An alternative foundation and the generalized continuum hypothesis

Abstract

In this paper I introduce a new and intuitive first-order foundational theory (where the concept of set is not primitive) and use it to show that the power set of an infinite set does not exist. In particular, proofs of uncountability of a set are essentially proofs of the non-existence of that specified set. In a certain sense, uncountability is shown to be a form of incompleteness. Also, the axiom of choice is shown to be a straightforward theorem. In view of the non-existence of a set of all real numbers or more generally the non-guaranteed existence of the completion of a metric space, topological concepts are re-introduced in the context of "extensions". Measure theory is also reformulated accordingly.