Bulletin of Symbolic Logic 18 (1):91-121 (2012)

Authors
Jouko A Vaananen
University of Helsinki
Abstract
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.
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DOI 10.2178/bsl/1327328440
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References found in this work BETA

Set Theory.Keith J. Devlin - 1981 - Journal of Symbolic Logic 46 (4):876-877.
Theory of Recursive Functions and Effective Computability.Hartley Rogers - 1971 - Journal of Symbolic Logic 36 (1):141-146.
Isomorphism and Higher Order Equivalence.M. Ajtai - 1979 - Annals of Mathematical Logic 16 (3):181.

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Citations of this work BETA

Internal Categoricity in Arithmetic and Set Theory.Jouko Väänänen & Tong Wang - 2015 - Notre Dame Journal of Formal Logic 56 (1):121-134.
Reverse Mathematics and Peano Categoricity.Stephen G. Simpson & Keita Yokoyama - 2013 - Annals of Pure and Applied Logic 164 (3):284-293.
Tracing Internal Categoricity.Jouko Väänänen - 2021 - Theoria 87 (4):986-1000.
WHAT CAN A CATEGORICITY THEOREM TELL US?Toby Meadows - 2013 - Review of Symbolic Logic (3):524-544.

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