Non-Paradoxes of Set Theory and the Diagonal Method

The requirement for axiomatic set theory rests on the belief that the unrestricted comprehension of sets can result in sets that are paradoxical and therefore unrestricted comprehension needs to be tamed. Some key paradoxes of Burali-Forti, Russell and Cantor are examined and shown not to be paradoxes. Additionally, a common technique known as the diagonal method is shown to never construct a contradictory sequence. Then it is shown that any infinite set can be placed in a bijection with its power set by explicitly constructing a bijection and that the real numbers and natural numbers have the same cardinality. Finally, semantic diagonal techniques are considered with particular reference to Godel's incompleteness theorem and the halting problem.