On 16 Jun., 02:39, "Peter Webb" <webbfam...@DIESPAMDIEoptusnet.com.au> wrote: > > Nevertheless your "definition" belongs to a countable set, hence it is > > no example to save Cantors "proof". > > > Either all entries of the lines of the list are defined and the > > diagonal is defined (in the same language) too. > > Yes. If you provide a list of Reals, then the diagonal is computable and > does not appear on the list.
Delicious. Cantor shows that the countable set of computable reals is uncountable. > > > Then the proof shows > > that the countable set of defined reals is uncountable. > > No, it shows that all "definable" (computable) Reals cannot be explicitly > listed. This is *not* the same as being uncountable.
It is impossible to explicitly list anything infinite. You can *define* a list by a_n = 1/n. But that is not the same as explicitly list 1/1, 1/2, 1/3, and so on. > > > Or it does not > > show anything at all. > > It shows that all "definable" (computable) Reals cannot be explicitly > listed. This is a well known proof in set theory. This is *not* the same as > being uncountable.
Cantor "proved" that the reals are uncountable because they cannot be listed. Now you say: Unlistability has nothing to do with uncountability. > > > To switch "languages" is the most lame argument one could think of. > > The diagonal argument does not switch languages. And it cannot be > > applied at all because the list of all finite defiitions does not > > contain infinite entries. Those however are required for the diagonal > > argument. > > No, that paragraph above is close to gibberish. Cantor said and proved that > any purported list of all Reals cannot contain all Reals. His proof is > simple and clear, provides an explicit construction of at least one missing > Real, and does contain or require any concepts of uncomputable numbers, or > use of the Axiom of Choice.
The proof is invalid. What is proved is: Any finite initial segment of a list does not contain the finite diagonal number. That is correct.
It is impossible at all to obtain a number from an infinite sequence because the number cannot be known unless the sequence has been finished. But an infinite sequence is never finished. For example: You cannot find out what number I have in mind when writing 0.1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111and so on.
You may guess that I mean 1/9 but I could also intend to write a 2 at a later position or to switch to zeros.
I am deeply impressed how few people see that from a finite definition D one can obtain an infinite sequence S: D ==> S but that the arrow cannot be reversed: S ==> D is wrong unless the S is completely given. It is never complete, however. > > Perhaps if you were to identify the step in Cantor's proof that you consider > wrong, then we might gain some idea as to what you are actually objecting > to?
Of course. Cantor assumes that the list represents finished infinity. But infinity cannot be finished by definition. Without finished infinity, there is always a further line of the list.
