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Topic: Using classes instead of sets
Replies: 1   Last Post: Mar 30, 2013 5:55 AM

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 Paul Posts: 780 Registered: 7/12/10
Re: Using classes instead of sets
Posted: Mar 30, 2013 5:55 AM
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On Friday, March 29, 2013 4:28:28 PM UTC, Frederick Williams wrote:
> Timothy Murphy wrote:
>

> >
>
> > Paul wrote:
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> >
>
> > > However, the surreal numbers don't form a set since they contain a copy of
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> > > the ordinals. They form a class. Since I can't see a problem with the
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> > > non-setness of surreal numbers, I wonder why definitions of other
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> > > mathematical structures aren't more general and why the above categories
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> > > are defined on sets rather than classes.
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> >
>
> > I feel Paul raised a very interesting question,
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> > which to my mind has not been answered yet.
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> >
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> > Conway's definition of a surreal number x is, roughly speaking,
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> >
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> > x = < L | R >
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> >
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> > where L and R are sets of surreal numbers
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> > subject to the condition that y in L & z in R => y < z.
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> >
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> > This is not a complete definition;
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> > one has to define the order on the surreal numbers at the same time.
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> >
>
> > But it seems to me that the difficulty - if there is one -
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> > lies in this recursive definition of a surreal number.
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> >
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> > Is there an axiomatic set theory that allows recursive definitions
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> > of this kind?
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>
>
> Surreal numbers may be defined in set theory. One definition is that a
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> surreal number is a function with domain some ordinal and codomain {1,
>
> -1}.
>

Yes. This definition is uncontroversial but the L | R definition needs more work to see that it is legitimate. This can be seen by showing that the two definitions are equivalent, which is done by Gonshor.
So the L | R definition is actually completely ok. If it seems that there is a "difficulty" with it, the issue is only in which details have been left out by the author we're learning this from, not with the theory itself.

Actually (in reply to Timothy Murphy), I feel that quasi did answer my questions as follows: 1) Why not have a class, instead of a set, as an underlying object for structures like groups and rings? Quasi gives 4 reasons.
2) If there's a problem with using a class as above, why is the definition of the surreal numbers (which do not form a set) ok?
Answer: There is no real problem with using a class. Surreal numbers are perfectly well-defined within standard models of set theory.

Paul Epstein

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