"K_h" <KHolmes@SX729.com> wrote in message news:xPOdncxHLvYyw4DRnZ2dnUVZ_tWdnZ2d@giganews.com... > > "Peter Webb" <webbfamily@DIESPAMDIEoptusnet.com.au> wrote in message > news:4c1c65f4$0$2162$afc38c87@news.optusnet.com.au... >> >>>>>> Perhaps if you could point out to me why you believe Cantor's proof >>>>>> that not all Reals can be listed (as it appears you do) but you don't >>>>>> believe my proof that not all computable Reals can be listed. They >>>>>> appear identical to me. >>>>> >>>>> All computable reals can be listed, but there is no finite algorithm >>>>> for doing so. An "infinite algorithm" could list every computable >>>>> real. An anti-diagonal, then, could be generated from this list but >>>>> the algorithm creating the anti-diagonal is implicitly relying on the >>>>> "infinite algorithm" underlying the list. In that sense the >>>>> anti-diagonal is not computable. >>>> >>>> Agreed. >>>> >>>>> The set of all reals are a different story. Even with an "infinite >>>>> algorithm" generating a list of reals, there is no way such a list >>>>> could contain every real. For a proof, do a google search on Cantor's >>>>> theorem. >>>>> >>>> >>>> No, Cantor's diagonal construction does not prove this, and nor is any >>>> of this machinery part of his proof. >>>> >>>> Cantor's proof applied to computable Reals proves that the computable >>>> Reals cannot be listed. >>> >>> You have contradicted yourself. I wrote "All computable reals can be >>> listed,..." and your reply was "Agreed". >> >> Typo, sorry. >> >> My whole argument is that they cannot be listed in their entirety, or we >> could use a Cantor construction to produce a computable Real not on the >> list. > > You are wrongly assuming that only computable sets exist.
No.
Show me where I use this assumption in my proof.
> That leaves out many sets. For example, if A is a subset of N, with > |A|>1, most of the functions from N to A don't exist by your thinking. > Because of your wrong assumption you falsely conclude that your > anti-diagonal produces a computable real from the non-computable list > containing all computable reals. >
Cantor's original proof requires the list to be provided in advance, such that the nth digit of the nth item on the list is known.
