ABSTRACT
Set theory suffers from the ambitions of its friends. The Logicists hoped to reduce the whole of mathematics to set theory, regarding it as part of logic, with sets being the successors of Frege’s difficult concept of Umfang, usually translated ‘extension’,1 was construed as meaning ‘set’, and cardinal numbers were construed by the Logicists as sets of equinumerous sets. After Russell showed naive set theory to be inconsistent, the axiomatized versions had a much weaker claim to be accounted part of logic. Most of modern set theory is better described as transfinite arithmetic. And whereas Frege’s programme of defining the natural numbers 0, 1, 2, etc.
