Andreas Blass states that proper classes do not exist and emphasizes that this is only his philosophical opinion, and not a mathematical fact.(link text).

Is it really not a mathematical fact? I think there are some mathematical results that justify his philosophical opinion. Levy and Vaught prove that Ackermann's set theory proves the existence of the classes {V}, P{V}, PP{V},....(Pacific Journal of Mathematics, 11:1045-1062, 1961). Furthermore, Reinhardt proves that Ackermann's set theory equals ZF (Annals of Mathematical Logic, 2:189-249). My understanding of these results is that anything we can prove in Ackermann's set theory, we can prove in ZF as well. There is no need to assume the existence of the classes P{V}, PP{V}, PPP{V},...because there is nothing new math fact to obtain.

Is my understanding correct?

Andreas Blass states that proper classes do not exist and emphasizes that this is only his philosophical opinion, and not a mathematical fact.(link text).

Is it really not a mathematical fact? I think there are some mathematical results that justify his philosophical opinion. Levy and Vaught prove that Ackermann's set theory proves the existence of the classes {V}, P{V}, PP{V},....(Pacific Journal of Mathematics, 11:1045-1062, 1961). Furthermore, Reinhardt proves that Ackermann's set theory equals ZF (Annals of Mathematical Logic, 2:189-249). My understanding of these results is that anything we can prove in Ackermann's set theory, we can prove in ZF as well. There is no need to assume the existence of the classes P{V}, PP{V}, PPP{V},...because there is nothing new math fact to obtain.

Is my understanding correct?