How can I quantify over the class of all cardinalities?

Question

I'd like to quantify over all cardinalities of sets.

My end goal is to make a category-theoretic arguement:

For all cardinalities of sets, in the category of sets with maps as morphisms: the subclass of objects with that given cardinality is closed under the class of isomorphisms, thereby forming a subcategory, for each set cardinality, where the morphisms are isomorphisms.

Can I quantify over a class that isn't a set?

Is there a good way to specify this class of cardinalities?

Is there a better way to go about this?

Answer 1

Your best bet is to prove something like $$\forall x \forall y(x \cong y \to |x| = |y|)$$ This is possible without quantifying over all cardinals (and, depending on your definition of cardinality, is a somewhat trivial result).

The fact that the class of sets of a given cardinality is closed under isomorphisms (bijections) is immediate: if $\kappa$ is a cardinal then, given any set $x$ of cardinality $\kappa$, if $y$ is another set and $x \cong y$ then $y$ also has cardinality $\kappa$, so lies in the class. (This is a very verbose rewording of the above expression though.)

Answer 2

In $\sf ZFC$ classes are really just a shorthand for formulas. That is when we say that $a\in A$ we really say that some $\varphi(x)$ is a formula in the language of set theory and $\sf ZFC\vdash\varphi(a)$.

So to quantify over a class $A$ we really just have a formula $\varphi$ and we say that $\forall x(\varphi(x)\rightarrow\ldots)$, or in simpler terms $\forall x\in A(\ldots)$.

Similarly the cardinals are a well-defined class in $\sf ZFC$. It is the class of ordinals which do not have an injection into a smaller ordinal; but since smaller ordinals are elements of larger ordinals, it means that we can say that there is no injection from $x$ into any of its elements.

In $\sf ZF$ the definition is slightly longer and more complicated, but we can do that as well. You just have to write the definition of a cardinal, which is longer, and then you take the class of all those satisfying this definition.

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I suddenly realized a possible source for confusion. When we say "quantify over something" we have two meanings, which are almost always clear from context:

1. That something is the object which is being quantified. For example "Quantify over $x$", meaning that $\forall x\varphi(x)$ and so on.

2. That something is the collection of objects which we are interested in. For example, "Quantify over all the real numbers" - $\forall x\in\Bbb R$

Classes are not objects, in $\sf ZFC$ anyway, so we cannot quantify over them in the first sense, but in the second sense we can, because of what I wrote above. Quantifying over all the cardinals is in the second sense of the term, rather than the first.