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Truth Definitions, Skolem Functions and Axiomatic Set Theory

Published online by Cambridge University Press:  15 January 2014

Jaakko Hintikka
Jaakko Hintikka*
Affiliation:
Department of Philosophy, Boston University, 745 Commonwealth Avenue, Boston, MA 02215, USA.E-mail: [email protected]
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Extract

§1. The mission of axiomatic set theory. What is set theory needed for in the foundations of mathematics? Why cannot we transact whatever foundational business we have to transact in terms of our ordinary logic without resorting to set theory? There are many possible answers, but most of them are likely to be variations of the same theme. The core area of ordinary logic is by a fairly common consent the received first-order logic. Why cannot it take care of itself? What is it that it cannot do? A large part of every answer is probably that first-order logic cannot handle its own model theory and other metatheory. For instance, a first-order language does not allow the codification of the most important semantical concept, viz. the notion of truth, for that language in that language itself, as shown already in Tarski (1935). In view of such negative results it is generally thought that one of the most important missions of set theory is to provide the wherewithal for a model theory of logic. For instance Gregory H. Moore (1994, p. 635) asserts in his encyclopedia article “Logic and set theory” that

Set theory influenced logic, both through its semantics, by expanding the possible models of various theories and by the formal definition of a model; and through its syntax, by allowing for logical languages in which formulas can be infinite in length or in which the number of symbols is uncountable.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 4 , Issue 3 , September 1998 , pp. 303 - 337
Copyright
Copyright © Association for Symbolic Logic 1998

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References

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