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From time to time, I see proper classes being endowed with algebraic structure. The ordinals with addition is one example, but I've seen a lot more, most of which have been above my head. The standard definitions of standard algebraic structures impose the requirement that the underlying class be a set though. I'm not exactly sure I know why that is. What kind of problems does removing this requirement cause? How much of algebra over sets carries over to algebra over classes?

I'm asking this because I don't know anything about proper classes other than what the first introductory course in foundations told me, plus some other bits. This makes me uneasy whenever I see or hear someone do anything with proper classes. I'm often left unconvinced about the rigor of such considerations because of this uneasiness.

Bartek
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Set theories that explicitly embrace proper classes as part of the first-order theory do so by allowing formulas to define and mention classes in restricted ways. Provided the algebraic relations on a proper class, e.g. an ordering in the case of ordinals, observe these restrictions, there is no lack of rigor in the presentation.

In some respects it does tie our hands, and so warrants caution in how we define operations. For example an operation that converts nonempty collections of ordinals into the least such ordinal in such a collection cannot be realized as a function using Kuratowski ordered pairs, since this would entail proper classes being elements of something. However the mapping relation can be expressed as a first-order formula, which is likely all that one needs to make a rigorous argument.

hardmath
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