Hmm, that does make it seem like I may be confused here. Possible repaired statement would then use GCH in some form rather than just CH and that should go through then since GCH will imply that for all infinite S, CH(S) and CH(2^S), which will allow one to use Caicedo’s argument. Does that at least go through?
It is widely reported that the (weak) CH is the that every uncountable subset of the reals is bijective with the reals, while strong CH is that the reals are bijective with ℵ_1. I think you and Buie are simply confusing the two statements.
Yes, Caicedo mentions that GCH implies AC. This is a theorem of Siepinski, who proved it locally. Specker strengthened the local statement to CH(S)+CH(2^S) ⇒ S is well-orderable. It is open whether CH(S) ⇒ S is well-orderable.
Ok. That makes sense. I think we’re on the same page then now, and you are correct that Buie and I were confuse about the precise version of the statements in question.
Hmm, that does make it seem like I may be confused here. Possible repaired statement would then use GCH in some form rather than just CH and that should go through then since GCH will imply that for all infinite S, CH(S) and CH(2^S), which will allow one to use Caicedo’s argument. Does that at least go through?
I think you are likely correct here.
Yes, Caicedo mentions that GCH implies AC. This is a theorem of Siepinski, who proved it locally. Specker strengthened the local statement to CH(S)+CH(2^S) ⇒ S is well-orderable. It is open whether CH(S) ⇒ S is well-orderable.
Ok. That makes sense. I think we’re on the same page then now, and you are correct that Buie and I were confuse about the precise version of the statements in question.