Bulletin of Symbolic Logic

Second order logic or set theory?

Jouko Väänänen

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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.

Article information

Source
Bull. Symbolic Logic, Volume 18, Issue 1 (2012), 91-121.

Dates
First available in Project Euclid: 23 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.bsl/1327328440

Digital Object Identifier
doi:10.2178/bsl/1327328440

Mathematical Reviews number (MathSciNet)
MR2798269

Zentralblatt MATH identifier
1252.03024

Citation

Väänänen, Jouko. Second order logic or set theory?. Bull. Symbolic Logic 18 (2012), no. 1, 91--121. doi:10.2178/bsl/1327328440. https://projecteuclid.org/euclid.bsl/1327328440


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