David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Bulletin of Symbolic Logic 18 (1):91-121 (2012)
We try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to be in manifest conflict with each other. However, our conclusion is that it is very difficult to see any real difference between the two. We analyze a phenomenon we call internal categoricity which extends the familiar categoricity results of second order logic to Henkin models and show that set theory enjoys the same kind of internal categoricity. Thus the existence of non-standard models, which is usually taken as a property of first order set theory, and categoricity, which is usually taken as a property of second order axiomatizations, can coherently coexist when put into their proper context. We also take a fresh look at complete second order axiomatizations and give a hierarchy result for second order characterizable structures. Finally we consider the problem of existence in mathematics from both points of view and find that second order logic depends on what we call large domain assumptions, which come quite close to the meaning of the axioms of set theory
|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
Stephen G. Simpson & Keita Yokoyama (2013). Reverse Mathematics and Peano Categoricity. Annals of Pure and Applied Logic 164 (3):284-293.
Jouko Väänänen (2015). Second‐Order Logic and Set Theory. Philosophy Compass 10 (7):463-478.
Catarina Dutilh Novaes (forthcoming). Axiomatizations of Arithmetic and the First-Order/Second-Order Divide. Synthese.
Jouko Väänänen (2015). Categoricity and Consistency in Second-Order Logic. Inquiry 58 (1):20-27.
Toby Meadows (2013). WHAT CAN A CATEGORICITY THEOREM TELL US? Review of Symbolic Logic (3):524-544.
Similar books and articles
Gabriel Uzquiano (2002). Categoricity Theorems and Conceptions of Set. Journal of Philosophical Logic 31 (2):181-196.
Jouko Vaananen (2001). Second-Order Logic and Foundations of Mathematics. Bulletin of Symbolic Logic 7 (4):504-520.
Ignacio Jané (1993). A Critical Appraisal of Second-Order Logic. History and Philosophy of Logic 14 (1):67-86.
S. Shapiro (2012). Higher-Order Logic or Set Theory: A False Dilemma. Philosophia Mathematica 20 (3):305-323.
Gregory H. Moore (1980). Beyond First-Order Logic: The Historical Interplay Between Mathematical Logic and Axiomatic Set Theory. History and Philosophy of Logic 1 (1-2):95-137.
Jaakko Hintikka (2009). A Proof of Nominalism: An Exercise in Successful Reduction in Logic. In A. Hieke & H. Leitgeb (eds.), Reduction - Abstraction - Analysis. Ontos
Agustin Rayo (1999). Toward a Theory of Second-Order Consequence. Notre Dame Journal of Formal Logic 40 (3):315-325.
Yuri Gurevich & Saharon Shelah (1983). Interpreting Second-Order Logic in the Monadic Theory of Order. Journal of Symbolic Logic 48 (3):816-828.
B. Hale (2013). Properties and the Interpretation of Second-Order Logic. Philosophia Mathematica 21 (2):133-156.
Helen Morris Cartwright (1993). On Plural Reference and Elementary Set Theory. Synthese 96 (2):201 - 254.
Ignacio Jané (1988). Lógica Y Ontología. Theoria 4 (1):81-106.
John E. Hutchinson (1976). Order Types of Ordinals in Models of Set Theory. Journal of Symbolic Logic 41 (2):489-502.
Steve Awodey, Carsten Butz & Alex Simpson (2007). Relating First-Order Set Theories and Elementary Toposes. Bulletin of Symbolic Logic 13 (3):340-358.
Added to index2012-01-24
Total downloads40 ( #83,639 of 1,726,249 )
Recent downloads (6 months)5 ( #147,227 of 1,726,249 )
How can I increase my downloads?