Re: Cardinality of the surreals

In article <1133377225.327815.205780@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> "Hero" <Hero.van.Jindelt@xxxxxx> writes:
> Peter Webb wrote:
....
> > Definition: If L and R are two sets of surreal numbers and no member of R is
> > less than or equal to any member of L then { L | R } is a surreal number.
> >
> > Let S be the set of all surreal numbers.
> >
> > Let L = {x: x is an element of S and x<1}
> > Let R = {x: x is an element of S and x>=1}
> >
> > Then {L|R} is a surreal number which is not part of S.
>
> For me, it seems that surreal numbers are ordered pairs of numbers.

In that case you are wrong. You may note that L and R are sets, not
single numbers.

> As
> they are defined as subsets of real numbers with respect to the usual
> ordering,

You are wrong. They start with positing a number {|} and with rules to
construct new numbers. Also ordering and arithmetic operations are
defined on those numbers. That is all, real numbers are not mentioned
when the surreals are constructed. That you can embed the real numbers
in the surreal numbers is something different. But in the surreal numbers
there are numbers larger than any real number...
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