David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jonathan Jenkins Ichikawa
Jack Alan Reynolds
Learn more about PhilPapers
“Small” large cardinal notions in the language of ZFC are those large cardinal notions that are consistent with V = L. Besides their original formulation in classical set theory, we have a variety of analogue notions in systems of admissible set theory, admissible recursion theory, constructive set theory, constructive type theory, explicit mathematics and recursive ordinal notations (as used in proof theory). On the face of it, it is surprising that such distinctively set-theoretical notions have analogues in such disaparate and relatively constructive contexts. There must be an underlying reason why that is possible (and, incidentally, why “large” large cardinal notions have not led to comparable analogues). My long term aim is to develop a common language in which such notions can be expressed and can be interpreted both in their original classical form and in their analogue form in each of these special constructive and semi-constructive cases. This is a program in progress. What is done here, to begin with, is to show how that can be done to a considerable extent in the settings of classical and admissible set theory (and thence, admissible recursion theory). The approach taken here is to expand the language of set theory to allow us to talk about (possibly partial) operations applicable both to sets and to operations and to formulate the large cardinal notions in question in terms of operational closure conditions; at the same time only minimal existence axioms are posited for sets. The resulting system, called Operational Set Theory, is a partial adaptation to the set-theoretical framework of the explicit mathematics framework Feferman (1975). The..
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library||
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
Bart Jacobs (1989). The Inconsistency of Higher Order Extensions of Martin-Löf's Type Theory. Journal of Philosophical Logic 18 (4):399 - 422.
Mitch Rudominer (1999). The Largest Countable Inductive Set is a Mouse Set. Journal of Symbolic Logic 64 (2):443-459.
Sy D. Friedman (1979). HC of an Admissible Set. Journal of Symbolic Logic 44 (1):95-102.
Jeremy Avigad (2002). An Ordinal Analysis of Admissible Set Theory Using Recursion on Ordinal Notations. Journal of Mathematical Logic 2 (1):91-112.
Andrés Villaveces (1998). Chains of End Elementary Extensions of Models of Set Theory. Journal of Symbolic Logic 63 (3):1116-1136.
Robert S. Lubarsky & Michael Rathjen (2008). On the Constructive Dedekind Reals. Logic and Analysis 1 (2):131-152.
John P. Burgess (2004). E Pluribus Unum: Plural Logic and Set Theory. Philosophia Mathematica 12 (3):193-221.
Bairj Donabedian (2003). The Natural Realm of Social Law. Sociological Theory 21 (2):175-190.
Itay Neeman & Jindřich Zapletal (2001). Proper Forcing and L(ℝ). Journal of Symbolic Logic 66 (2):801-810.
Mark F. Sharlow (1987). Proper Classes Via the Iterative Conception of Set. Journal of Symbolic Logic 52 (3):636-650.
Added to index2009-01-28
Total downloads29 ( #143,805 of 1,934,803 )
Recent downloads (6 months)1 ( #434,672 of 1,934,803 )
How can I increase my downloads?