With the discovery of such mathematical paradoxes as the Burli-Forti paradox, Russell’s paradox, Cantor’s paradox and Skolem’s paradox by early 1930’s as Bunch notes, Hilbert’s program did not succeed such that “disagreement about how to eliminate contradictions were replaced by discussions of how to live with contradictions in mathematics.” Attempts to avoid the paradoxes led to other paradoxical notions but most mathematicians rejected these notions. Thus the present situation is that mathematics cannot be formulated, except in axiomatic theory, without contradictions without the loss of useful results. With regard to axiomatic theory, this cannot be proven to be consistent with the result that paradoxes can occur at any time.