**Some Thoughts About the Axiom of Choice**

The axiom of choice is the most controversial axiom of mathematics.
Together with the continuum hypothesis it formes the first of
Hilbert's problems. Though the logical status of these two statements
has been settled by the work of Goedel and Cohen, the question of
their truth (or, if you prefer, validity) remains unsettled.
This situation leads to several fundamental questions regarding the
foundation of mathematics. E.g.:
Should the axiom of choice be regarded as valid - as done in almost
all modern mathematical textbooks ?
If not, should or can it be replaced by some alternative axiom ?
Is there only one mathematical truth ?
If familiar theorems collapse under the rejection of the axiom of
choice, could it also happen that desirable statements that are false
in the presence of the axiom of choice become true in its absence ?
The purpose of the talk is to present a short history of the problems
involved, to outline the relationship between the axiom of choice and
the continuum hypothesis and to discuss such questions as those
raised above.

*Horst Herrlich*