It is for infinitary algebra in Sections 5.6 and 6.2 that this axiom will be most missed in this book. We respond to this difficulty by examining what infinitary operations are of interest in practice. In the countable case the following assumption, formulated by Paul Bernays, is often more directly applicable. Similarly, we get Dependent Choice by repeatedly applying the choice function to the seed. Dependent Choice does not imply excluded middle or vice versa .
Category theory provides the technology for creating new worlds.
This is quite simple so long as we do not require them to be classical. Worlds have also been created with convenient but exotic properties. More recently, synthetic domain theory has similarly postulated that all functions are to be computable . Excluded middle was traditionally regarded as a true fact about the real world, so in order to investigate intuitionistic logic it was necessary to build fictional worlds where excluded middle does not hold. The point of view of this book is that these worlds are not exotic but quite normal, and their logic is perfectly typical, just as algebraic extensions of the rationals have come to be seen as ordinary number domains with straightforward arithmetic structure.