Introduction	  	  These	   two	   special	   issues	  of	   the	  Notre	  Dame	   Journal	   of	   Formal	  Logic	   feature	  work	  related	   to	   the	   talks	   and	   presentations	   from	   the	   summer	   school	   “Set	   Theory	   and	  Higher-­‐Order	   Logic:	   Foundational	   Issues	   and	  Mathematical	   Developments,”	  which	  took	  place	  at	   the	   Institute	  of	  Philosophy	   in	  London	  on	  August	  1-­‐6,	  2011.	  The	   local	  organizers	   were	   part	   of	   Øystein	   Linnebo’s	   European	   Research	   Council-­‐funded	  project	   Plurals,	   Predicates,	   and	   Paradox.	   Further	   organization	   and	   support	   was	  generously	   provided	   by	   the	   New	   Frontiers	   of	   Infinity	   Project	   (European	   Science	  Foundation),	   the	   Ideals	   of	   Proof	   Project	   (Agence	   nationale	   de	   la	   recherche),	   the	  Philosophy	   Department	   and	   Logic	   Group	   of	   the	   University	   of	   Notre	   Dame,	   the	  Munich	   Center	   for	   Mathematical	   Philosophy	   (Alexander	   von	   Humboldt	   Stiftung),	  and	  the	  Philosophy	  Department	  of	  Birkbeck,	  University	  of	  London.	  	  	  The	   summer	   school	  was	  well-­‐attended	   (circa	   80	   total	   participants),	   and	  was	   split	  into	   two	   components,	   with	   five	   days	   of	   introductory	   and	   advanced	   tutorials,	  followed	   by	   a	   two-­‐day	   workshop.	   We’re	   thankful	   to	   the	   speakers	   for	   the	   large	  amounts	  of	  time	  which	  they	  devoted	  to	  the	  summer	  school,	  and	  are	  very	  happy	  that	  so	  many	   of	   them	   could	   contribute	   to	   these	   special	   issues.	   The	   other	   contributors	  spoke	  on	  separate	  occasions	  to	  the	  Plurals,	  Predicates,	  and	  Paradox	  project	  on	  the	  topic	  of	   set	   theory	  and	  higher-­‐order	   logic.	  One	  can	   find	   the	  slides	   for	   the	   tutorials	  and	  talks	  for	  the	  summer	  school	  on	  each	  of	  the	  following	  websites:	  	  http://www.bbk.ac.uk/philosophy/our-­‐research/ppp/summer-­‐school	  http://www.oysteinlinnebo.org/ppp/summer-­‐school	  	  The	   topic	   of	   set	   theory	   and	   higher-­‐order	   logic	   has	   of	   course	   been	   at	   the	   heart	   of	  classical	   developments	   in	   the	   foundations	   of	   set	   theory.	   For	   instance,	   there	   is	  Gödel’s	  famous	  idea	  that	  one	  can	  conceive	  of	  the	  cumulative	  hierarchy	  of	  sets	  as	  an	  extension	  of	  second-­‐order	  logic,	  third-­‐order	  logic,	  etc.	  into	  the	  transfinite.	  Likewise,	  there	   is	  Kreisel’s	   important	  observation	   that	  Zermelo’s	  quasi-­‐categoricity	   theorem	  indicates	   that	   the	   continuum	   hypothesis	   is	   decided	   by	   certain	   varieties	   of	   the	  semantics	   for	   second-­‐order	   logic,	   in	   spite	   of	   its	   deductive	   independence	   from	   the	  first-­‐order	   axioms	   of	   set	   theory.	   Finally,	   there	   is	   the	   old	   idea	   that	   large	   cardinal	  axioms	  themselves	  can	  be	  motivated	  by	  so-­‐called	  reflection	  principles,	  which	  posit	  that	  the	  universe	  can’t	  be	  distinguished	  from	  its	  initial	  segments	  by	  various	  first-­‐	  or	  higher-­‐order	  resources.	  	  While	  ideas	  such	  as	  these	  are	  staples	  of	  classical	  foundations	  of	  set	  theory,	   it’s	  our	  hope	  that	  these	  special	  issues	  and	  the	  summer	  school	  help	  to	  underscore	  that	  there	  is	   still	   much	   left	   to	   say	   on	   the	   topic	   of	   set-­‐theory	   and	   higher-­‐order	   logic.	   For	  instance,	   our	   special	   issues	   include	   an	   introduction	   to	   large	   cardinals	   as	   strong	  axioms	   of	   infinity	   and	   sketches	   how	   they	   imply	   increasing	   levels	   of	   the	   axiom	   of	  definable	  determinacy.	  Likewise,	  one	  can	  consult	   the	  above	  websites	   for	  the	  slides	  from	   the	   tutorials	   and	   talks	   of	   Antonelli,	   Bagaria,	   Hamkins,	   Horsten,	   Incurvati,	  Leitgeb	  and	  our	  contributors	  on	  a	  multitude	  of	   recent	   topics	   like	  Omega	   logic,	   the	  multiverse	  conception	  of	  set	  theory,	  groundedness,	  and	  formal	  theories	  of	  truth.	  	  Many	  of	  the	  papers	  in	  this	  special	  issue	  —	  like	  much	  work	  in	  the	  philosophy	  of	  set	  theory	  —	  involves	  grappling	  in	  one	  way	  or	  another	  with	  the	  paradoxes.	  Russell,	  and	  Dummett	   	   after	   him,	   suggested	   that	   the	   general	   mechanism	   behind	   many	   of	   the	  paradoxes	  was	  that	  of	  indefinite	  extensibility.	  Uzquiano	  in	  his	  contribution	  proposes	  a	   new	  way	   of	   understanding	   the	   Russellian	   and	   Dummettian	   notion	   of	   indefinite	  extensibility.	   While	   it	   has	   long	   been	   noted	   that	   Russell’s	   paradox	   is	   structurally	  similar	   to	   Cantor’s	   theorem	   on	   the	   cardinality	   of	   the	   power	   set,	   the	   article	   by	  Meadows	  points	  out	  a	  similarity	  between	  the	  proof	  of	  Cantor’s	  theorem	  and	  certain	  elements	   of	   Cohen’s	   method	   of	   forcing.	   Russell	   originally	   used	   his	   paradox	   to	  demonstrate	   the	   inconsistency	   of	   Frege’s	   Grundgesetze,	   and	   two	   of	   our	  contributions	   pertain	   to	   the	   Grundgesetze.	   Payne	   articulates	   a	   consistent	   modal	  rendition	   of	   a	   variant	   of	   Frege’s	   system,	   and	   Cruz-­‐Filipe	   and	   Ferreira	   study	   the	  strength	  of	   a	  version	  of	   the	  Grundgesetze	   in	  which	   the	   comprehension	   schema	   for	  the	  ambient	  higher-­‐order	  logic	  is	  restricted	  to	  its	  so-­‐called	  predicative	  instances.	  	  Higher-­‐order	  logic	  also	  features	  prominently	  in	  several	  of	  the	  other	  more	  technical	  papers	   in	   these	   special	   issues.	   Philosophical	   logicians	   and	   set	   theorists	   have	   long	  used	  logics	  wherein	  the	  space	  of	   truth-­‐values	   is	  a	  Boolean	  algebra	  with	  more	  than	  just	   two	   elements,	   and	   the	   paper	   of	   Ikegami	   and	   Väänänen	   devises	   a	   related	  Boolean-­‐valued	   	   semantics	   for	   second-­‐order	   logic.	   In	   their	   joint	   paper,	   Väänänen	  and	  Wang	  note	  that	  the	  classical	  categoricity	  theorems	  of	  Dedekind	  and	  Zermelo	  can	  be	  done	   in	  a	  weak	  background	  second-­‐order	   logic.	  Visser	   in	  his	   tutorial	  discussed	  the	  general	  question	  of	  how	  much	  consistency	  strength	  is	  added	  by	  including	  a	  layer	  of	  predicative	  classes	  on	  top	  of	  a	  theory,	  and	  in	  his	  contribution	  he	  studies	  another	  measure	   of	   the	   strength	   of	   a	   theory,	   namely	   the	   arithmetics	   of	   the	   theory.	   Like	  Visser’s	   tutorial,	   the	  paper	  of	  Mathias	  and	  Bowler	   is	  a	  contribution	  to	  the	  study	  of	  the	  fine-­‐grained	  properties	  of	  weak	  theories	  of	  classes	  and	  set.	  In	  particular,	  Mathias	  and	  Bowler	  here	  lay	  the	  groundwork	  for	  a	  weak	  set	  theory	  which	  is	  yet	  still	  strong	  enough	  to	  do	  some	  forcing.	  	  	  Salvatore	  Florio	  Øystein	  Linnebo	  Sean	  Walsh	  Philip	  Welch	  

English

Introduction:Set Theory and Higher-Order Logic: Fundational Issues and Mathematical Development

  IntroductionFlorio, Salvatore; Linnebo, Øystein; Walsh, Sean; Welch, PhilipLicense:None: All rights reservedDocument VersionPeer reviewed versionCitation for published version (Harvard):Florio, S, Linnebo, Ø, Walsh, S & Welch, P 2015, 'Introduction: Set Theory and Higher-Order Logic: FundationalIssues and Mathematical Development', Notre Dame Journal of Formal Logic, vol. 56, no. 1.Link to publication on Research at Birmingham portalGeneral rightsUnless a licence is specified above, all rights (including copyright and moral rights) in this document are retained by the authors and/or thecopyright holders. The express permission of the copyright holder must be obtained for any use of this material other than for purposespermitted by law.•	Users may freely distribute the URL that is used to identify this publication.•	Users may download and/or print one copy of the publication from the University of Birmingham research portal for the purpose of privatestudy or non-commercial research.•	User may use extracts from the document in line with the concept of ‘fair dealing’ under the Copyright, Designs and Patents Act 1988 (?)•	Users may not further distribute the material nor use it for the purposes of commercial gain.Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document.When citing, please reference the published version.Take down policyWhile the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has beenuploaded in error or has been deemed to be commercially or otherwise sensitive.If you believe that this is the case for this document, please contact UBIRA@lists.bham.ac.uk providing details and we will remove access tothe work immediately and investigate.Download date: 01. Feb. 2019Introduction	  	  These	   two	   special	   issues	  of	   the	  Notre	  Dame	   Journal	   of	   Formal	  Logic	   feature	  work	  related	   to	   the	   talks	   and	   presentations	   from	   the	   summer	   school	   “Set	   Theory	   and	  Higher-­‐Order	   Logic:	   Foundational	   Issues	   and	  Mathematical	   Developments,”	  which	  took	  place	  at	   the	   Institute	  of	  Philosophy	   in	  London	  on	  August	  1-­‐6,	  2011.	  The	   local	  organizers	   were	   part	   of	   Øystein	   Linnebo’s	   European	   Research	   Council-­‐funded	  project	   Plurals,	   Predicates,	   and	   Paradox.	   Further	   organization	   and	   support	   was	  generously	   provided	   by	   the	   New	   Frontiers	   of	   Infinity	   Project	   (European	   Science	  Foundation),	   the	   Ideals	   of	   Proof	   Project	   (Agence	   nationale	   de	   la	   recherche),	   the	  Philosophy	   Department	   and	   Logic	   Group	   of	   the	   University	   of	   Notre	   Dame,	   the	  Munich	   Center	   for	   Mathematical	   Philosophy	   (Alexander	   von	   Humboldt	   Stiftung),	  and	  the	  Philosophy	  Department	  of	  Birkbeck,	  University	  of	  London.	  	  	  The	   summer	   school	  was	  well-­‐attended	   (circa	   80	   total	   participants),	   and	  was	   split	  into	   two	   components,	   with	   five	   days	   of	   introductory	   and	   advanced	   tutorials,	  followed	   by	   a	   two-­‐day	   workshop.	   We’re	   thankful	   to	   the	   speakers	   for	   the	   large	  amounts	  of	  time	  which	  they	  devoted	  to	  the	  summer	  school,	  and	  are	  very	  happy	  that	  so	  many	   of	   them	   could	   contribute	   to	   these	   special	   issues.	   The	   other	   contributors	  spoke	  on	  separate	  occasions	  to	  the	  Plurals,	  Predicates,	  and	  Paradox	  project	  on	  the	  topic	  of	   set	   theory	  and	  higher-­‐order	   logic.	  One	  can	   find	   the	  slides	   for	   the	   tutorials	  and	  talks	  for	  the	  summer	  school	  on	  each	  of	  the	  following	  websites:	  	  http://www.bbk.ac.uk/philosophy/our-­‐research/ppp/summer-­‐school	  http://www.oysteinlinnebo.org/ppp/summer-­‐school	  	  The	   topic	   of	   set	   theory	   and	   higher-­‐order	   logic	   has	   of	   course	   been	   at	   the	   heart	   of	  classical	   developments	   in	   the	   foundations	   of	   set	   theory.	   For	   instance,	   there	   is	  Gödel’s	  famous	  idea	  that	  one	  can	  conceive	  of	  the	  cumulative	  hierarchy	  of	  sets	  as	  an	  extension	  of	  second-­‐order	  logic,	  third-­‐order	  logic,	  etc.	  into	  the	  transfinite.	  Likewise,	  there	   is	  Kreisel’s	   important	  observation	   that	  Zermelo’s	  quasi-­‐categoricity	   theorem	  indicates	   that	   the	   continuum	   hypothesis	   is	   decided	   by	   certain	   varieties	   of	   the	  semantics	   for	   second-­‐order	   logic,	   in	   spite	   of	   its	   deductive	   independence	   from	   the	  first-­‐order	   axioms	   of	   set	   theory.	   Finally,	   there	   is	   the	   old	   idea	   that	   large	   cardinal	  axioms	  themselves	  can	  be	  motivated	  by	  so-­‐called	  reflection	  principles,	  which	  posit	  that	  the	  universe	  can’t	  be	  distinguished	  from	  its	  initial	  segments	  by	  various	  first-­‐	  or	  higher-­‐order	  resources.	  	  While	  ideas	  such	  as	  these	  are	  staples	  of	  classical	  foundations	  of	  set	  theory,	   it’s	  our	  hope	  that	  these	  special	  issues	  and	  the	  summer	  school	  help	  to	  underscore	  that	  there	  is	   still	   much	   left	   to	   say	   on	   the	   topic	   of	   set-­‐theory	   and	   higher-­‐order	   logic.	   For	  instance,	   our	   special	   issues	   include	   an	   introduction	   to	   large	   cardinals	   as	   strong	  axioms	   of	   infinity	   and	   sketches	   how	   they	   imply	   increasing	   levels	   of	   the	   axiom	   of	  definable	  determinacy.	  Likewise,	  one	  can	  consult	   the	  above	  websites	   for	  the	  slides	  from	   the	   tutorials	   and	   talks	   of	   Antonelli,	   Bagaria,	   Hamkins,	   Horsten,	   Incurvati,	  Leitgeb	  and	  our	  contributors	  on	  a	  multitude	  of	   recent	   topics	   like	  Omega	   logic,	   the	  multiverse	  conception	  of	  set	  theory,	  groundedness,	  and	  formal	  theories	  of	  truth.	  	  Many	  of	  the	  papers	  in	  this	  special	  issue	  —	  like	  much	  work	  in	  the	  philosophy	  of	  set	  theory	  —	  involves	  grappling	  in	  one	  way	  or	  another	  with	  the	  paradoxes.	  Russell,	  and	  Dummett	   	   after	   him,	   suggested	   that	   the	   general	   mechanism	   behind	   many	   of	   the	  paradoxes	  was	  that	  of	  indefinite	  extensibility.	  Uzquiano	  in	  his	  contribution	  proposes	  a	   new	  way	   of	   understanding	   the	   Russellian	   and	   Dummettian	   notion	   of	   indefinite	  extensibility.	   While	   it	   has	   long	   been	   noted	   that	   Russell’s	   paradox	   is	   structurally	  similar	   to	   Cantor’s	   theorem	   on	   the	   cardinality	   of	   the	   power	   set,	   the	   article	   by	  Meadows	  points	  out	  a	  similarity	  between	  the	  proof	  of	  Cantor’s	  theorem	  and	  certain	  elements	   of	   Cohen’s	   method	   of	   forcing.	   Russell	   originally	   used	   his	   paradox	   to	  demonstrate	   the	   inconsistency	   of	   Frege’s	   Grundgesetze,	   and	   two	   of	   our	  contributions	   pertain	   to	   the	   Grundgesetze.	   Payne	   articulates	   a	   consistent	   modal	  rendition	   of	   a	   variant	   of	   Frege’s	   system,	   and	   Cruz-­‐Filipe	   and	   Ferreira	   study	   the	  strength	  of	   a	  version	  of	   the	  Grundgesetze	   in	  which	   the	   comprehension	   schema	   for	  the	  ambient	  higher-­‐order	  logic	  is	  restricted	  to	  its	  so-­‐called	  predicative	  instances.	  	  Higher-­‐order	  logic	  also	  features	  prominently	  in	  several	  of	  the	  other	  more	  technical	  papers	   in	   these	   special	   issues.	   Philosophical	   logicians	   and	   set	   theorists	   have	   long	  used	  logics	  wherein	  the	  space	  of	   truth-­‐values	   is	  a	  Boolean	  algebra	  with	  more	  than	  just	   two	   elements,	   and	   the	   paper	   of	   Ikegami	   and	   Väänänen	   devises	   a	   related	  Boolean-­‐valued	   	   semantics	   for	   second-­‐order	   logic.	   In	   their	   joint	   paper,	   Väänänen	  and	  Wang	  note	  that	  the	  classical	  categoricity	  theorems	  of	  Dedekind	  and	  Zermelo	  can	  be	  done	   in	  a	  weak	  background	  second-­‐order	   logic.	  Visser	   in	  his	   tutorial	  discussed	  the	  general	  question	  of	  how	  much	  consistency	  strength	  is	  added	  by	  including	  a	  layer	  of	  predicative	  classes	  on	  top	  of	  a	  theory,	  and	  in	  his	  contribution	  he	  studies	  another	  measure	   of	   the	   strength	   of	   a	   theory,	   namely	   the	   arithmetics	   of	   the	   theory.	   Like	  Visser’s	   tutorial,	   the	  paper	  of	  Mathias	  and	  Bowler	   is	  a	  contribution	  to	  the	  study	  of	  the	  fine-­‐grained	  properties	  of	  weak	  theories	  of	  classes	  and	  set.	  In	  particular,	  Mathias	  and	  Bowler	  here	  lay	  the	  groundwork	  for	  a	  weak	  set	  theory	  which	  is	  yet	  still	  strong	  enough	  to	  do	  some	  forcing.	  	  	  Salvatore	  Florio	  Øystein	  Linnebo	  Sean	  Walsh	  Philip	  Welch	  

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Introduction:Set Theory and Higher-Order Logic: Fundational Issues and Mathematical Development

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