Alternative theories and higher infinite

pp. 229-260

in: Daniel Parrochia, Pierre Neuville, Towards a general theory of classifications, Berlin, Springer, 2013

Abstract

We have presented in the previous chapter, a theory able to generate the infinite set of classifications as the continuum, each classification being in a one-to-one correspondence with a real number. But, as we know, there are, in the mathematics of the infinite, since the works of Cantor, Suslin and others, a lot of possible views of the continuum. Moreover, since the undecidability results of Cohen, there exist also a lot of possible set theories. So, it may be useful to ask some questions about what happens concerning the existence of classifications in those alternative theories, in particular, when they admit higher forms of the infinite. Though the risk is obvious, there, to end up at some undecidability results, several arguments speak for such an extension.