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Is a class of ordinals actually a set, or can it be a proper class? If it's a proper class, is f really a function or some generalized notion that allows classes?

What is the domain of an enumerating function? All the ordinals? (if so, f can't actually be a function)

What is an order type?

What is \kappa? Some ordinal, I assume?

In the definition of \kappa-continuous, should the result be true for every N less than \kappa?

Parting words from the person who closed the correction: 
Is a class of ordinals actually a set, or can it be a proper class? If it's a proper class, is f really a function or some generalized notion that allows classes? > Nope, it's defined as a subset, not a subclass. What is the domain of an enumerating f
Status: Accepted
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Comments

Thanks for the explanation, but the entry still seems to be just as unclear. Perhaps it is only intended to be comprehensible to specialists?

mathcam wrote:

> Nope, it's defined as a subset, not a subclass.

It shouldn't be defined as a subset, since it can be a proper class. This is why the domain of the enumerating function can be On (the class of all ordinals) - this wouldn't be possible if the subclass was a set.

Can you really define a function whose domain is a proper class? Won't there be too many functions? (I can't quite see how to explicitly construct Russell's paradox, but it seems possible.)

archibal:

> Can you really define a function
> whose domain is a proper class?

The way I would define a function, the domain is necessarily a set. But the article is using the word 'function' in a more general sense.

For an example, see my article AlephNumbers. Each ordinal $\alpha$ maps to a cardinal $\aleph_\alpha$. I haven't called this map a function, because as far as I'm concerned it isn't - but it's precisely the sort of thing that this article is talking about when it uses the word 'function' - it's the 'enumerating function' for the class of infinite cardinals. In ZFC there are no proper classes, so the 'function' is really just a suitable predicate.

> Won't there be too many functions?
> (I can't quite see how to explicitly construct
> Russell's paradox, but it seems possible.)

The problem doesn't arise in ZFC, since these 'functions' are not actual objects (sets), they are just a way of talking about things. I'm not sure what happens in set theories which have proper classes.

> archibal:
>
> > Can you really define a function
> > whose domain is a proper class?
>
.
.
.
> The problem doesn't arise in ZFC, since these 'functions'
> are not actual objects (sets), they are just a way of
> talking about things. I'm not sure what happens in set
> theories which have proper classes.

In ZFC and similar set theories, talk of proper classes is a sort of code that allows one to avoid frequent distracting references to the syntax of the formal languages in which these theories are couched. Azriel Levy provides rules for decoding in painful detail in his _Basic Set Theory_ ( available from Dover). In theories in which proper classes are "first class" objects, there is no problem with functions that are proper classes. What distinguishes sets from proper classes is that every set is a member of some class, while no proper class is a member of any class. The members of the identity function on On, for example, are just the pairs < a, a > of ordinals, and each of these is a set.

The last remark of the entry:

All these definitions can be easily extended to all ordinals: a class is \emph{closed} (resp. \emph{unbounded}) if it is $\kappa$-closed (unbounded) for all $\kappa$. A function is \emph{continuous} (resp. \emph{normal}) if it is $\kappa$-continuous (normal) for all $\kappa$.

is unnecessary since On is clearly a subclass of On.

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