a1 Department of Pure Mathematics, University of Sydney, Sydney, 2006, Australia
A Dedekind-finite set is one not equinumerous with any of its proper subsets; it is well known that the axiom of choice implies that all such sets are finite. In this paper we show that in the absence of the axiom of choice it is possible to construct Dedekind-finite sets which are large, in the sense that they can be mapped onto large ordinals; we extend the result to proper classes. It is also shown that the axiom of choice for countable sets is not implied by the assumption that all Dedekind-finite sets are finite.
(Received September 25 1972)
(Revised December 19 1972)
1 The results in §§3 and 4 of this paper are taken from the author's Ph.D. thesis (University of Bristol 1971, supervised by Dr F. Rowbottom). The author held a Monash University Travelling Scholarship while the research for the thesis was carried out.