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Topic: Using classes instead of sets
Replies: 26   Last Post: Apr 1, 2013 8:04 PM

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Frederick Williams

Posts: 2,164
Registered: 10/4/10
Re: Using classes instead of sets
Posted: Mar 28, 2013 9:45 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply wrote:

> Although this reply contains much interesting info, it doesn't answer my main concern. [Totally my fault for not expressing it directly enough, in the first place.] My point was meant to be that definitions of groups, rings, fields, topological spaces etc. generally begin with something like "Let X be a set.."
> Why? Why not say "Let X be a class..." It would be more general.

Often one studies all groups, or all groups of a certain kind. Are
those collections classes?

> Perhaps it might lead to Russell-style paradoxes. But, if that is the risk, why is it ok for surreal numbers to form a proper class which isn't a set?

Because there is just one structure of surreal numbers, people do not
speak of the class (or whatever) of all such structures or all such
structures of a certain kind.

Confession: my claim is a guess.

I suspect that when an algebra text says "Let X be a set...", the author
is not thinking in terms of a formal theory of sets or of sets and

Also, I have seen "Fields" with a capital "F" for (e.g.) the field of
surreal numbers.

When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

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