Uh, that's a lot more than "Platonism"... how was anyone supposed to guess you've been assuming CH?

Edit: To clarify -- apparently you've been thinking of this as "I can accept R, just not a well-ordering on it." Whereas I've been thinking of this as "Somehow Eliezer can accept R, but not a cardinal that's much smaller?!"

Edit again: Though I guess if we don't have choice and R isn't well-orderable than I guess omega_1 could be just incomparable to it for all I know. In any case I feel like the problem is stemming from this CH assumption rather than omega_1! I don't think you can easily get rid of a smallest uncountable ordinal (see other post on this topic -- throwing out replacement will alllow you to get rid of the von Neumann ordinal but not, I don't think, the ordinal in the general sense), but if all you want is for there to be no well-order on the continuum, you don't have to.