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So, I'm not really asking whether classes (like the "set" of all classes) is countable, because it's obviously not the case, since all sets are classes and sets are clearly uncountable. At the same time, classes are defined to be formulas and the set of all formulas in a formal language is countable. What am I missing?

Elvis
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  • Just because the collection of sentences is countable doesn't tell you how many elements of a set theoretic universe satisfy a particular sentence in that collection. The recipe is not the food. – Eric Towers Jan 28 '25 at 02:49
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    Do you mean whether the collection of all classes is countable or not? This depends on what exactly is meant by "class". If a class meant ${x|\varphi(x)}$ for some first-order formula $\varphi$ without parameters, then yes, the collection of all classes is countable externally, because there are only countably many first-order formulas without parameters. But, first, there's no reason the collection of all sets is uncountable externally (after all, if ZFC has a model, it must also have a countable model). And, even if the collection of all sets is uncountable externally, if classes are… – David Gao Jan 28 '25 at 03:08
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    … defined as before, not all sets are classes anymore. Only those sets that are definable without parameters are classes. And the collection of definable sets is indeed countable externally, so there's no contradiction. If you want all sets to be classes, then you need ${x|\varphi(x)}$ where $\varphi$ is a first-order formula with parameters to be a class as well. But because there are possibly uncountably many parameters to use, the (external) set of first-order formulas with parameters can be uncountable externally, so there is again no contradiction. – David Gao Jan 28 '25 at 03:11
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    (The main thing here is that, when talking about things like "classes", "formulas", and their countability/uncountability, one should exercise caution and be clear about definitions, as well as be careful with what happens internally vs. externally.) – David Gao Jan 28 '25 at 03:16
  • @DavidGao how can ZFC have a countable model if using it we can build $\Bbb R$, that is, an uncountable collection of sets? – Elvis Jan 28 '25 at 03:21
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    @Elvis This is what I meant when I said be careful with what happens internally vs. externally. In ZFC, you can of course prove $\mathbb{R}$ is uncountable, but this is an internal theorem. It just means there does not exist a bijection between the internal set $\mathbb{R}$ and the internal set $\mathbb{N}$, within the model. But there is no reason why there cannot be such a bijection outside the model. (Nor, for that matter, is there any reason why the internal $\mathbb{N}$ even has to be countable externally.) – David Gao Jan 28 '25 at 03:27
  • @DavidGao I'm very interested, but very ignorant, about this topic. Is there any resource where I can learn more, especially about this internal/esternal thing? (Right now I'm reading this.) – Elvis Jan 28 '25 at 03:30
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    @Elvis You're already reading a text on model theory. I'm not aware of any resources discussing this topic, nor do I think such resources are necessary. It's just a matter of thinking through what exactly model theory theorems imply in the context of ZFC, especially Löwenheim-Skolem and compactness theorems. (You can, of course, read a text on axiomatic set theory as well. I just don't know if any of them addresses this topic.) – David Gao Jan 28 '25 at 03:40
  • @DavidGao okay, thanks. Also, what do you mean by "externally countable" exactly? – Elvis Jan 28 '25 at 03:42
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    @Elvis You're studying a model $M$ of ZFC, so there must be some background theory $T$ (usually called metatheory) which allows such study. A class is a subset of the said model $M$, so it is possible to ask whether this set is countable, in the metatheory (instead of within the model $M$). "External" just means "in the metatheory", while "internal" just means "looking within the model $M$". – David Gao Jan 28 '25 at 03:47

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