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I was wondering about this. In Conway's "On Numbers and Games", he discusses the "surreal numbers", and in one point mentions that they are full of "gaps". That the surreal number line is riddled with gaps. Namely, what he mentions is that these gaps occur for "cuts" between proper classes of surreal numbers, whereas ordinary surreal numbers are cuts between sets.

He then goes and mentions how that we cannot collect these together, it would be an "illegal" (undefined?) object in conventional set theory. Which makes me wonder -- could there exist some greater, more powerful form of set theory that could enable this kind of "higher-order collection" to exist? And then we could talk about the properties of "all surreal number plus all gaps in a single continuum". Or is there a good, fundamental reason that this simply cannot be done? If so, what is it? And if it can be done, what kind of properties would this monster have, anyways?

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See here and here. –  Michael Greinecker Nov 10 '12 at 4:19
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up vote 3 down vote accepted

In ZFC proper classes are not objects, and therefore we cannot have a class of classes. This is why this is forbidden.

You can extend ZFC to allow classes and 2-classes (classes of classes), but this makes things complicated and very delicate. Instead a common cure for the problem is to assume there exists a set model of ZFC (e.g. if there is an inaccessible cardinal), and talk about the surreals of that model from an external fashion. Namely the collection of all cuts is a set in the universe, but not a definable collection within that model. Again, this is full of delicate points.

There is one trick we can use, as mentioned in my answer linked by Michael Greinecker in the comments. We can canonically "cut" a non-empty piece from each equivalence class, and collect those objects. This is known as Scott's trick and I wrote about it more here and here (and probably elsewhere on this site).

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