Definition of a function in NBG set theory

[echinuz]

Hi,
I have a situation where I want to define a function over all possible rings. For example, I would like to define a function that accepts a ring as an input and returns its additive identity. However, this seems impossible to do in ZF set theory since we can not define the domain of this function. In other words, we can not generate the set {x:isring(x)} since the axiom of separation requires us to restrict x to some set S, {x \in S:isring(x)}. In this case, we would need S to be the set of all sets which is forbidden. However, this sort of action does seem permitted in NBG set theory since class comprehensions allow us to create the class of all rings {x:isring(x)}. Assuming this is correct, we return to our original problem which was to define a function whose input is a ring. What is a function is NBG set theory? In ZF set theory, we typically define a function as a relation between two sets that has certain properties. What is the analogy with classes? Is there a good reference for these sort of constructs?

 

[Preno]

Hi,
I have a situation where I want to define a function over all possible rings. For example, I would like to define a function that accepts a ring as an input and returns its additive identity. However, this seems impossible to do in ZF set theory since we can not define the domain of this function. In other words, we can not generate the set {x:isring(x)} since the axiom of separation requires us to restrict x to some set S, {x \in S:isring(x)}. In this case, we would need S to be the set of all sets which is forbidden. However, this sort of action does seem permitted in NBG set theory since class comprehensions allow us to create the class of all rings {x:isring(x)}. Assuming this is correct, we return to our original problem which was to define a function whose input is a ring. What is a function is NBG set theory? In ZF set theory, we typically define a function as a relation between two sets that has certain properties. What is the analogy with classes? Is there a good reference for these sort of constructs?

You can define a class function in ZFC. Since NBG is a conservative extension of ZFC, you will get the same results.

In your example, you just define a class function from the class of rings to the class of sets. Afaik, functions are defined the same way in ZFC and NBG.