From the point of view of a category theorist, I would say there are many fundamentally non-equivalent way of handling size questions - but they are in a completely different direction than what is mentioned in the question. To me, the only real distinctions between different approaches to size depend on the answer to practical questions like "what can you do with classes", "what can you do with small sets", "How many different sizes do you need", etc...

I wouldn't say that the fact (mentioned in the question) that in one case the cardinality of a set is bounded, and in the other it is not is a real difference: In both cases, the cardinality of any set is bounded by the cardinality of the class of all sets - it is just that what you call a "cardinal" in both case is not the same thing - in one case it refer to a small cardinal and not in the other. But that is not a real distinction - we are talking about the same thing but giving it different names.

But here are what I would consider three fundamentally different formalisms:

1. If your "classes" are axiomatized within ZFC by a "formula as classes" paradigm or by NBG set theory. then you can't do much with classes, the category of classes and maps between them is pretty much just a category with finite limits (A little more than that to be honest, let's say a "Heyting pretopos" to be in line with the literature on algebraic set theory). In many cases that's enough, but not always.

2. If you use MK set theory then you can build new class by comprehension using quantification over all class.

3. If you use something like inaccessible cardinal or Grothendieck universes, then your category of classes is a fully formed set theory. You can form the class of functions between two classes, the class of subclass of a class, etc... Your category of classes is an elementary topos (and even more).

These corresponds to fundamentally different theory with different consistency strength : (1) is ZFC or equivalent, (2) is stronger than ZFC and (3) is even stronger. And the difference are significant in practice - I can't think of an example where (2) is significant, but if you want to talk about the category of all endofunctor of $Set$, you need something like (3), while (1) only let you talk about individual functor $Set \to Set$.

But the differences inside a single group are just what I would call "linguistical" differences as the one I mentioned before, for example on whether the word "sets" is going to mean "small set" or "set or class", or on whether we actually have a category of all small sets or just a category equivalent to that.

Note that I'm not saying at all there are only three way of handling size issue - I'm just giving three example, but there are many other parameter you can play on:

• You can decide the number of different sizes (just "set" and "classes", or "set/class/super class", a countable number of size, or maybe even more than that).

• you can change what the category of small set need to satisfy: For example if "small sets" just needs to form an elementary topos with NNO (so only satisfies bounded replacement) then even if you ask your classes to be a full model of ZFC, you have a theory not logically stronger than ZFC. If you are asking even less properties of your small set them taking "small" to mean $\kappa$-small for some regular cardinal $\kappa$ might be enough and you have a proper class of different size at your disposition just within ZFC, and more freedom to apply your results.

From the point of view of a category theorist, I would say there are many fundamentally non-equivalent way of handling size questions - but they are in a completely different direction than what is mentioned in the question. To me, the only real distinctions between different approaches to size depend on the answer to practical questions like "what can you do with classes", "what can you do with small sets", "How many different sizes do you need", etc...

I wouldn't say that the fact (mentioned in the question) that in one case the cardinality of a set is bounded, and in the other it is not is a real difference: In both cases, the cardinality of any set is bounded by the cardinality of the class of all sets - it is just that what you call a "cardinal" in both case is not the same thing - in one case it refer to a small cardinal and not in the other. But that is not a real distinction - we are talking about the same thing but giving it different names.

But here are what I would consider three fundamentally different formalisms:

1. If your "classes" are axiomatized within ZFC by a "formula as classes" paradigm or by NBG set theory. then you can't do much with classes, the category of classes and maps between them is pretty much just a category with finite limits (A little more than that to be honest, let's say a "Heyting pretopos" to be in line with the literature on algebraic set theory). In many cases that's enough, but not always.

2. If you use MK set theory then you can build new class by comprehension using quantification over all class.

3. If you use something like inaccessible cardinal or Grothendieck universes, then your category of classes is a fully formed set theory. You can form the class of functions between two classes, the class of subclass of a class, etc... Your category of classes is an elementary topos (and even more).

These corresponds to fundamentally different theory with different consistency strength : (1) is ZFC or equivalent, (2) is stronger than ZFC and (3) is even stronger. And the difference are significant in practice - I can't think of an example where (2) is significant, but if you want to talk about the category of all endofunctor of $\mathsf{Set}$, you need something like (3), while (1) only let you talk about individual functor $\mathsf{Set} \to \mathsf{Set}$.

But the differences inside a single group are just what I would call "linguistical" differences as the one I mentioned before, for example on whether the word "sets" is going to mean "small set" or "set or class", or on whether we actually have a category of all small sets or just a category equivalent to that.

Note that I'm not saying at all there are only three way of handling size issue - I'm just giving three example, but there are many other parameter you can play on:

• You can decide the number of different sizes (just "set" and "classes", or "set/class/super class", a countable number of size, or maybe even more than that).

• you can change what the category of small set need to satisfy: For example if "small sets" just needs to form an elementary topos with NNO (so only satisfies bounded replacement) then even if you ask your classes to be a full model of ZFC, you have a theory not logically stronger than ZFC. If you are asking even less properties of your small set them taking "small" to mean $\kappa$-small for some regular cardinal $\kappa$ might be enough and you have a proper class of different size at your disposition just within ZFC, and more freedom to apply your results.

From the point of view of a category theorist, I would say there are many fundamentally non-equivalent way of handling size questions - but they are in a completely different direction than what is mentioned in the question. To me, the only real distinctions between different approaches to size depend on the answer to practical questions like "what can you do with classes", "what can you do with small sets", "How many different sizes do you need", etc...

I wouldn't say that the fact (mentioned in the question) that in one case the cardinality of a set is bounded, and in the other it is not is a real difference: In both cases, the cardinality of any set is bounded by the cardinality of the class of all sets - it is just that what you call a "cardinal" in both case is not the same thing - in one case it refer to a small cardinal and in not in the other, but that is not a real distinction - we are talking about the same thing but giving it different names.

But here is what I would consider three fundamentally different formalism

1. If your "classes" are axiomatized within ZFC by a "formula as classes" paradigm or by NBG set theory. then you can't do much with classes, the category of classes and maps between them is pretty much just a category with finite limits (A little more than that to be honest, let's say a "Heyting pretopos" to be in line with the literature on algebraic set theory). In many cases that's enough, but not always.

2. If you use MK set theory then you can build new class by comprehension using quantification over all class.

3. If you use something like inaccessible cardinal or Grothendieck universes, then your category of classes is a fully formed set theory. You can form the class of functions between two classes, the class of subclass of a class, etc... Your category of classes is an elementary topos (and even more).

These corresponds to fundamentally different theory with different consistency strength : (1) is ZFC or equivalent, (2) is stronger than ZFC and (3) is even stronger. And the difference are significant in practice - I can't think of an example where (2) is significant, but if you want to talk about the category of all endofunctor of $Set$, you need something like (3), while (1) only let you talk about individual functor $Set \to Set$.

But the differences inside a single group are just what I would call "linguistical" differences as the one I mentioned before, for example on whether the word "sets" is going to mean "small set" or "set or class", or on whether we actually have a category of all small sets or just a category equivalent to that.

Note that I'm not saying at all there are only three way of handling size issue - I'm just giving three example, but there are many other parameter you can play on:

• You can decide the number of different sizes (just "set" and "classes", or "set/class/super class", a countable number of size, or maybe even more than that).

• you can change what the category of small set need to satisfy: For example if "small sets" just needs to form an elementary topos with NNO (so only satisfies bounded replacement) then even if you ask your classes to be a full model of ZFC, you have a theory not logically stronger than ZFC. If you are asking even less properties of your small set them taking "small" to mean $\kappa$-small for some regular cardinal $\kappa$ might be enough and you have a proper class of different size at your disposition just within ZFC, and more freedom to apply your results.

From the point of view of a category theorist, I would say there are many fundamentally non-equivalent way of handling size questions - but they are in a completely different direction than what is mentioned in the question. To me, the only real distinctions between different approaches to size depend on the answer to practical questions like "what can you do with classes", "what can you do with small sets", "How many different sizes do you need", etc...

I wouldn't say that the fact (mentioned in the question) that in one case the cardinality of a set is bounded, and in the other it is not is a real difference: In both cases, the cardinality of any set is bounded by the cardinality of the class of all sets - it is just that what you call a "cardinal" in both case is not the same thing - in one case it refer to a small cardinal and not in the other. But that is not a real distinction - we are talking about the same thing but giving it different names.

But here are what I would consider three fundamentally different formalisms:

1. If your "classes" are axiomatized within ZFC by a "formula as classes" paradigm or by NBG set theory. then you can't do much with classes, the category of classes and maps between them is pretty much just a category with finite limits (A little more than that to be honest, let's say a "Heyting pretopos" to be in line with the literature on algebraic set theory). In many cases that's enough, but not always.

2. If you use MK set theory then you can build new class by comprehension using quantification over all class.

3. If you use something like inaccessible cardinal or Grothendieck universes, then your category of classes is a fully formed set theory. You can form the class of functions between two classes, the class of subclass of a class, etc... Your category of classes is an elementary topos (and even more).

These corresponds to fundamentally different theory with different consistency strength : (1) is ZFC or equivalent, (2) is stronger than ZFC and (3) is even stronger. And the difference are significant in practice - I can't think of an example where (2) is significant, but if you want to talk about the category of all endofunctor of $Set$, you need something like (3), while (1) only let you talk about individual functor $Set \to Set$.

But the differences inside a single group are just what I would call "linguistical" differences as the one I mentioned before, for example on whether the word "sets" is going to mean "small set" or "set or class", or on whether we actually have a category of all small sets or just a category equivalent to that.

Note that I'm not saying at all there are only three way of handling size issue - I'm just giving three example, but there are many other parameter you can play on:

• You can decide the number of different sizes (just "set" and "classes", or "set/class/super class", a countable number of size, or maybe even more than that).

• you can change what the category of small set need to satisfy: For example if "small sets" just needs to form an elementary topos with NNO (so only satisfies bounded replacement) then even if you ask your classes to be a full model of ZFC, you have a theory not logically stronger than ZFC. If you are asking even less properties of your small set them taking "small" to mean $\kappa$-small for some regular cardinal $\kappa$ might be enough and you have a proper class of different size at your disposition just within ZFC, and more freedom to apply your results.