[FOM] 622: Adventures in Formalization 6

Mitchell Spector spector at alum.mit.edu
Wed Oct 28 04:04:10 EDT 2015

Addition on Conway's surreals, restricted to ordinals, is not the same as the usual addition on 
ordinals.  Addition of surreals is commutative, while the usual addition of ordinals is not commutative.

In fact, addition of surreals, restricted to ordinals, is the same as the so-called natural addition 
of ordinals.  (I don't think I've ever seen natural addition of ordinals used in set theory.)

Mitchell Spector

Hendrik Boom wrote:
> On Tue, Oct 27, 2015 at 02:55:07PM -0700, Mitchell Spector wrote:
>> With regard to the discussion as to whether the set of natural
>> numbers should be a subset of the set of real numbers, or whether
>> it's good enough to have an isomorphism that lets us identify
>> natural numbers with certain real numbers:
>> It may worth pointing out that Conway's surreal numbers provide a
>> systematic approach which includes both the natural numbers and the
>> real numbers, and which makes N a subset of R.
> And those Conway numbers also contain the ordinals, and the
> nonnegative integers are a subset of those, too.  So one unifirmly
> defined system defines everything from reals and ordinals, even
> providing a meaning for dividing omega minus one by pi.
> Anyone know whether the addition he defines matches the usual one on
> ordinals?
> -- hendrik
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