How far can we go with the hypothesis of a bijective class function between the proper classes On and V?

Question

1- Building upon the classical first order logic with equality, and using the language of NBG, we work in the Set theory with (i): Extensionality (ii) Foundation; (iii): Union, (iv); The existence of a bijective class-function F between the proper class On of ordinal numbers and the universal class V of all sets. E will be the membership relation and G will be the inverse of F from V onto On.

2-The function F gives an isomorphism of well-ordered classe between (On, E) and (V ,W), where F(a)WF(b) is equivalent with aEb. E is setlike, and for z=F(b), the class of W predecessors of z contains exactly the sets F(a) with aEb. But replacement seems necessary to prove that W is setlike.

3- But, using Foundation, so that every set x has a rank r(x) that is an ordinal a, we can modify W to build a setlike well-order WS on V,defined so that xWSy iff r(x) < r(y) or r(x)=r(y) and xWy. Every class A is well-orderd by the relativisation of W and WS on A, and W and WS agree on every rank level set W(a). The order WS on V is the well-ordered sum of the well-ordered sets W(a).

4- For every class A, let F(A)= inf (A/W), then F(A) is a member set of A, and F is a global choice function on V, whose restriction to every class B is a choice function on B.

5- Now, let A be a class and G/A be the restriction of the function G on A and let G be the image of A by G/A. G is a class of ordinal numbers, and so is the class H(A) that is the union of G. Two cases are possible for H(A).

6- If H(A)=On, H(A) is a proper class of ordinals, so an unbounded and cofinal class in (On,E). Because H(A) is the union of the class of ordinals G, H(A) must be included in G, that is included in On, so that G must be On. If G=On, clearly H(A)=On, so that both are equivalents.

7- If H(A) is not =On, then there is an ordinal b such that H(A)=b, so that H(A) is a set.

8- If all of this is correct, we have that a class of ordinals is a proper class if it is an unbounded and cofinal class in (On,E), and moreover two such classes must be bijective.

9- Now consider the bijective function G/A from A onto G. If G is a proper class, G is bijective with On, that is bijective with V by G. So that A must be bijective with V, and A cannot be a set, because being bijective with On, the well-ordered class A cannot be with any ordinal number. So A is also a proper class.

10- We then have that every proper class is bijective with V (and On), so that two proper classes are always bijective.

11- Let B be a set and J a surjective function from B onto a class A. Suppose that A is a proper class, then we obtain a surjection from B onto V, so that using global choice we can build a bijection between B and V, so B would be a proper class. This proves that the class A cannot be a proper class, and must be a set. We have that for every function defined on a set A, the image B of A must be a set. This is the replacement axiom.

QUESTION : What is wrong in this development ?

Answer 1

I/ I will work in NBG set theory, as developped in Gödel's "The consistency of the continuum hypothesis", without the axiom E of "global choice, so using the axiom groups A, B, C and D, and adding to these the axiom F of "restricted axiom of limitation of size", meaning the existence of a bijection between On and V, where On is the proper class of all ordinal numbers and V is the proper class of all sets. II/ Ideally, I would like to work using only Extensionality, Fondation, the axiom G of Universes (Tarski-Grothendieck axiom, from which the axioms of Power set, Infinity and choice can be derived) and the axiom F (from which the axioms of global setlike well-order, global choice, replkacement,and so separation and empty set, and also union as azriel Lévy proved, and also full Limitation of size, see my point IV, can be derived), but I do not know how to avoid a vicious circle regarding replacement and union. III/ If full limitation of size is true, then so is restricted limitation of size. IV/ Now suppose axiom F is true andlet F be a bijective function from On onto V, G the inverse of F bijective function from V onto On, and E the membership relation on classes (and sets).
By foundation, every set x has a rank r(x) that is an ordinal number a. Let R(A) be the class of the ranks of the class A, V(a) be the SET of all sets whose rank is r(x) <= a and U(a) be the set of all sets with rank r(x) = a. Let W be the relation on V defined by xWy iff f(x)EF(y); F is an isomorphism of well-ordered classes between <On, E> and <V,W>, and moreover E being setlike, W must be setlike. Another setlike wellordered relation on V is WS defined as xWSy iff (r(x) < r(y) OR (r(x)=r(y) AND xWy). W and WS agree on every set U(a), their relativizations to On are E, and WS is the well-ordree sum of the well-orders on the sets u(a) indexed by On. If A is a set, R(A) is a set by replacement, so a bouded class in On. So that if A is a proper class, R(A) must be an unbounded class, so a cofinal class in On. But every ordinal class (transitive class well-ordered by E) must be an (set) ordinal number or all of the class On. So the class of ordinal numbers R(A),well-ordered by On, must be isomorphic with <On,E>, giving a surjection from A into O. Every class A is setlike well-ordered by the restriction of WS to A, giving the global axiom of setlike well-order, and if we take L(A) to be the least WS member set of A, L(A) is a set member in the class A, giving the axiom of gobal choice. Now let <A,WS> be a proper setlike well-ordered class, and let G(A) be the image of the proper class A in the class On. Let H be the function from V into VxOn defined by H(x) = <x,r(x)>. H is injective, and if A is a proper class R(A) is a proper class, so that H(A) must be a proper class, and G(A), being the first projection of H(A) must be a proper class by contraposition of the axiom of replacement. Now, G(A) being a proper class of ordinals is cofinal in On. Let K be the enumeration function of G(A), defined by the recursion K(a) = the least ordinal in On that is not the image of an ordinal b in G(A) such that b<a, for every a in G(A). K is an increasing injection from G(A) into On, whose image is an initial section in <On,E> that ca

Answer 2

K is an increasing function from; G(A) into On, so an injection, whose image in an initial section of <On,E> that cannot be a set. So that K must be a bijection from K(A) onto On. Now, chaining the bijections G from A to On and K, we obtain a bijection from A to On. And F being a bijection between On and V, we get that for every proper class A we can build a bijections between A and On and A and V, so that given any two proper classes A and B, we can build a bijection between A and B, thus obtaining the full axiom of limitation of sieze.