Re: What does it take to identify the field and the continuum of reals?

Vaughan Pratt wrote in part: [about Conway's surreal numbers]

Now I conjecture that stopping just before day \epsilon_0, for the sake of getting a set, produces a field closed under all functions whose n-th bit is computable in time f(n) where f is definable in Peano arithmetic. And it may well be that Conway already has a field just before day \omega^\omega, I haven't checked but surely someone has by now (I'm very bad at keeping up with these things, sorry).

It's my understanding that you don't get a field until (just before) day K for K an inaccessible cardinal number. I'm not 100% certain about this, however! -- Toby Bartels