**Akihiro Kanamori**

Boston University

## "The Empty Set, the Singleton, and the Ordered Pair"

For the modern set theorist the empty set 0 , the singleton {a} ,
and the ordered pair <x,y> are at the beginning of the systematic, axiomatic
development of set theory, both as a field of mathematics and as a unifying
framework for ongoing mathematics. These notions are the simplest building
blocks in the abstract, generative conception of sets advanced by the initial
axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long
before the complexities of Power Set, Replacement, and Choice are broached in
the formal elaboration of the `set of'{} operation. So it is surprising that,
while these notions are unproblematic today, they were once sources of
considerable concern and confusion among leading pioneers of mathematical
logic like Frege, Russell, Dedekind, and Peano. In the development of modern
mathematical logic out of the turbulence of 19th Century logic, the emergence
of the empty set, the singleton, and the ordered pair as clear and
elementary set-theoretic concepts serves as a motif that reflects, if not
illuminates, larger and more significant developments in mathematical logic:
the shift from the intensional to the extensional viewpoint, the development
of type distinctions, the logical vs. the iterative conception of set, and
the emergence of various concepts and principles as distinctively set-theoretic
rather than purely logical. Here there is a loose analogy with Tarski's
recursive definition of truth for formal languages: The mathematical interest
lie mainly in the procedure of recursion and the attendant formal semantics
in model theory, whereas the philosophical interest lies mainly in the
basis of the recursion, truth and meaning at the level of basic predication.

## Tuesday, March 5, 2002

SST 777

4:30 p.m.

*Refreshments will be served*

©