Study on the Proof of the Existence of Transfinite Cardinals Strictly Smaller than with an Ensuing Solution to the Twin Prime Conjecture

In this paper the author submits a proof using the Power Set relation for the existence of a transfinite cardinal strictly smaller than , the cardinality of the Naturals. Further, it can be established taking these arguments to their logical conclusion that even smaller transfinite cardinals exist. In addition, as a lemma using these new found and revolutionary concepts, the author conjectures that some outstanding unresolved problems in number theory can be brought to heel. Specifically, a proof of the twin prime conjecture is given.